Linear.Quaternion:$ccosh from linear-1.19.1.3

Percentage Accurate: 88.9% → 99.9%

Time: 9.9s

Alternatives: 13

Speedup: 1.0×

• 50.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 88.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.0%

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

   1. associate-*l/ 99.9%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

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

Alternative 2: 86.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sinh y \leq -\infty:\\ \;\;\;\;\sinh y\\ \mathbf{elif}\;\sinh y \leq 2 \cdot 10^{-12}:\\ \;\;\;\;\sin x \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sinh y) (- INFINITY))
   (sinh y)
   (if (<= (sinh y) 2e-12) (* (sin x) (/ y x)) (sinh y))))

C

double code(double x, double y) {
	double tmp;
	if (sinh(y) <= -((double) INFINITY)) {
		tmp = sinh(y);
	} else if (sinh(y) <= 2e-12) {
		tmp = sin(x) * (y / x);
	} else {
		tmp = sinh(y);
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (Math.sinh(y) <= -Double.POSITIVE_INFINITY) {
		tmp = Math.sinh(y);
	} else if (Math.sinh(y) <= 2e-12) {
		tmp = Math.sin(x) * (y / x);
	} else {
		tmp = Math.sinh(y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if math.sinh(y) <= -math.inf:
		tmp = math.sinh(y)
	elif math.sinh(y) <= 2e-12:
		tmp = math.sin(x) * (y / x)
	else:
		tmp = math.sinh(y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (sinh(y) <= Float64(-Inf))
		tmp = sinh(y);
	elseif (sinh(y) <= 2e-12)
		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) <= -Inf)
		tmp = sinh(y);
	elseif (sinh(y) <= 2e-12)
		tmp = sin(x) * (y / x);
	else
		tmp = sinh(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[Sinh[y], $MachinePrecision], (-Infinity)], N[Sinh[y], $MachinePrecision], If[LessEqual[N[Sinh[y], $MachinePrecision], 2e-12], 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) < -inf.0 or 1.99999999999999996e-12 < (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 77.3%

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

   if -inf.0 < (sinh.f64 y) < 1.99999999999999996e-12

   1. Initial program 71.2%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 70.9%

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

      1. associate-/l* 99.6%

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

      2. associate-/r/ 99.4%

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

   6. Simplified 99.4%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sinh y \leq -\infty:\\ \;\;\;\;\sinh y\\ \mathbf{elif}\;\sinh y \leq 2 \cdot 10^{-12}:\\ \;\;\;\;\sin x \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \]

Alternative 3: 86.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sinh y \leq -\infty:\\ \;\;\;\;\sinh y\\ \mathbf{elif}\;\sinh y \leq 2 \cdot 10^{-12}:\\ \;\;\;\;\frac{\sin x}{x} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sinh y) (- INFINITY))
   (sinh y)
   (if (<= (sinh y) 2e-12) (* (/ (sin x) x) y) (sinh y))))

C

double code(double x, double y) {
	double tmp;
	if (sinh(y) <= -((double) INFINITY)) {
		tmp = sinh(y);
	} else if (sinh(y) <= 2e-12) {
		tmp = (sin(x) / x) * y;
	} else {
		tmp = sinh(y);
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (Math.sinh(y) <= -Double.POSITIVE_INFINITY) {
		tmp = Math.sinh(y);
	} else if (Math.sinh(y) <= 2e-12) {
		tmp = (Math.sin(x) / x) * y;
	} else {
		tmp = Math.sinh(y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if math.sinh(y) <= -math.inf:
		tmp = math.sinh(y)
	elif math.sinh(y) <= 2e-12:
		tmp = (math.sin(x) / x) * y
	else:
		tmp = math.sinh(y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (sinh(y) <= Float64(-Inf))
		tmp = sinh(y);
	elseif (sinh(y) <= 2e-12)
		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) <= -Inf)
		tmp = sinh(y);
	elseif (sinh(y) <= 2e-12)
		tmp = (sin(x) / x) * y;
	else
		tmp = sinh(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[Sinh[y], $MachinePrecision], (-Infinity)], N[Sinh[y], $MachinePrecision], If[LessEqual[N[Sinh[y], $MachinePrecision], 2e-12], 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) < -inf.0 or 1.99999999999999996e-12 < (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 77.3%

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

   if -inf.0 < (sinh.f64 y) < 1.99999999999999996e-12

   1. Initial program 71.2%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 70.9%

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

      1. associate-/l* 99.6%

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

   6. Simplified 99.6%

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

      1. clear-num 99.3%

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

      2. associate-/r/ 99.6%

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

      3. clear-num 99.6%

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

   8. Applied egg-rr 99.6%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sinh y \leq -\infty:\\ \;\;\;\;\sinh y\\ \mathbf{elif}\;\sinh y \leq 2 \cdot 10^{-12}:\\ \;\;\;\;\frac{\sin x}{x} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \]

Alternative 4: 86.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sinh y \leq -\infty:\\ \;\;\;\;\sinh y\\ \mathbf{elif}\;\sinh y \leq 2 \cdot 10^{-12}:\\ \;\;\;\;\frac{y}{\frac{x}{\sin x}}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sinh y) (- INFINITY))
   (sinh y)
   (if (<= (sinh y) 2e-12) (/ y (/ x (sin x))) (sinh y))))

C

double code(double x, double y) {
	double tmp;
	if (sinh(y) <= -((double) INFINITY)) {
		tmp = sinh(y);
	} else if (sinh(y) <= 2e-12) {
		tmp = y / (x / sin(x));
	} else {
		tmp = sinh(y);
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (Math.sinh(y) <= -Double.POSITIVE_INFINITY) {
		tmp = Math.sinh(y);
	} else if (Math.sinh(y) <= 2e-12) {
		tmp = y / (x / Math.sin(x));
	} else {
		tmp = Math.sinh(y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if math.sinh(y) <= -math.inf:
		tmp = math.sinh(y)
	elif math.sinh(y) <= 2e-12:
		tmp = y / (x / math.sin(x))
	else:
		tmp = math.sinh(y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (sinh(y) <= Float64(-Inf))
		tmp = sinh(y);
	elseif (sinh(y) <= 2e-12)
		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) <= -Inf)
		tmp = sinh(y);
	elseif (sinh(y) <= 2e-12)
		tmp = y / (x / sin(x));
	else
		tmp = sinh(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[Sinh[y], $MachinePrecision], (-Infinity)], N[Sinh[y], $MachinePrecision], If[LessEqual[N[Sinh[y], $MachinePrecision], 2e-12], 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) < -inf.0 or 1.99999999999999996e-12 < (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 77.3%

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

   if -inf.0 < (sinh.f64 y) < 1.99999999999999996e-12

   1. Initial program 71.2%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 70.9%

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

      1. associate-/l* 99.6%

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

   6. Simplified 99.6%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sinh y \leq -\infty:\\ \;\;\;\;\sinh y\\ \mathbf{elif}\;\sinh y \leq 2 \cdot 10^{-12}:\\ \;\;\;\;\frac{y}{\frac{x}{\sin x}}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \]

Alternative 5: 74.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sinh y \leq -\infty:\\ \;\;\;\;\sinh y\\ \mathbf{elif}\;\sinh y \leq 10^{-114}:\\ \;\;\;\;\frac{\frac{y}{x}}{x \cdot 0.16666666666666666 + \frac{1}{x}}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sinh y) (- INFINITY))
   (sinh y)
   (if (<= (sinh y) 1e-114)
     (/ (/ y x) (+ (* x 0.16666666666666666) (/ 1.0 x)))
     (sinh y))))

C

double code(double x, double y) {
	double tmp;
	if (sinh(y) <= -((double) INFINITY)) {
		tmp = sinh(y);
	} else if (sinh(y) <= 1e-114) {
		tmp = (y / x) / ((x * 0.16666666666666666) + (1.0 / x));
	} else {
		tmp = sinh(y);
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (Math.sinh(y) <= -Double.POSITIVE_INFINITY) {
		tmp = Math.sinh(y);
	} else if (Math.sinh(y) <= 1e-114) {
		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) <= -math.inf:
		tmp = math.sinh(y)
	elif math.sinh(y) <= 1e-114:
		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) <= Float64(-Inf))
		tmp = sinh(y);
	elseif (sinh(y) <= 1e-114)
		tmp = Float64(Float64(y / 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) <= -Inf)
		tmp = sinh(y);
	elseif (sinh(y) <= 1e-114)
		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], (-Infinity)], N[Sinh[y], $MachinePrecision], If[LessEqual[N[Sinh[y], $MachinePrecision], 1e-114], N[(N[(y / x), $MachinePrecision] / N[(N[(x * 0.16666666666666666), $MachinePrecision] + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sinh[y], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (sinh.f64 y) < -inf.0 or 1.0000000000000001e-114 < (sinh.f64 y)

   1. Initial program 98.1%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 x around 0 78.0%

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

   if -inf.0 < (sinh.f64 y) < 1.0000000000000001e-114

   1. Initial program 67.8%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 67.5%

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

      1. associate-/l* 99.5%

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

      2. associate-/r/ 99.4%

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

   6. Simplified 99.4%

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

      1. associate-/r/ 99.5%

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

      2. div-inv 99.3%

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

      3. associate-/r* 99.2%

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

   8. Applied egg-rr 99.2%

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

   9. Taylor expanded in x around 0 81.8%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sinh y \leq -\infty:\\ \;\;\;\;\sinh y\\ \mathbf{elif}\;\sinh y \leq 10^{-114}:\\ \;\;\;\;\frac{\frac{y}{x}}{x \cdot 0.16666666666666666 + \frac{1}{x}}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \]

Alternative 6: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.0%

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

   1. associate-*r/ 99.9%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

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

Alternative 7: 55.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{+179}:\\ \;\;\;\;\frac{y \cdot \left(-y\right)}{y \cdot \left(x \cdot \left(x \cdot -0.16666666666666666\right)\right) - y}\\ \mathbf{elif}\;y \leq -7.5 \cdot 10^{+166} \lor \neg \left(y \leq 1.25 \cdot 10^{+169}\right):\\ \;\;\;\;\sqrt{y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{\frac{x}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.2e+179)
   (/ (* y (- y)) (- (* y (* x (* x -0.16666666666666666))) y))
   (if (or (<= y -7.5e+166) (not (<= y 1.25e+169)))
     (sqrt (* y y))
     (* x (/ 1.0 (/ x y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.2e+179) {
		tmp = (y * -y) / ((y * (x * (x * -0.16666666666666666))) - y);
	} else if ((y <= -7.5e+166) || !(y <= 1.25e+169)) {
		tmp = sqrt((y * y));
	} else {
		tmp = x * (1.0 / (x / y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-2.2d+179)) then
        tmp = (y * -y) / ((y * (x * (x * (-0.16666666666666666d0)))) - y)
    else if ((y <= (-7.5d+166)) .or. (.not. (y <= 1.25d+169))) then
        tmp = sqrt((y * y))
    else
        tmp = x * (1.0d0 / (x / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -2.2e+179) {
		tmp = (y * -y) / ((y * (x * (x * -0.16666666666666666))) - y);
	} else if ((y <= -7.5e+166) || !(y <= 1.25e+169)) {
		tmp = Math.sqrt((y * y));
	} else {
		tmp = x * (1.0 / (x / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2.2e+179:
		tmp = (y * -y) / ((y * (x * (x * -0.16666666666666666))) - y)
	elif (y <= -7.5e+166) or not (y <= 1.25e+169):
		tmp = math.sqrt((y * y))
	else:
		tmp = x * (1.0 / (x / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.2e+179)
		tmp = Float64(Float64(y * Float64(-y)) / Float64(Float64(y * Float64(x * Float64(x * -0.16666666666666666))) - y));
	elseif ((y <= -7.5e+166) || !(y <= 1.25e+169))
		tmp = sqrt(Float64(y * y));
	else
		tmp = Float64(x * Float64(1.0 / Float64(x / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -2.2e+179)
		tmp = (y * -y) / ((y * (x * (x * -0.16666666666666666))) - y);
	elseif ((y <= -7.5e+166) || ~((y <= 1.25e+169)))
		tmp = sqrt((y * y));
	else
		tmp = x * (1.0 / (x / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.2e+179], N[(N[(y * (-y)), $MachinePrecision] / N[(N[(y * N[(x * N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[y, -7.5e+166], N[Not[LessEqual[y, 1.25e+169]], $MachinePrecision]], N[Sqrt[N[(y * y), $MachinePrecision]], $MachinePrecision], N[(x * N[(1.0 / N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.2e179

   1. Initial program 100.0%

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

      1. associate-*r/ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 6.7%

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

      1. associate-/l* 6.7%

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

      2. associate-/r/ 59.2%

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

   6. Simplified 59.2%

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

   7. Taylor expanded in x around 0 14.5%

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

      1. +-commutative 14.5%

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

      2. remove-double-neg 14.5%

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

      3. unsub-neg 14.5%

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

      4. *-commutative 14.5%

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

      5. associate-*l* 14.5%

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

      6. fma-neg 14.5%

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

      7. unpow2 14.5%

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

      8. remove-double-neg 14.5%

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

   9. Simplified 14.5%

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

       1. fma-udef 14.5%

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

       2. flip-+ 47.6%

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

       3. *-commutative 47.6%

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

       4. associate-*l* 47.6%

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

       5. *-commutative 47.6%

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

       6. associate-*l* 47.6%

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

       7. *-commutative 47.6%

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

       8. associate-*l* 47.6%

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

   11. Applied egg-rr 47.6%

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

   12. Taylor expanded in x around 0 66.7%

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

       1. unpow2 66.7%

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

       2. neg-mul-1 66.7%

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

       3. distribute-rgt-neg-in 66.7%

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

   14. Simplified 66.7%

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

   if -2.2e179 < y < -7.50000000000000029e166 or 1.25000000000000004e169 < y

   1. Initial program 100.0%

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

      1. associate-*r/ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 5.3%

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

   5. Taylor expanded in x around 0 22.5%

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

      1. div-inv 22.5%

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

      2. associate-*l* 4.6%

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

      3. div-inv 4.6%

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

      4. *-inverses 4.6%

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

      5. *-commutative 4.6%

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

      6. *-un-lft-identity 4.6%

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

      7. add-sqr-sqrt 4.5%

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

      8. sqrt-unprod 80.6%

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

   7. Applied egg-rr 80.6%

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

   if -7.50000000000000029e166 < y < 1.25000000000000004e169

   1. Initial program 82.5%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 44.8%

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

   5. Taylor expanded in x around 0 25.8%

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

      1. div-inv 26.2%

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

      2. *-commutative 26.2%

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

      3. associate-*l* 58.3%

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

   7. Applied egg-rr 58.3%

      \[\leadsto \color{blue}{x \cdot \left(y \cdot \frac{1}{x}\right)} \]
   8. Step-by-step derivation

      1. div-inv 58.4%

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

      2. clear-num 59.1%

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

   9. Applied egg-rr 59.1%

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

4. Final simplification 62.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{+179}:\\ \;\;\;\;\frac{y \cdot \left(-y\right)}{y \cdot \left(x \cdot \left(x \cdot -0.16666666666666666\right)\right) - y}\\ \mathbf{elif}\;y \leq -7.5 \cdot 10^{+166} \lor \neg \left(y \leq 1.25 \cdot 10^{+169}\right):\\ \;\;\;\;\sqrt{y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{\frac{x}{y}}\\ \end{array} \]

Alternative 8: 48.7% accurate, 13.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.7 \cdot 10^{+137}:\\ \;\;\;\;\frac{x}{\frac{x}{y}}\\ \mathbf{elif}\;x \leq -3.1 \cdot 10^{+91}:\\ \;\;\;\;x \cdot \left(x \cdot \left(y \cdot -0.16666666666666666\right)\right)\\ \mathbf{elif}\;x \leq 9 \cdot 10^{-92}:\\ \;\;\;\;x \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;y + \left(y \cdot -0.16666666666666666\right) \cdot \left(x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -3.7e+137)
   (/ x (/ x y))
   (if (<= x -3.1e+91)
     (* x (* x (* y -0.16666666666666666)))
     (if (<= x 9e-92)
       (* x (/ y x))
       (+ y (* (* y -0.16666666666666666) (* x x)))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -3.7e+137) {
		tmp = x / (x / y);
	} else if (x <= -3.1e+91) {
		tmp = x * (x * (y * -0.16666666666666666));
	} else if (x <= 9e-92) {
		tmp = x * (y / x);
	} else {
		tmp = y + ((y * -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 (x <= (-3.7d+137)) then
        tmp = x / (x / y)
    else if (x <= (-3.1d+91)) then
        tmp = x * (x * (y * (-0.16666666666666666d0)))
    else if (x <= 9d-92) then
        tmp = x * (y / x)
    else
        tmp = y + ((y * (-0.16666666666666666d0)) * (x * x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -3.7e+137) {
		tmp = x / (x / y);
	} else if (x <= -3.1e+91) {
		tmp = x * (x * (y * -0.16666666666666666));
	} else if (x <= 9e-92) {
		tmp = x * (y / x);
	} else {
		tmp = y + ((y * -0.16666666666666666) * (x * x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -3.7e+137:
		tmp = x / (x / y)
	elif x <= -3.1e+91:
		tmp = x * (x * (y * -0.16666666666666666))
	elif x <= 9e-92:
		tmp = x * (y / x)
	else:
		tmp = y + ((y * -0.16666666666666666) * (x * x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -3.7e+137)
		tmp = Float64(x / Float64(x / y));
	elseif (x <= -3.1e+91)
		tmp = Float64(x * Float64(x * Float64(y * -0.16666666666666666)));
	elseif (x <= 9e-92)
		tmp = Float64(x * Float64(y / x));
	else
		tmp = Float64(y + Float64(Float64(y * -0.16666666666666666) * Float64(x * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -3.7e+137)
		tmp = x / (x / y);
	elseif (x <= -3.1e+91)
		tmp = x * (x * (y * -0.16666666666666666));
	elseif (x <= 9e-92)
		tmp = x * (y / x);
	else
		tmp = y + ((y * -0.16666666666666666) * (x * x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -3.7e+137], N[(x / N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -3.1e+91], N[(x * N[(x * N[(y * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 9e-92], N[(x * N[(y / x), $MachinePrecision]), $MachinePrecision], N[(y + N[(N[(y * -0.16666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -3.7000000000000002e137

   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 56.6%

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

   5. Taylor expanded in x around 0 25.8%

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

      1. associate-/l* 4.9%

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

      2. associate-/r/ 41.3%

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

   7. Applied egg-rr 41.3%

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

      1. associate-*l/ 25.8%

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

      2. *-commutative 25.8%

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

      3. associate-/l* 43.5%

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

   9. Applied egg-rr 43.5%

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

   if -3.7000000000000002e137 < x < -3.09999999999999998e91

   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 27.3%

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

      1. associate-/l* 27.5%

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

      2. associate-/r/ 27.3%

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

   6. Simplified 27.3%

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

   7. Taylor expanded in x around 0 51.5%

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

      1. +-commutative 51.5%

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

      2. remove-double-neg 51.5%

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

      3. unsub-neg 51.5%

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

      4. *-commutative 51.5%

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

      5. associate-*l* 51.5%

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

      6. fma-neg 51.5%

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

      7. unpow2 51.5%

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

      8. remove-double-neg 51.5%

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

   9. Simplified 51.5%

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

       1. fma-udef 51.5%

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

       2. flip-+ 13.1%

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

       3. *-commutative 13.1%

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

       4. associate-*l* 13.1%

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

       5. *-commutative 13.1%

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

       6. associate-*l* 13.1%

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

       7. *-commutative 13.1%

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

       8. associate-*l* 13.1%

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

   11. Applied egg-rr 13.1%

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

   12. Taylor expanded in x around inf 51.5%

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

       1. *-commutative 51.5%

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

       2. *-commutative 51.5%

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

       3. unpow2 51.5%

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

       4. associate-*l* 51.5%

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

       5. *-commutative 51.5%

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

       6. associate-*r* 51.5%

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

       7. *-commutative 51.5%

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

       8. associate-*l* 51.5%

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

   14. Simplified 51.5%

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

   if -3.09999999999999998e91 < x < 9.0000000000000001e-92

   1. Initial program 75.1%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 30.1%

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

   5. Taylor expanded in x around 0 23.9%

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

      1. associate-/l* 48.2%

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

      2. associate-/r/ 75.1%

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

   7. Applied egg-rr 75.1%

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

   if 9.0000000000000001e-92 < x

   1. Initial program 98.6%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 41.3%

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

      1. associate-/l* 42.6%

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

      2. associate-/r/ 42.6%

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

   6. Simplified 42.6%

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

   7. Taylor expanded in x around 0 40.5%

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

      1. associate-*r* 40.5%

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

      2. unpow2 40.5%

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

   9. Simplified 40.5%

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

4. Final simplification 60.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.7 \cdot 10^{+137}:\\ \;\;\;\;\frac{x}{\frac{x}{y}}\\ \mathbf{elif}\;x \leq -3.1 \cdot 10^{+91}:\\ \;\;\;\;x \cdot \left(x \cdot \left(y \cdot -0.16666666666666666\right)\right)\\ \mathbf{elif}\;x \leq 9 \cdot 10^{-92}:\\ \;\;\;\;x \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;y + \left(y \cdot -0.16666666666666666\right) \cdot \left(x \cdot x\right)\\ \end{array} \]

Alternative 9: 48.8% accurate, 13.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.7 \cdot 10^{+137}:\\ \;\;\;\;\frac{x}{\frac{x}{y}}\\ \mathbf{elif}\;x \leq -1.45 \cdot 10^{+90}:\\ \;\;\;\;x \cdot \left(x \cdot \left(y \cdot -0.16666666666666666\right)\right)\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-92}:\\ \;\;\;\;x \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;y + x \cdot \left(y \cdot \left(x \cdot -0.16666666666666666\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -3.7e+137)
   (/ x (/ x y))
   (if (<= x -1.45e+90)
     (* x (* x (* y -0.16666666666666666)))
     (if (<= x 9.5e-92)
       (* x (/ y x))
       (+ y (* x (* y (* x -0.16666666666666666))))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -3.7e+137) {
		tmp = x / (x / y);
	} else if (x <= -1.45e+90) {
		tmp = x * (x * (y * -0.16666666666666666));
	} else if (x <= 9.5e-92) {
		tmp = x * (y / x);
	} else {
		tmp = y + (x * (y * (x * -0.16666666666666666)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-3.7d+137)) then
        tmp = x / (x / y)
    else if (x <= (-1.45d+90)) then
        tmp = x * (x * (y * (-0.16666666666666666d0)))
    else if (x <= 9.5d-92) then
        tmp = x * (y / x)
    else
        tmp = y + (x * (y * (x * (-0.16666666666666666d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -3.7e+137) {
		tmp = x / (x / y);
	} else if (x <= -1.45e+90) {
		tmp = x * (x * (y * -0.16666666666666666));
	} else if (x <= 9.5e-92) {
		tmp = x * (y / x);
	} else {
		tmp = y + (x * (y * (x * -0.16666666666666666)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -3.7e+137:
		tmp = x / (x / y)
	elif x <= -1.45e+90:
		tmp = x * (x * (y * -0.16666666666666666))
	elif x <= 9.5e-92:
		tmp = x * (y / x)
	else:
		tmp = y + (x * (y * (x * -0.16666666666666666)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -3.7e+137)
		tmp = Float64(x / Float64(x / y));
	elseif (x <= -1.45e+90)
		tmp = Float64(x * Float64(x * Float64(y * -0.16666666666666666)));
	elseif (x <= 9.5e-92)
		tmp = Float64(x * Float64(y / x));
	else
		tmp = Float64(y + Float64(x * Float64(y * Float64(x * -0.16666666666666666))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -3.7e+137)
		tmp = x / (x / y);
	elseif (x <= -1.45e+90)
		tmp = x * (x * (y * -0.16666666666666666));
	elseif (x <= 9.5e-92)
		tmp = x * (y / x);
	else
		tmp = y + (x * (y * (x * -0.16666666666666666)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -3.7e+137], N[(x / N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.45e+90], N[(x * N[(x * N[(y * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 9.5e-92], N[(x * N[(y / x), $MachinePrecision]), $MachinePrecision], N[(y + N[(x * N[(y * N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -3.7000000000000002e137

   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 56.6%

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

   5. Taylor expanded in x around 0 25.8%

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

      1. associate-/l* 4.9%

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

      2. associate-/r/ 41.3%

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

   7. Applied egg-rr 41.3%

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

      1. associate-*l/ 25.8%

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

      2. *-commutative 25.8%

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

      3. associate-/l* 43.5%

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

   9. Applied egg-rr 43.5%

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

   if -3.7000000000000002e137 < x < -1.4500000000000001e90

   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 27.3%

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

      1. associate-/l* 27.5%

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

      2. associate-/r/ 27.3%

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

   6. Simplified 27.3%

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

   7. Taylor expanded in x around 0 51.5%

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

      1. +-commutative 51.5%

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

      2. remove-double-neg 51.5%

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

      3. unsub-neg 51.5%

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

      4. *-commutative 51.5%

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

      5. associate-*l* 51.5%

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

      6. fma-neg 51.5%

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

      7. unpow2 51.5%

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

      8. remove-double-neg 51.5%

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

   9. Simplified 51.5%

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

       1. fma-udef 51.5%

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

       2. flip-+ 13.1%

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

       3. *-commutative 13.1%

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

       4. associate-*l* 13.1%

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

       5. *-commutative 13.1%

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

       6. associate-*l* 13.1%

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

       7. *-commutative 13.1%

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

       8. associate-*l* 13.1%

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

   11. Applied egg-rr 13.1%

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

   12. Taylor expanded in x around inf 51.5%

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

       1. *-commutative 51.5%

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

       2. *-commutative 51.5%

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

       3. unpow2 51.5%

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

       4. associate-*l* 51.5%

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

       5. *-commutative 51.5%

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

       6. associate-*r* 51.5%

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

       7. *-commutative 51.5%

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

       8. associate-*l* 51.5%

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

   14. Simplified 51.5%

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

   if -1.4500000000000001e90 < x < 9.49999999999999946e-92

   1. Initial program 75.1%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 30.1%

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

   5. Taylor expanded in x around 0 23.9%

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

      1. associate-/l* 48.2%

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

      2. associate-/r/ 75.1%

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

   7. Applied egg-rr 75.1%

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

   if 9.49999999999999946e-92 < x

   1. Initial program 98.6%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 41.3%

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

      1. associate-/l* 42.6%

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

      2. associate-/r/ 42.6%

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

   6. Simplified 42.6%

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

   7. Taylor expanded in x around 0 40.5%

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

      1. +-commutative 40.5%

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

      2. remove-double-neg 40.5%

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

      3. unsub-neg 40.5%

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

      4. *-commutative 40.5%

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

      5. associate-*l* 40.5%

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

      6. fma-neg 40.5%

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

      7. unpow2 40.5%

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

      8. remove-double-neg 40.5%

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

   9. Simplified 40.5%

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

       1. fma-udef 40.5%

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

       2. flip-+ 15.1%

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

       3. *-commutative 15.1%

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

       4. associate-*l* 15.1%

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

       5. *-commutative 15.1%

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

       6. associate-*l* 15.1%

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

       7. *-commutative 15.1%

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

       8. associate-*l* 15.1%

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

   11. Applied egg-rr 15.1%

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

       1. flip-+ 40.5%

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

       2. associate-*l* 40.5%

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

       3. *-commutative 40.5%

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

   13. Applied egg-rr 40.5%

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

4. Final simplification 60.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.7 \cdot 10^{+137}:\\ \;\;\;\;\frac{x}{\frac{x}{y}}\\ \mathbf{elif}\;x \leq -1.45 \cdot 10^{+90}:\\ \;\;\;\;x \cdot \left(x \cdot \left(y \cdot -0.16666666666666666\right)\right)\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-92}:\\ \;\;\;\;x \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;y + x \cdot \left(y \cdot \left(x \cdot -0.16666666666666666\right)\right)\\ \end{array} \]

Alternative 10: 49.2% accurate, 15.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{+136}:\\ \;\;\;\;\frac{x}{\frac{x}{y}}\\ \mathbf{elif}\;x \leq -1.55 \cdot 10^{+90} \lor \neg \left(x \leq 48\right):\\ \;\;\;\;-0.16666666666666666 \cdot \left(y \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -4.8e+136)
   (/ x (/ x y))
   (if (or (<= x -1.55e+90) (not (<= x 48.0)))
     (* -0.16666666666666666 (* y (* x x)))
     (* x (/ y x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -4.8e+136) {
		tmp = x / (x / y);
	} else if ((x <= -1.55e+90) || !(x <= 48.0)) {
		tmp = -0.16666666666666666 * (y * (x * 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 <= (-4.8d+136)) then
        tmp = x / (x / y)
    else if ((x <= (-1.55d+90)) .or. (.not. (x <= 48.0d0))) then
        tmp = (-0.16666666666666666d0) * (y * (x * 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 <= -4.8e+136) {
		tmp = x / (x / y);
	} else if ((x <= -1.55e+90) || !(x <= 48.0)) {
		tmp = -0.16666666666666666 * (y * (x * x));
	} else {
		tmp = x * (y / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -4.8e+136:
		tmp = x / (x / y)
	elif (x <= -1.55e+90) or not (x <= 48.0):
		tmp = -0.16666666666666666 * (y * (x * x))
	else:
		tmp = x * (y / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -4.8e+136)
		tmp = Float64(x / Float64(x / y));
	elseif ((x <= -1.55e+90) || !(x <= 48.0))
		tmp = Float64(-0.16666666666666666 * Float64(y * Float64(x * x)));
	else
		tmp = Float64(x * Float64(y / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -4.8e+136)
		tmp = x / (x / y);
	elseif ((x <= -1.55e+90) || ~((x <= 48.0)))
		tmp = -0.16666666666666666 * (y * (x * x));
	else
		tmp = x * (y / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -4.8e+136], N[(x / N[(x / y), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[x, -1.55e+90], N[Not[LessEqual[x, 48.0]], $MachinePrecision]], N[(-0.16666666666666666 * N[(y * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(y / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.8000000000000001e136

   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 56.6%

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

   5. Taylor expanded in x around 0 25.8%

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

      1. associate-/l* 4.9%

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

      2. associate-/r/ 41.3%

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

   7. Applied egg-rr 41.3%

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

      1. associate-*l/ 25.8%

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

      2. *-commutative 25.8%

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

      3. associate-/l* 43.5%

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

   9. Applied egg-rr 43.5%

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

   if -4.8000000000000001e136 < x < -1.54999999999999994e90 or 48 < x

   1. Initial program 99.9%

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

      1. associate-*r/ 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around 0 33.0%

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

      1. associate-/l* 33.1%

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

      2. associate-/r/ 33.0%

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

   6. Simplified 33.0%

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

   7. Taylor expanded in x around 0 34.4%

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

      1. +-commutative 34.4%

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

      2. remove-double-neg 34.4%

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

      3. unsub-neg 34.4%

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

      4. *-commutative 34.4%

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

      5. associate-*l* 34.4%

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

      6. fma-neg 34.4%

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

      7. unpow2 34.4%

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

      8. remove-double-neg 34.4%

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

   9. Simplified 34.4%

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

   10. Taylor expanded in x around inf 34.4%

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

       1. unpow2 34.4%

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

   12. Simplified 34.4%

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

   if -1.54999999999999994e90 < x < 48

   1. Initial program 78.2%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 34.0%

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

   5. Taylor expanded in x around 0 28.3%

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

      1. associate-/l* 49.6%

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

      2. associate-/r/ 72.5%

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

   7. Applied egg-rr 72.5%

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

4. Final simplification 59.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{+136}:\\ \;\;\;\;\frac{x}{\frac{x}{y}}\\ \mathbf{elif}\;x \leq -1.55 \cdot 10^{+90} \lor \neg \left(x \leq 48\right):\\ \;\;\;\;-0.16666666666666666 \cdot \left(y \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{x}\\ \end{array} \]

Alternative 11: 49.2% accurate, 15.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{+136}:\\ \;\;\;\;\frac{x}{\frac{x}{y}}\\ \mathbf{elif}\;x \leq -1.45 \cdot 10^{+90} \lor \neg \left(x \leq 48\right):\\ \;\;\;\;x \cdot \left(x \cdot \left(y \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -4.8e+136)
   (/ x (/ x y))
   (if (or (<= x -1.45e+90) (not (<= x 48.0)))
     (* x (* x (* y -0.16666666666666666)))
     (* x (/ y x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -4.8e+136) {
		tmp = x / (x / y);
	} else if ((x <= -1.45e+90) || !(x <= 48.0)) {
		tmp = x * (x * (y * -0.16666666666666666));
	} 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 <= (-4.8d+136)) then
        tmp = x / (x / y)
    else if ((x <= (-1.45d+90)) .or. (.not. (x <= 48.0d0))) then
        tmp = x * (x * (y * (-0.16666666666666666d0)))
    else
        tmp = x * (y / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -4.8e+136) {
		tmp = x / (x / y);
	} else if ((x <= -1.45e+90) || !(x <= 48.0)) {
		tmp = x * (x * (y * -0.16666666666666666));
	} else {
		tmp = x * (y / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -4.8e+136:
		tmp = x / (x / y)
	elif (x <= -1.45e+90) or not (x <= 48.0):
		tmp = x * (x * (y * -0.16666666666666666))
	else:
		tmp = x * (y / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -4.8e+136)
		tmp = Float64(x / Float64(x / y));
	elseif ((x <= -1.45e+90) || !(x <= 48.0))
		tmp = Float64(x * Float64(x * Float64(y * -0.16666666666666666)));
	else
		tmp = Float64(x * Float64(y / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -4.8e+136)
		tmp = x / (x / y);
	elseif ((x <= -1.45e+90) || ~((x <= 48.0)))
		tmp = x * (x * (y * -0.16666666666666666));
	else
		tmp = x * (y / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -4.8e+136], N[(x / N[(x / y), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[x, -1.45e+90], N[Not[LessEqual[x, 48.0]], $MachinePrecision]], N[(x * N[(x * N[(y * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(y / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.8000000000000001e136

   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 56.6%

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

   5. Taylor expanded in x around 0 25.8%

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

      1. associate-/l* 4.9%

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

      2. associate-/r/ 41.3%

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

   7. Applied egg-rr 41.3%

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

      1. associate-*l/ 25.8%

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

      2. *-commutative 25.8%

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

      3. associate-/l* 43.5%

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

   9. Applied egg-rr 43.5%

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

   if -4.8000000000000001e136 < x < -1.4500000000000001e90 or 48 < x

   1. Initial program 99.9%

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

      1. associate-*r/ 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around 0 33.0%

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

      1. associate-/l* 33.1%

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

      2. associate-/r/ 33.0%

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

   6. Simplified 33.0%

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

   7. Taylor expanded in x around 0 34.4%

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

      1. +-commutative 34.4%

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

      2. remove-double-neg 34.4%

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

      3. unsub-neg 34.4%

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

      4. *-commutative 34.4%

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

      5. associate-*l* 34.4%

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

      6. fma-neg 34.4%

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

      7. unpow2 34.4%

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

      8. remove-double-neg 34.4%

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

   9. Simplified 34.4%

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

       1. fma-udef 34.4%

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

       2. flip-+ 2.4%

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

       3. *-commutative 2.4%

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

       4. associate-*l* 2.4%

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

       5. *-commutative 2.4%

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

       6. associate-*l* 2.4%

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

       7. *-commutative 2.4%

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

       8. associate-*l* 2.4%

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

   11. Applied egg-rr 2.4%

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

   12. Taylor expanded in x around inf 34.4%

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

       1. *-commutative 34.4%

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

       2. *-commutative 34.4%

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

       3. unpow2 34.4%

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

       4. associate-*l* 34.5%

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

       5. *-commutative 34.5%

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

       6. associate-*r* 34.5%

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

       7. *-commutative 34.5%

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

       8. associate-*l* 34.5%

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

   14. Simplified 34.5%

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

   if -1.4500000000000001e90 < x < 48

   1. Initial program 78.2%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 34.0%

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

   5. Taylor expanded in x around 0 28.3%

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

      1. associate-/l* 49.6%

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

      2. associate-/r/ 72.5%

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

   7. Applied egg-rr 72.5%

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

4. Final simplification 60.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{+136}:\\ \;\;\;\;\frac{x}{\frac{x}{y}}\\ \mathbf{elif}\;x \leq -1.45 \cdot 10^{+90} \lor \neg \left(x \leq 48\right):\\ \;\;\;\;x \cdot \left(x \cdot \left(y \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{x}\\ \end{array} \]

Alternative 12: 49.7% 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 86.0%

   \[\frac{\sin x \cdot \sinh y}{x} \]
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 36.9%

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

5. Taylor expanded in x around 0 25.7%

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

   1. associate-/l* 33.0%

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

   2. associate-/r/ 55.6%

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

7. Applied egg-rr 55.6%

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

8. Final simplification 55.6%

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

Alternative 13: 27.7% accurate, 205.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 y)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return y

Julia

function code(x, y)
	return y
end

MATLAB

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

Wolfram

code[x_, y_] := y

Derivation

1. Initial program 86.0%

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

   1. associate-*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 36.9%

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

   1. associate-/l* 50.8%

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

   2. associate-/r/ 65.1%

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

6. Simplified 65.1%

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

7. Taylor expanded in x around 0 33.0%

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

8. Final simplification 33.0%

   \[\leadsto y \]

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 2023174 
(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))