Linear.Quaternion:$ccosh from linear-1.19.1.3

Percentage Accurate: 88.4% → 99.9%

Time: 11.9s

Alternatives: 15

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: 88.4% 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 92.1%

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

   1. associate-*l/ 99.9%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

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

Alternative 2: 86.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sinh y \leq -0.001:\\ \;\;\;\;\sinh y\\ \mathbf{elif}\;\sinh y \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\sin x \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sinh y) -0.001)
   (sinh y)
   (if (<= (sinh y) 5e-6) (* (sin x) (/ y x)) (sinh y))))

C

double code(double x, double y) {
	double tmp;
	if (sinh(y) <= -0.001) {
		tmp = sinh(y);
	} else if (sinh(y) <= 5e-6) {
		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) <= (-0.001d0)) then
        tmp = sinh(y)
    else if (sinh(y) <= 5d-6) 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) <= -0.001) {
		tmp = Math.sinh(y);
	} else if (Math.sinh(y) <= 5e-6) {
		tmp = Math.sin(x) * (y / x);
	} else {
		tmp = Math.sinh(y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if math.sinh(y) <= -0.001:
		tmp = math.sinh(y)
	elif math.sinh(y) <= 5e-6:
		tmp = math.sin(x) * (y / x)
	else:
		tmp = math.sinh(y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (sinh(y) <= -0.001)
		tmp = sinh(y);
	elseif (sinh(y) <= 5e-6)
		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) <= -0.001)
		tmp = sinh(y);
	elseif (sinh(y) <= 5e-6)
		tmp = sin(x) * (y / x);
	else
		tmp = sinh(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[Sinh[y], $MachinePrecision], -0.001], N[Sinh[y], $MachinePrecision], If[LessEqual[N[Sinh[y], $MachinePrecision], 5e-6], 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) < -1e-3 or 5.00000000000000041e-6 < (sinh.f64 y)

   1. Initial program 99.9%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 71.6%

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

   if -1e-3 < (sinh.f64 y) < 5.00000000000000041e-6

   1. Initial program 82.5%

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

      1. associate-*r/ 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in y around 0 82.0%

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

      1. associate-/l* 99.3%

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

      2. associate-/r/ 99.1%

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

   6. Simplified 99.1%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sinh y \leq -0.001:\\ \;\;\;\;\sinh y\\ \mathbf{elif}\;\sinh y \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\sin x \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \]

Alternative 3: 86.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sinh y \leq -0.001:\\ \;\;\;\;\sinh y\\ \mathbf{elif}\;\sinh y \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{\sin x}{x} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sinh y) -0.001)
   (sinh y)
   (if (<= (sinh y) 5e-6) (* (/ (sin x) x) y) (sinh y))))

C

double code(double x, double y) {
	double tmp;
	if (sinh(y) <= -0.001) {
		tmp = sinh(y);
	} else if (sinh(y) <= 5e-6) {
		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) <= (-0.001d0)) then
        tmp = sinh(y)
    else if (sinh(y) <= 5d-6) 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) <= -0.001) {
		tmp = Math.sinh(y);
	} else if (Math.sinh(y) <= 5e-6) {
		tmp = (Math.sin(x) / x) * y;
	} else {
		tmp = Math.sinh(y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if math.sinh(y) <= -0.001:
		tmp = math.sinh(y)
	elif math.sinh(y) <= 5e-6:
		tmp = (math.sin(x) / x) * y
	else:
		tmp = math.sinh(y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (sinh(y) <= -0.001)
		tmp = sinh(y);
	elseif (sinh(y) <= 5e-6)
		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) <= -0.001)
		tmp = sinh(y);
	elseif (sinh(y) <= 5e-6)
		tmp = (sin(x) / x) * y;
	else
		tmp = sinh(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[Sinh[y], $MachinePrecision], -0.001], N[Sinh[y], $MachinePrecision], If[LessEqual[N[Sinh[y], $MachinePrecision], 5e-6], 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) < -1e-3 or 5.00000000000000041e-6 < (sinh.f64 y)

   1. Initial program 99.9%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 71.6%

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

   if -1e-3 < (sinh.f64 y) < 5.00000000000000041e-6

   1. Initial program 82.5%

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

      1. associate-*l/ 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in y around 0 99.2%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sinh y \leq -0.001:\\ \;\;\;\;\sinh y\\ \mathbf{elif}\;\sinh y \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{\sin x}{x} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \]

Alternative 4: 75.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sinh y \leq -0.001:\\ \;\;\;\;\sinh y\\ \mathbf{elif}\;\sinh y \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{y}{1 + \left(x \cdot x\right) \cdot 0.16666666666666666}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sinh y) -0.001)
   (sinh y)
   (if (<= (sinh y) 5e-6)
     (/ y (+ 1.0 (* (* x x) 0.16666666666666666)))
     (sinh y))))

C

double code(double x, double y) {
	double tmp;
	if (sinh(y) <= -0.001) {
		tmp = sinh(y);
	} else if (sinh(y) <= 5e-6) {
		tmp = y / (1.0 + ((x * x) * 0.16666666666666666));
	} 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) <= (-0.001d0)) then
        tmp = sinh(y)
    else if (sinh(y) <= 5d-6) then
        tmp = y / (1.0d0 + ((x * x) * 0.16666666666666666d0))
    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) <= -0.001) {
		tmp = Math.sinh(y);
	} else if (Math.sinh(y) <= 5e-6) {
		tmp = y / (1.0 + ((x * x) * 0.16666666666666666));
	} else {
		tmp = Math.sinh(y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if math.sinh(y) <= -0.001:
		tmp = math.sinh(y)
	elif math.sinh(y) <= 5e-6:
		tmp = y / (1.0 + ((x * x) * 0.16666666666666666))
	else:
		tmp = math.sinh(y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (sinh(y) <= -0.001)
		tmp = sinh(y);
	elseif (sinh(y) <= 5e-6)
		tmp = Float64(y / Float64(1.0 + Float64(Float64(x * x) * 0.16666666666666666)));
	else
		tmp = sinh(y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (sinh(y) <= -0.001)
		tmp = sinh(y);
	elseif (sinh(y) <= 5e-6)
		tmp = y / (1.0 + ((x * x) * 0.16666666666666666));
	else
		tmp = sinh(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[Sinh[y], $MachinePrecision], -0.001], N[Sinh[y], $MachinePrecision], If[LessEqual[N[Sinh[y], $MachinePrecision], 5e-6], N[(y / N[(1.0 + N[(N[(x * x), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sinh[y], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (sinh.f64 y) < -1e-3 or 5.00000000000000041e-6 < (sinh.f64 y)

   1. Initial program 99.9%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 71.6%

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

   if -1e-3 < (sinh.f64 y) < 5.00000000000000041e-6

   1. Initial program 82.5%

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

      1. associate-*r/ 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in y around 0 82.0%

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

      1. associate-/l* 99.3%

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

   6. Simplified 99.3%

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

   7. Taylor expanded in x around 0 72.7%

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

      1. *-commutative 72.7%

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

      2. unpow2 72.7%

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

   9. Simplified 72.7%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sinh y \leq -0.001:\\ \;\;\;\;\sinh y\\ \mathbf{elif}\;\sinh y \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{y}{1 + \left(x \cdot x\right) \cdot 0.16666666666666666}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \]

Alternative 5: 99.9% 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 92.1%

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

   1. associate-*r/ 99.5%

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

3. Simplified 99.5%

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

4. Final simplification 99.5%

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

Alternative 6: 86.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 + \left(x \cdot x\right) \cdot -0.16666666666666666\right) \cdot \sinh y\\ \mathbf{if}\;y \leq -0.0015:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 0.0028:\\ \;\;\;\;\frac{y}{\frac{x}{\sin x}}\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+100}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (+ 1.0 (* (* x x) -0.16666666666666666)) (sinh y))))
   (if (<= y -0.0015)
     t_0
     (if (<= y 0.0028) (/ y (/ x (sin x))) (if (<= y 2e+100) t_0 (sinh y))))))

C

double code(double x, double y) {
	double t_0 = (1.0 + ((x * x) * -0.16666666666666666)) * sinh(y);
	double tmp;
	if (y <= -0.0015) {
		tmp = t_0;
	} else if (y <= 0.0028) {
		tmp = y / (x / sin(x));
	} else if (y <= 2e+100) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 + ((x * x) * (-0.16666666666666666d0))) * sinh(y)
    if (y <= (-0.0015d0)) then
        tmp = t_0
    else if (y <= 0.0028d0) then
        tmp = y / (x / sin(x))
    else if (y <= 2d+100) then
        tmp = t_0
    else
        tmp = sinh(y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (1.0 + ((x * x) * -0.16666666666666666)) * Math.sinh(y);
	double tmp;
	if (y <= -0.0015) {
		tmp = t_0;
	} else if (y <= 0.0028) {
		tmp = y / (x / Math.sin(x));
	} else if (y <= 2e+100) {
		tmp = t_0;
	} else {
		tmp = Math.sinh(y);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (1.0 + ((x * x) * -0.16666666666666666)) * math.sinh(y)
	tmp = 0
	if y <= -0.0015:
		tmp = t_0
	elif y <= 0.0028:
		tmp = y / (x / math.sin(x))
	elif y <= 2e+100:
		tmp = t_0
	else:
		tmp = math.sinh(y)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(1.0 + Float64(Float64(x * x) * -0.16666666666666666)) * sinh(y))
	tmp = 0.0
	if (y <= -0.0015)
		tmp = t_0;
	elseif (y <= 0.0028)
		tmp = Float64(y / Float64(x / sin(x)));
	elseif (y <= 2e+100)
		tmp = t_0;
	else
		tmp = sinh(y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (1.0 + ((x * x) * -0.16666666666666666)) * sinh(y);
	tmp = 0.0;
	if (y <= -0.0015)
		tmp = t_0;
	elseif (y <= 0.0028)
		tmp = y / (x / sin(x));
	elseif (y <= 2e+100)
		tmp = t_0;
	else
		tmp = sinh(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(1.0 + N[(N[(x * x), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] * N[Sinh[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.0015], t$95$0, If[LessEqual[y, 0.0028], N[(y / N[(x / N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2e+100], t$95$0, N[Sinh[y], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.0015 or 0.00279999999999999997 < y < 2.00000000000000003e100

   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 x around 0 75.0%

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

      1. *-commutative 75.0%

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

      2. unpow2 75.0%

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

   6. Simplified 75.0%

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

   if -0.0015 < y < 0.00279999999999999997

   1. Initial program 82.5%

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

      1. associate-*r/ 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in y around 0 82.0%

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

      1. associate-/l* 99.3%

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

   6. Simplified 99.3%

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

   if 2.00000000000000003e100 < 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 82.2%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.0015:\\ \;\;\;\;\left(1 + \left(x \cdot x\right) \cdot -0.16666666666666666\right) \cdot \sinh y\\ \mathbf{elif}\;y \leq 0.0028:\\ \;\;\;\;\frac{y}{\frac{x}{\sin x}}\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+100}:\\ \;\;\;\;\left(1 + \left(x \cdot x\right) \cdot -0.16666666666666666\right) \cdot \sinh y\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \]

Alternative 7: 60.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 6 \cdot \frac{y}{x \cdot x}\\ \mathbf{if}\;y \leq -3.5 \cdot 10^{+17}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 75:\\ \;\;\;\;\frac{y}{1 + \left(x \cdot x\right) \cdot 0.16666666666666666}\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{+149}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{y \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* 6.0 (/ y (* x x)))))
   (if (<= y -3.5e+17)
     t_0
     (if (<= y 75.0)
       (/ y (+ 1.0 (* (* x x) 0.16666666666666666)))
       (if (<= y 2.4e+149) t_0 (sqrt (* y y)))))))

C

double code(double x, double y) {
	double t_0 = 6.0 * (y / (x * x));
	double tmp;
	if (y <= -3.5e+17) {
		tmp = t_0;
	} else if (y <= 75.0) {
		tmp = y / (1.0 + ((x * x) * 0.16666666666666666));
	} else if (y <= 2.4e+149) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = 6.0d0 * (y / (x * x))
    if (y <= (-3.5d+17)) then
        tmp = t_0
    else if (y <= 75.0d0) then
        tmp = y / (1.0d0 + ((x * x) * 0.16666666666666666d0))
    else if (y <= 2.4d+149) then
        tmp = t_0
    else
        tmp = sqrt((y * y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 6.0 * (y / (x * x));
	double tmp;
	if (y <= -3.5e+17) {
		tmp = t_0;
	} else if (y <= 75.0) {
		tmp = y / (1.0 + ((x * x) * 0.16666666666666666));
	} else if (y <= 2.4e+149) {
		tmp = t_0;
	} else {
		tmp = Math.sqrt((y * y));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 6.0 * (y / (x * x))
	tmp = 0
	if y <= -3.5e+17:
		tmp = t_0
	elif y <= 75.0:
		tmp = y / (1.0 + ((x * x) * 0.16666666666666666))
	elif y <= 2.4e+149:
		tmp = t_0
	else:
		tmp = math.sqrt((y * y))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(6.0 * Float64(y / Float64(x * x)))
	tmp = 0.0
	if (y <= -3.5e+17)
		tmp = t_0;
	elseif (y <= 75.0)
		tmp = Float64(y / Float64(1.0 + Float64(Float64(x * x) * 0.16666666666666666)));
	elseif (y <= 2.4e+149)
		tmp = t_0;
	else
		tmp = sqrt(Float64(y * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 6.0 * (y / (x * x));
	tmp = 0.0;
	if (y <= -3.5e+17)
		tmp = t_0;
	elseif (y <= 75.0)
		tmp = y / (1.0 + ((x * x) * 0.16666666666666666));
	elseif (y <= 2.4e+149)
		tmp = t_0;
	else
		tmp = sqrt((y * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(6.0 * N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -3.5e+17], t$95$0, If[LessEqual[y, 75.0], N[(y / N[(1.0 + N[(N[(x * x), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.4e+149], t$95$0, N[Sqrt[N[(y * y), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.5e17 or 75 < y < 2.40000000000000012e149

   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 3.9%

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

      1. associate-/l* 3.9%

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

   6. Simplified 3.9%

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

   7. Taylor expanded in x around 0 3.0%

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

      1. *-commutative 3.0%

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

      2. unpow2 3.0%

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

   9. Simplified 3.0%

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

   10. Taylor expanded in x around inf 30.5%

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

       1. unpow2 30.5%

          \[\leadsto 6 \cdot \frac{y}{\color{blue}{x \cdot x}} \]

   12. Simplified 30.5%

       \[\leadsto \color{blue}{6 \cdot \frac{y}{x \cdot x}} \]

   if -3.5e17 < y < 75

   1. Initial program 83.6%

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

      1. associate-*r/ 98.9%

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

   3. Simplified 98.9%

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

   4. Taylor expanded in y around 0 77.9%

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

      1. associate-/l* 94.1%

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

   6. Simplified 94.1%

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

   7. Taylor expanded in x around 0 69.0%

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

      1. *-commutative 69.0%

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

      2. unpow2 69.0%

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

   9. Simplified 69.0%

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

   if 2.40000000000000012e149 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 5.2%

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

   3. Taylor expanded in x around 0 35.1%

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

      1. *-commutative 35.1%

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

   5. Simplified 35.1%

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

      1. associate-/l* 5.7%

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

      2. *-inverses 5.7%

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

      3. /-rgt-identity 5.7%

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

      4. add-sqr-sqrt 5.7%

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

      5. sqrt-unprod 80.1%

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

   7. Applied egg-rr 80.1%

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

4. Final simplification 55.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.5 \cdot 10^{+17}:\\ \;\;\;\;6 \cdot \frac{y}{x \cdot x}\\ \mathbf{elif}\;y \leq 75:\\ \;\;\;\;\frac{y}{1 + \left(x \cdot x\right) \cdot 0.16666666666666666}\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{+149}:\\ \;\;\;\;6 \cdot \frac{y}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{y \cdot y}\\ \end{array} \]

Alternative 8: 86.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.00155:\\ \;\;\;\;\sinh y\\ \mathbf{elif}\;y \leq 0.00045:\\ \;\;\;\;\frac{y}{\frac{x}{\sin x}}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -0.00155)
   (sinh y)
   (if (<= y 0.00045) (/ y (/ x (sin x))) (sinh y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -0.00155) {
		tmp = sinh(y);
	} else if (y <= 0.00045) {
		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 (y <= (-0.00155d0)) then
        tmp = sinh(y)
    else if (y <= 0.00045d0) 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 (y <= -0.00155) {
		tmp = Math.sinh(y);
	} else if (y <= 0.00045) {
		tmp = y / (x / Math.sin(x));
	} else {
		tmp = Math.sinh(y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -0.00155:
		tmp = math.sinh(y)
	elif y <= 0.00045:
		tmp = y / (x / math.sin(x))
	else:
		tmp = math.sinh(y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -0.00155)
		tmp = sinh(y);
	elseif (y <= 0.00045)
		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 (y <= -0.00155)
		tmp = sinh(y);
	elseif (y <= 0.00045)
		tmp = y / (x / sin(x));
	else
		tmp = sinh(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -0.00155], N[Sinh[y], $MachinePrecision], If[LessEqual[y, 0.00045], N[(y / N[(x / N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sinh[y], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.00154999999999999995 or 4.4999999999999999e-4 < y

   1. Initial program 99.9%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 71.6%

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

   if -0.00154999999999999995 < y < 4.4999999999999999e-4

   1. Initial program 82.5%

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

      1. associate-*r/ 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in y around 0 82.0%

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

      1. associate-/l* 99.3%

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

   6. Simplified 99.3%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.00155:\\ \;\;\;\;\sinh y\\ \mathbf{elif}\;y \leq 0.00045:\\ \;\;\;\;\frac{y}{\frac{x}{\sin x}}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \]

Alternative 9: 48.8% accurate, 11.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{y}{x}\\ t_1 := y + y \cdot \left(x \cdot \left(x \cdot -0.16666666666666666\right)\right)\\ \mathbf{if}\;x \leq 10^{+108}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 5.3 \cdot 10^{+216}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{+238}:\\ \;\;\;\;\frac{x \cdot y}{x}\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{+269}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* x (/ y x))) (t_1 (+ y (* y (* x (* x -0.16666666666666666))))))
   (if (<= x 1e+108)
     t_0
     (if (<= x 5.3e+216)
       t_1
       (if (<= x 9.5e+238) (/ (* x y) x) (if (<= x 5.2e+269) t_1 t_0))))))

C

double code(double x, double y) {
	double t_0 = x * (y / x);
	double t_1 = y + (y * (x * (x * -0.16666666666666666)));
	double tmp;
	if (x <= 1e+108) {
		tmp = t_0;
	} else if (x <= 5.3e+216) {
		tmp = t_1;
	} else if (x <= 9.5e+238) {
		tmp = (x * y) / x;
	} else if (x <= 5.2e+269) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = x * (y / x)
    t_1 = y + (y * (x * (x * (-0.16666666666666666d0))))
    if (x <= 1d+108) then
        tmp = t_0
    else if (x <= 5.3d+216) then
        tmp = t_1
    else if (x <= 9.5d+238) then
        tmp = (x * y) / x
    else if (x <= 5.2d+269) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x * (y / x);
	double t_1 = y + (y * (x * (x * -0.16666666666666666)));
	double tmp;
	if (x <= 1e+108) {
		tmp = t_0;
	} else if (x <= 5.3e+216) {
		tmp = t_1;
	} else if (x <= 9.5e+238) {
		tmp = (x * y) / x;
	} else if (x <= 5.2e+269) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x * (y / x)
	t_1 = y + (y * (x * (x * -0.16666666666666666)))
	tmp = 0
	if x <= 1e+108:
		tmp = t_0
	elif x <= 5.3e+216:
		tmp = t_1
	elif x <= 9.5e+238:
		tmp = (x * y) / x
	elif x <= 5.2e+269:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x * Float64(y / x))
	t_1 = Float64(y + Float64(y * Float64(x * Float64(x * -0.16666666666666666))))
	tmp = 0.0
	if (x <= 1e+108)
		tmp = t_0;
	elseif (x <= 5.3e+216)
		tmp = t_1;
	elseif (x <= 9.5e+238)
		tmp = Float64(Float64(x * y) / x);
	elseif (x <= 5.2e+269)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x * (y / x);
	t_1 = y + (y * (x * (x * -0.16666666666666666)));
	tmp = 0.0;
	if (x <= 1e+108)
		tmp = t_0;
	elseif (x <= 5.3e+216)
		tmp = t_1;
	elseif (x <= 9.5e+238)
		tmp = (x * y) / x;
	elseif (x <= 5.2e+269)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x * N[(y / x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(y + N[(y * N[(x * N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 1e+108], t$95$0, If[LessEqual[x, 5.3e+216], t$95$1, If[LessEqual[x, 9.5e+238], N[(N[(x * y), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 5.2e+269], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 1e108 or 5.2e269 < x

   1. Initial program 90.9%

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

   2. Taylor expanded in y around 0 40.7%

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

   3. Taylor expanded in x around 0 25.0%

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

      1. *-commutative 25.0%

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

   5. Simplified 25.0%

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

      1. associate-/l* 26.7%

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

      2. associate-/r/ 48.7%

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

   7. Applied egg-rr 48.7%

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

   if 1e108 < x < 5.30000000000000002e216 or 9.5000000000000003e238 < x < 5.2e269

   1. Initial program 99.8%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 35.7%

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

      1. associate-/l* 35.7%

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

   6. Simplified 35.7%

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

   7. Taylor expanded in x around 0 41.2%

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

      1. associate-*r* 41.2%

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

      2. *-commutative 41.2%

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

      3. unpow2 41.2%

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

      4. associate-*r* 41.2%

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

   9. Simplified 41.2%

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

   if 5.30000000000000002e216 < x < 9.5000000000000003e238

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 2.9%

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

   3. Taylor expanded in x around 0 67.9%

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

      1. *-commutative 67.9%

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

   5. Simplified 67.9%

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

4. Final simplification 48.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 10^{+108}:\\ \;\;\;\;x \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq 5.3 \cdot 10^{+216}:\\ \;\;\;\;y + y \cdot \left(x \cdot \left(x \cdot -0.16666666666666666\right)\right)\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{+238}:\\ \;\;\;\;\frac{x \cdot y}{x}\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{+269}:\\ \;\;\;\;y + y \cdot \left(x \cdot \left(x \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{x}\\ \end{array} \]

Alternative 10: 48.8% accurate, 11.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{y}{x}\\ \mathbf{if}\;x \leq 1.1 \cdot 10^{+108}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 5.3 \cdot 10^{+216}:\\ \;\;\;\;y + \left(x \cdot -0.16666666666666666\right) \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{+238}:\\ \;\;\;\;\frac{x \cdot y}{x}\\ \mathbf{elif}\;x \leq 1.32 \cdot 10^{+270}:\\ \;\;\;\;y + y \cdot \left(x \cdot \left(x \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* x (/ y x))))
   (if (<= x 1.1e+108)
     t_0
     (if (<= x 5.3e+216)
       (+ y (* (* x -0.16666666666666666) (* x y)))
       (if (<= x 9.5e+238)
         (/ (* x y) x)
         (if (<= x 1.32e+270)
           (+ y (* y (* x (* x -0.16666666666666666))))
           t_0))))))

C

double code(double x, double y) {
	double t_0 = x * (y / x);
	double tmp;
	if (x <= 1.1e+108) {
		tmp = t_0;
	} else if (x <= 5.3e+216) {
		tmp = y + ((x * -0.16666666666666666) * (x * y));
	} else if (x <= 9.5e+238) {
		tmp = (x * y) / x;
	} else if (x <= 1.32e+270) {
		tmp = y + (y * (x * (x * -0.16666666666666666)));
	} else {
		tmp = t_0;
	}
	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 = x * (y / x)
    if (x <= 1.1d+108) then
        tmp = t_0
    else if (x <= 5.3d+216) then
        tmp = y + ((x * (-0.16666666666666666d0)) * (x * y))
    else if (x <= 9.5d+238) then
        tmp = (x * y) / x
    else if (x <= 1.32d+270) then
        tmp = y + (y * (x * (x * (-0.16666666666666666d0))))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x * (y / x);
	double tmp;
	if (x <= 1.1e+108) {
		tmp = t_0;
	} else if (x <= 5.3e+216) {
		tmp = y + ((x * -0.16666666666666666) * (x * y));
	} else if (x <= 9.5e+238) {
		tmp = (x * y) / x;
	} else if (x <= 1.32e+270) {
		tmp = y + (y * (x * (x * -0.16666666666666666)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x * (y / x)
	tmp = 0
	if x <= 1.1e+108:
		tmp = t_0
	elif x <= 5.3e+216:
		tmp = y + ((x * -0.16666666666666666) * (x * y))
	elif x <= 9.5e+238:
		tmp = (x * y) / x
	elif x <= 1.32e+270:
		tmp = y + (y * (x * (x * -0.16666666666666666)))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x * Float64(y / x))
	tmp = 0.0
	if (x <= 1.1e+108)
		tmp = t_0;
	elseif (x <= 5.3e+216)
		tmp = Float64(y + Float64(Float64(x * -0.16666666666666666) * Float64(x * y)));
	elseif (x <= 9.5e+238)
		tmp = Float64(Float64(x * y) / x);
	elseif (x <= 1.32e+270)
		tmp = Float64(y + Float64(y * Float64(x * Float64(x * -0.16666666666666666))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x * (y / x);
	tmp = 0.0;
	if (x <= 1.1e+108)
		tmp = t_0;
	elseif (x <= 5.3e+216)
		tmp = y + ((x * -0.16666666666666666) * (x * y));
	elseif (x <= 9.5e+238)
		tmp = (x * y) / x;
	elseif (x <= 1.32e+270)
		tmp = y + (y * (x * (x * -0.16666666666666666)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x * N[(y / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 1.1e+108], t$95$0, If[LessEqual[x, 5.3e+216], N[(y + N[(N[(x * -0.16666666666666666), $MachinePrecision] * N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 9.5e+238], N[(N[(x * y), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 1.32e+270], N[(y + N[(y * N[(x * N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if x < 1.1000000000000001e108 or 1.31999999999999992e270 < x

   1. Initial program 90.9%

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

   2. Taylor expanded in y around 0 40.7%

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

   3. Taylor expanded in x around 0 25.0%

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

      1. *-commutative 25.0%

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

   5. Simplified 25.0%

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

      1. associate-/l* 26.7%

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

      2. associate-/r/ 48.7%

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

   7. Applied egg-rr 48.7%

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

   if 1.1000000000000001e108 < x < 5.30000000000000002e216

   1. Initial program 99.8%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 44.9%

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

      1. associate-/l* 45.0%

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

   6. Simplified 45.0%

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

   7. Taylor expanded in x around 0 36.8%

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

      1. +-commutative 36.8%

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

      2. fma-def 36.8%

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

      3. unpow2 36.8%

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

      4. associate-*l* 36.9%

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

      5. *-commutative 36.9%

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

   9. Simplified 36.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.16666666666666666, x \cdot \left(y \cdot x\right), y\right)} \]
   10. Step-by-step derivation

       1. fma-udef 36.9%

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

       2. associate-*r* 36.9%

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

       3. *-commutative 36.9%

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

       4. *-commutative 36.9%

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

   11. Applied egg-rr 36.9%

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

   if 5.30000000000000002e216 < x < 9.5000000000000003e238

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 2.9%

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

   3. Taylor expanded in x around 0 67.9%

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

      1. *-commutative 67.9%

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

   5. Simplified 67.9%

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

   if 9.5000000000000003e238 < x < 1.31999999999999992e270

   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 2.5%

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

      1. associate-/l* 2.5%

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

   6. Simplified 2.5%

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

   7. Taylor expanded in x around 0 57.1%

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

      1. associate-*r* 57.1%

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

      2. *-commutative 57.1%

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

      3. unpow2 57.1%

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

      4. associate-*r* 57.1%

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

   9. Simplified 57.1%

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

4. Final simplification 48.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.1 \cdot 10^{+108}:\\ \;\;\;\;x \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq 5.3 \cdot 10^{+216}:\\ \;\;\;\;y + \left(x \cdot -0.16666666666666666\right) \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{+238}:\\ \;\;\;\;\frac{x \cdot y}{x}\\ \mathbf{elif}\;x \leq 1.32 \cdot 10^{+270}:\\ \;\;\;\;y + y \cdot \left(x \cdot \left(x \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{x}\\ \end{array} \]

Alternative 11: 54.1% accurate, 15.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 6 \cdot \frac{y}{x \cdot x}\\ t_1 := x \cdot \frac{y}{x}\\ \mathbf{if}\;y \leq -13:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 75:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{+149}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 6.4 \cdot 10^{+272}:\\ \;\;\;\;\frac{x \cdot y}{x}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* 6.0 (/ y (* x x)))) (t_1 (* x (/ y x))))
   (if (<= y -13.0)
     t_0
     (if (<= y 75.0)
       t_1
       (if (<= y 1.3e+149) t_0 (if (<= y 6.4e+272) (/ (* x y) x) t_1))))))

C

double code(double x, double y) {
	double t_0 = 6.0 * (y / (x * x));
	double t_1 = x * (y / x);
	double tmp;
	if (y <= -13.0) {
		tmp = t_0;
	} else if (y <= 75.0) {
		tmp = t_1;
	} else if (y <= 1.3e+149) {
		tmp = t_0;
	} else if (y <= 6.4e+272) {
		tmp = (x * y) / x;
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = 6.0d0 * (y / (x * x))
    t_1 = x * (y / x)
    if (y <= (-13.0d0)) then
        tmp = t_0
    else if (y <= 75.0d0) then
        tmp = t_1
    else if (y <= 1.3d+149) then
        tmp = t_0
    else if (y <= 6.4d+272) then
        tmp = (x * y) / x
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 6.0 * (y / (x * x));
	double t_1 = x * (y / x);
	double tmp;
	if (y <= -13.0) {
		tmp = t_0;
	} else if (y <= 75.0) {
		tmp = t_1;
	} else if (y <= 1.3e+149) {
		tmp = t_0;
	} else if (y <= 6.4e+272) {
		tmp = (x * y) / x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 6.0 * (y / (x * x))
	t_1 = x * (y / x)
	tmp = 0
	if y <= -13.0:
		tmp = t_0
	elif y <= 75.0:
		tmp = t_1
	elif y <= 1.3e+149:
		tmp = t_0
	elif y <= 6.4e+272:
		tmp = (x * y) / x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y)
	t_0 = Float64(6.0 * Float64(y / Float64(x * x)))
	t_1 = Float64(x * Float64(y / x))
	tmp = 0.0
	if (y <= -13.0)
		tmp = t_0;
	elseif (y <= 75.0)
		tmp = t_1;
	elseif (y <= 1.3e+149)
		tmp = t_0;
	elseif (y <= 6.4e+272)
		tmp = Float64(Float64(x * y) / x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 6.0 * (y / (x * x));
	t_1 = x * (y / x);
	tmp = 0.0;
	if (y <= -13.0)
		tmp = t_0;
	elseif (y <= 75.0)
		tmp = t_1;
	elseif (y <= 1.3e+149)
		tmp = t_0;
	elseif (y <= 6.4e+272)
		tmp = (x * y) / x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(6.0 * N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x * N[(y / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -13.0], t$95$0, If[LessEqual[y, 75.0], t$95$1, If[LessEqual[y, 1.3e+149], t$95$0, If[LessEqual[y, 6.4e+272], N[(N[(x * y), $MachinePrecision] / x), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -13 or 75 < y < 1.29999999999999989e149

   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 3.9%

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

      1. associate-/l* 3.9%

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

   6. Simplified 3.9%

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

   7. Taylor expanded in x around 0 3.0%

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

      1. *-commutative 3.0%

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

      2. unpow2 3.0%

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

   9. Simplified 3.0%

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

   10. Taylor expanded in x around inf 30.0%

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

       1. unpow2 30.0%

          \[\leadsto 6 \cdot \frac{y}{\color{blue}{x \cdot x}} \]

   12. Simplified 30.0%

       \[\leadsto \color{blue}{6 \cdot \frac{y}{x \cdot x}} \]

   if -13 < y < 75 or 6.3999999999999997e272 < y

   1. Initial program 84.2%

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

   2. Taylor expanded in y around 0 75.3%

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

   3. Taylor expanded in x around 0 28.6%

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

      1. *-commutative 28.6%

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

   5. Simplified 28.6%

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

      1. associate-/l* 43.4%

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

      2. associate-/r/ 68.7%

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

   7. Applied egg-rr 68.7%

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

   if 1.29999999999999989e149 < y < 6.3999999999999997e272

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 4.2%

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

   3. Taylor expanded in x around 0 38.1%

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

      1. *-commutative 38.1%

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

   5. Simplified 38.1%

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

4. Final simplification 50.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -13:\\ \;\;\;\;6 \cdot \frac{y}{x \cdot x}\\ \mathbf{elif}\;y \leq 75:\\ \;\;\;\;x \cdot \frac{y}{x}\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{+149}:\\ \;\;\;\;6 \cdot \frac{y}{x \cdot x}\\ \mathbf{elif}\;y \leq 6.4 \cdot 10^{+272}:\\ \;\;\;\;\frac{x \cdot y}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{x}\\ \end{array} \]

Alternative 12: 55.1% accurate, 15.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 6 \cdot \frac{y}{x \cdot x}\\ \mathbf{if}\;y \leq -3.5 \cdot 10^{+17}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 75:\\ \;\;\;\;\frac{y}{1 + \left(x \cdot x\right) \cdot 0.16666666666666666}\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{+149}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{+272}:\\ \;\;\;\;\frac{x \cdot y}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* 6.0 (/ y (* x x)))))
   (if (<= y -3.5e+17)
     t_0
     (if (<= y 75.0)
       (/ y (+ 1.0 (* (* x x) 0.16666666666666666)))
       (if (<= y 2.4e+149)
         t_0
         (if (<= y 5.5e+272) (/ (* x y) x) (* x (/ y x))))))))

C

double code(double x, double y) {
	double t_0 = 6.0 * (y / (x * x));
	double tmp;
	if (y <= -3.5e+17) {
		tmp = t_0;
	} else if (y <= 75.0) {
		tmp = y / (1.0 + ((x * x) * 0.16666666666666666));
	} else if (y <= 2.4e+149) {
		tmp = t_0;
	} else if (y <= 5.5e+272) {
		tmp = (x * y) / x;
	} else {
		tmp = x * (y / 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 = 6.0d0 * (y / (x * x))
    if (y <= (-3.5d+17)) then
        tmp = t_0
    else if (y <= 75.0d0) then
        tmp = y / (1.0d0 + ((x * x) * 0.16666666666666666d0))
    else if (y <= 2.4d+149) then
        tmp = t_0
    else if (y <= 5.5d+272) then
        tmp = (x * y) / x
    else
        tmp = x * (y / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 6.0 * (y / (x * x));
	double tmp;
	if (y <= -3.5e+17) {
		tmp = t_0;
	} else if (y <= 75.0) {
		tmp = y / (1.0 + ((x * x) * 0.16666666666666666));
	} else if (y <= 2.4e+149) {
		tmp = t_0;
	} else if (y <= 5.5e+272) {
		tmp = (x * y) / x;
	} else {
		tmp = x * (y / x);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 6.0 * (y / (x * x))
	tmp = 0
	if y <= -3.5e+17:
		tmp = t_0
	elif y <= 75.0:
		tmp = y / (1.0 + ((x * x) * 0.16666666666666666))
	elif y <= 2.4e+149:
		tmp = t_0
	elif y <= 5.5e+272:
		tmp = (x * y) / x
	else:
		tmp = x * (y / x)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(6.0 * Float64(y / Float64(x * x)))
	tmp = 0.0
	if (y <= -3.5e+17)
		tmp = t_0;
	elseif (y <= 75.0)
		tmp = Float64(y / Float64(1.0 + Float64(Float64(x * x) * 0.16666666666666666)));
	elseif (y <= 2.4e+149)
		tmp = t_0;
	elseif (y <= 5.5e+272)
		tmp = Float64(Float64(x * y) / x);
	else
		tmp = Float64(x * Float64(y / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 6.0 * (y / (x * x));
	tmp = 0.0;
	if (y <= -3.5e+17)
		tmp = t_0;
	elseif (y <= 75.0)
		tmp = y / (1.0 + ((x * x) * 0.16666666666666666));
	elseif (y <= 2.4e+149)
		tmp = t_0;
	elseif (y <= 5.5e+272)
		tmp = (x * y) / x;
	else
		tmp = x * (y / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(6.0 * N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -3.5e+17], t$95$0, If[LessEqual[y, 75.0], N[(y / N[(1.0 + N[(N[(x * x), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.4e+149], t$95$0, If[LessEqual[y, 5.5e+272], N[(N[(x * y), $MachinePrecision] / x), $MachinePrecision], N[(x * N[(y / x), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -3.5e17 or 75 < y < 2.40000000000000012e149

   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 3.9%

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

      1. associate-/l* 3.9%

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

   6. Simplified 3.9%

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

   7. Taylor expanded in x around 0 3.0%

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

      1. *-commutative 3.0%

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

      2. unpow2 3.0%

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

   9. Simplified 3.0%

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

   10. Taylor expanded in x around inf 30.5%

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

       1. unpow2 30.5%

          \[\leadsto 6 \cdot \frac{y}{\color{blue}{x \cdot x}} \]

   12. Simplified 30.5%

       \[\leadsto \color{blue}{6 \cdot \frac{y}{x \cdot x}} \]

   if -3.5e17 < y < 75

   1. Initial program 83.6%

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

      1. associate-*r/ 98.9%

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

   3. Simplified 98.9%

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

   4. Taylor expanded in y around 0 77.9%

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

      1. associate-/l* 94.1%

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

   6. Simplified 94.1%

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

   7. Taylor expanded in x around 0 69.0%

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

      1. *-commutative 69.0%

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

      2. unpow2 69.0%

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

   9. Simplified 69.0%

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

   if 2.40000000000000012e149 < y < 5.4999999999999998e272

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 4.2%

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

   3. Taylor expanded in x around 0 38.1%

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

      1. *-commutative 38.1%

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

   5. Simplified 38.1%

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

   if 5.4999999999999998e272 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 9.4%

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

   3. Taylor expanded in x around 0 23.0%

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

      1. *-commutative 23.0%

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

   5. Simplified 23.0%

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

      1. associate-/l* 10.0%

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

      2. associate-/r/ 87.0%

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

   7. Applied egg-rr 87.0%

      \[\leadsto \color{blue}{\frac{y}{x} \cdot x} \]
3. Recombined 4 regimes into one program.

4. Final simplification 51.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.5 \cdot 10^{+17}:\\ \;\;\;\;6 \cdot \frac{y}{x \cdot x}\\ \mathbf{elif}\;y \leq 75:\\ \;\;\;\;\frac{y}{1 + \left(x \cdot x\right) \cdot 0.16666666666666666}\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{+149}:\\ \;\;\;\;6 \cdot \frac{y}{x \cdot x}\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{+272}:\\ \;\;\;\;\frac{x \cdot y}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{x}\\ \end{array} \]

Alternative 13: 47.9% accurate, 22.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.65 \cdot 10^{+122} \lor \neg \left(x \leq 2 \cdot 10^{+288}\right):\\ \;\;\;\;x \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x 1.65e+122) (not (<= x 2e+288))) (* x (/ y x)) (/ (* x y) x)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= 1.65e+122) || !(x <= 2e+288)) {
		tmp = x * (y / x);
	} else {
		tmp = (x * y) / 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 <= 1.65d+122) .or. (.not. (x <= 2d+288))) then
        tmp = x * (y / x)
    else
        tmp = (x * y) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= 1.65e+122) || !(x <= 2e+288)) {
		tmp = x * (y / x);
	} else {
		tmp = (x * y) / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= 1.65e+122) or not (x <= 2e+288):
		tmp = x * (y / x)
	else:
		tmp = (x * y) / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= 1.65e+122) || !(x <= 2e+288))
		tmp = Float64(x * Float64(y / x));
	else
		tmp = Float64(Float64(x * y) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= 1.65e+122) || ~((x <= 2e+288)))
		tmp = x * (y / x);
	else
		tmp = (x * y) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, 1.65e+122], N[Not[LessEqual[x, 2e+288]], $MachinePrecision]], N[(x * N[(y / x), $MachinePrecision]), $MachinePrecision], N[(N[(x * y), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.6499999999999999e122 or 2e288 < x

   1. Initial program 91.0%

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

   2. Taylor expanded in y around 0 40.7%

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

   3. Taylor expanded in x around 0 23.9%

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

      1. *-commutative 23.9%

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

   5. Simplified 23.9%

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

      1. associate-/l* 26.5%

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

      2. associate-/r/ 47.8%

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

   7. Applied egg-rr 47.8%

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

   if 1.6499999999999999e122 < x < 2e288

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 32.3%

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

   3. Taylor expanded in x around 0 29.1%

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

      1. *-commutative 29.1%

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

   5. Simplified 29.1%

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

4. Final simplification 45.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.65 \cdot 10^{+122} \lor \neg \left(x \leq 2 \cdot 10^{+288}\right):\\ \;\;\;\;x \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{x}\\ \end{array} \]

Alternative 14: 50.3% 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 92.1%

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

2. Taylor expanded in y around 0 39.6%

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

3. Taylor expanded in x around 0 24.6%

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

   1. *-commutative 24.6%

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

5. Simplified 24.6%

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

   1. associate-/l* 23.5%

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

   2. associate-/r/ 43.9%

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

7. Applied egg-rr 43.9%

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

8. Final simplification 43.9%

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

Alternative 15: 27.9% 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 92.1%

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

   1. associate-*r/ 99.5%

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

3. Simplified 99.5%

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

4. Taylor expanded in y around 0 39.6%

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

   1. associate-/l* 47.4%

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

6. Simplified 47.4%

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

7. Taylor expanded in x around 0 23.5%

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

8. Final simplification 23.5%

   \[\leadsto y \]

Developer target: 99.9% 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 2023293 
(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))