Linear.Quaternion:$ccosh from linear-1.19.1.3

Percentage Accurate: 89.4% → 99.9%

Time: 8.3s

Alternatives: 11

Speedup: 1.0×

• 50.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 89.4% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.7%

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

   1. *-commutative 87.7%

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

   2. associate-/l* 99.9%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

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

Alternative 2: 84.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sinh y \leq -\infty \lor \neg \left(\sinh y \leq 10^{-58}\right):\\ \;\;\;\;\sinh y\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \frac{y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= (sinh y) (- INFINITY)) (not (<= (sinh y) 1e-58)))
   (sinh y)
   (* (sin x) (/ y x))))

C

double code(double x, double y) {
	double tmp;
	if ((sinh(y) <= -((double) INFINITY)) || !(sinh(y) <= 1e-58)) {
		tmp = sinh(y);
	} else {
		tmp = sin(x) * (y / x);
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if ((Math.sinh(y) <= -Double.POSITIVE_INFINITY) || !(Math.sinh(y) <= 1e-58)) {
		tmp = Math.sinh(y);
	} else {
		tmp = Math.sin(x) * (y / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (math.sinh(y) <= -math.inf) or not (math.sinh(y) <= 1e-58):
		tmp = math.sinh(y)
	else:
		tmp = math.sin(x) * (y / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((sinh(y) <= Float64(-Inf)) || !(sinh(y) <= 1e-58))
		tmp = sinh(y);
	else
		tmp = Float64(sin(x) * Float64(y / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((sinh(y) <= -Inf) || ~((sinh(y) <= 1e-58)))
		tmp = sinh(y);
	else
		tmp = sin(x) * (y / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[N[Sinh[y], $MachinePrecision], (-Infinity)], N[Not[LessEqual[N[Sinh[y], $MachinePrecision], 1e-58]], $MachinePrecision]], N[Sinh[y], $MachinePrecision], N[(N[Sin[x], $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sinh.f64 y) < -inf.0 or 1e-58 < (sinh.f64 y)

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 72.1%

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

   if -inf.0 < (sinh.f64 y) < 1e-58

   1. Initial program 71.3%

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

      1. *-commutative 71.3%

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

      2. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around 0 71.3%

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

      1. associate-/l* 99.9%

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

   6. Simplified 99.9%

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

      1. associate-/r/ 99.7%

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

   8. Applied egg-rr 99.7%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sinh y \leq -\infty \lor \neg \left(\sinh y \leq 10^{-58}\right):\\ \;\;\;\;\sinh y\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \frac{y}{x}\\ \end{array} \]

Alternative 3: 84.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sinh y \leq -\infty \lor \neg \left(\sinh y \leq 10^{-58}\right):\\ \;\;\;\;\sinh y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\sin x}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= (sinh y) (- INFINITY)) (not (<= (sinh y) 1e-58)))
   (sinh y)
   (* y (/ (sin x) x))))

C

double code(double x, double y) {
	double tmp;
	if ((sinh(y) <= -((double) INFINITY)) || !(sinh(y) <= 1e-58)) {
		tmp = sinh(y);
	} else {
		tmp = y * (sin(x) / x);
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if ((Math.sinh(y) <= -Double.POSITIVE_INFINITY) || !(Math.sinh(y) <= 1e-58)) {
		tmp = Math.sinh(y);
	} else {
		tmp = y * (Math.sin(x) / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (math.sinh(y) <= -math.inf) or not (math.sinh(y) <= 1e-58):
		tmp = math.sinh(y)
	else:
		tmp = y * (math.sin(x) / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((sinh(y) <= Float64(-Inf)) || !(sinh(y) <= 1e-58))
		tmp = sinh(y);
	else
		tmp = Float64(y * Float64(sin(x) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((sinh(y) <= -Inf) || ~((sinh(y) <= 1e-58)))
		tmp = sinh(y);
	else
		tmp = y * (sin(x) / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[N[Sinh[y], $MachinePrecision], (-Infinity)], N[Not[LessEqual[N[Sinh[y], $MachinePrecision], 1e-58]], $MachinePrecision]], N[Sinh[y], $MachinePrecision], N[(y * N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sinh.f64 y) < -inf.0 or 1e-58 < (sinh.f64 y)

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 72.1%

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

   if -inf.0 < (sinh.f64 y) < 1e-58

   1. Initial program 71.3%

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

      1. *-commutative 71.3%

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

      2. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around 0 71.3%

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

      1. associate-/l* 99.9%

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

   6. Simplified 99.9%

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

      1. associate-/l* 71.3%

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

      2. *-commutative 71.3%

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

      3. associate-*l/ 99.8%

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

   8. Applied egg-rr 99.8%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sinh y \leq -\infty \lor \neg \left(\sinh y \leq 10^{-58}\right):\\ \;\;\;\;\sinh y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\sin x}{x}\\ \end{array} \]

Alternative 4: 84.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sinh y \leq -\infty \lor \neg \left(\sinh y \leq 10^{-58}\right):\\ \;\;\;\;\sinh y\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{x}{\sin x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= (sinh y) (- INFINITY)) (not (<= (sinh y) 1e-58)))
   (sinh y)
   (/ y (/ x (sin x)))))

C

double code(double x, double y) {
	double tmp;
	if ((sinh(y) <= -((double) INFINITY)) || !(sinh(y) <= 1e-58)) {
		tmp = sinh(y);
	} else {
		tmp = y / (x / sin(x));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if ((Math.sinh(y) <= -Double.POSITIVE_INFINITY) || !(Math.sinh(y) <= 1e-58)) {
		tmp = Math.sinh(y);
	} else {
		tmp = y / (x / Math.sin(x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (math.sinh(y) <= -math.inf) or not (math.sinh(y) <= 1e-58):
		tmp = math.sinh(y)
	else:
		tmp = y / (x / math.sin(x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((sinh(y) <= Float64(-Inf)) || !(sinh(y) <= 1e-58))
		tmp = sinh(y);
	else
		tmp = Float64(y / Float64(x / sin(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((sinh(y) <= -Inf) || ~((sinh(y) <= 1e-58)))
		tmp = sinh(y);
	else
		tmp = y / (x / sin(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[N[Sinh[y], $MachinePrecision], (-Infinity)], N[Not[LessEqual[N[Sinh[y], $MachinePrecision], 1e-58]], $MachinePrecision]], N[Sinh[y], $MachinePrecision], N[(y / N[(x / N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sinh.f64 y) < -inf.0 or 1e-58 < (sinh.f64 y)

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 72.1%

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

   if -inf.0 < (sinh.f64 y) < 1e-58

   1. Initial program 71.3%

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

      1. *-commutative 71.3%

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

      2. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around 0 71.3%

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

      1. associate-/l* 99.9%

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

   6. Simplified 99.9%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sinh y \leq -\infty \lor \neg \left(\sinh y \leq 10^{-58}\right):\\ \;\;\;\;\sinh y\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{x}{\sin x}}\\ \end{array} \]

Alternative 5: 75.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sinh y \leq -\infty \lor \neg \left(\sinh y \leq 10^{-58}\right):\\ \;\;\;\;\sinh y\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{1 + x \cdot \left(x \cdot 0.16666666666666666\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= (sinh y) (- INFINITY)) (not (<= (sinh y) 1e-58)))
   (sinh y)
   (/ y (+ 1.0 (* x (* x 0.16666666666666666))))))

C

double code(double x, double y) {
	double tmp;
	if ((sinh(y) <= -((double) INFINITY)) || !(sinh(y) <= 1e-58)) {
		tmp = sinh(y);
	} else {
		tmp = y / (1.0 + (x * (x * 0.16666666666666666)));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if ((Math.sinh(y) <= -Double.POSITIVE_INFINITY) || !(Math.sinh(y) <= 1e-58)) {
		tmp = Math.sinh(y);
	} else {
		tmp = y / (1.0 + (x * (x * 0.16666666666666666)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (math.sinh(y) <= -math.inf) or not (math.sinh(y) <= 1e-58):
		tmp = math.sinh(y)
	else:
		tmp = y / (1.0 + (x * (x * 0.16666666666666666)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((sinh(y) <= Float64(-Inf)) || !(sinh(y) <= 1e-58))
		tmp = sinh(y);
	else
		tmp = Float64(y / Float64(1.0 + Float64(x * Float64(x * 0.16666666666666666))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((sinh(y) <= -Inf) || ~((sinh(y) <= 1e-58)))
		tmp = sinh(y);
	else
		tmp = y / (1.0 + (x * (x * 0.16666666666666666)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[N[Sinh[y], $MachinePrecision], (-Infinity)], N[Not[LessEqual[N[Sinh[y], $MachinePrecision], 1e-58]], $MachinePrecision]], N[Sinh[y], $MachinePrecision], N[(y / N[(1.0 + N[(x * N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sinh.f64 y) < -inf.0 or 1e-58 < (sinh.f64 y)

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 72.1%

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

   if -inf.0 < (sinh.f64 y) < 1e-58

   1. Initial program 71.3%

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

      1. *-commutative 71.3%

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

      2. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

      1. clear-num 99.8%

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

      2. associate-/r/ 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in y around 0 99.8%

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

   7. Taylor expanded in x around 0 76.7%

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

      1. *-commutative 76.7%

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

      2. distribute-rgt-in 76.8%

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

      3. *-commutative 76.8%

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

      4. lft-mult-inverse 76.9%

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

   9. Applied egg-rr 76.9%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sinh y \leq -\infty \lor \neg \left(\sinh y \leq 10^{-58}\right):\\ \;\;\;\;\sinh y\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{1 + x \cdot \left(x \cdot 0.16666666666666666\right)}\\ \end{array} \]

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 87.7%

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

   1. *-commutative 87.7%

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

   2. associate-/l* 99.9%

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

3. Simplified 99.9%

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

   1. associate-/r/ 99.9%

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

5. Applied egg-rr 99.9%

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

6. Final simplification 99.9%

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

Alternative 7: 55.3% accurate, 15.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y}{x \cdot \left(x \cdot 0.16666666666666666\right)}\\ \mathbf{if}\;y \leq -160000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 430:\\ \;\;\;\;\frac{x}{\frac{x}{y}}\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{+187}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{+248}:\\ \;\;\;\;\frac{y \cdot x}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ y (* x (* x 0.16666666666666666)))))
   (if (<= y -160000.0)
     t_0
     (if (<= y 430.0)
       (/ x (/ x y))
       (if (<= y 4.8e+187)
         t_0
         (if (<= y 1.1e+248) (/ (* y x) x) (* x (/ y x))))))))

C

double code(double x, double y) {
	double t_0 = y / (x * (x * 0.16666666666666666));
	double tmp;
	if (y <= -160000.0) {
		tmp = t_0;
	} else if (y <= 430.0) {
		tmp = x / (x / y);
	} else if (y <= 4.8e+187) {
		tmp = t_0;
	} else if (y <= 1.1e+248) {
		tmp = (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) :: t_0
    real(8) :: tmp
    t_0 = y / (x * (x * 0.16666666666666666d0))
    if (y <= (-160000.0d0)) then
        tmp = t_0
    else if (y <= 430.0d0) then
        tmp = x / (x / y)
    else if (y <= 4.8d+187) then
        tmp = t_0
    else if (y <= 1.1d+248) then
        tmp = (y * x) / x
    else
        tmp = x * (y / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = y / (x * (x * 0.16666666666666666));
	double tmp;
	if (y <= -160000.0) {
		tmp = t_0;
	} else if (y <= 430.0) {
		tmp = x / (x / y);
	} else if (y <= 4.8e+187) {
		tmp = t_0;
	} else if (y <= 1.1e+248) {
		tmp = (y * x) / x;
	} else {
		tmp = x * (y / x);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y / (x * (x * 0.16666666666666666))
	tmp = 0
	if y <= -160000.0:
		tmp = t_0
	elif y <= 430.0:
		tmp = x / (x / y)
	elif y <= 4.8e+187:
		tmp = t_0
	elif y <= 1.1e+248:
		tmp = (y * x) / x
	else:
		tmp = x * (y / x)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y / Float64(x * Float64(x * 0.16666666666666666)))
	tmp = 0.0
	if (y <= -160000.0)
		tmp = t_0;
	elseif (y <= 430.0)
		tmp = Float64(x / Float64(x / y));
	elseif (y <= 4.8e+187)
		tmp = t_0;
	elseif (y <= 1.1e+248)
		tmp = Float64(Float64(y * x) / x);
	else
		tmp = Float64(x * Float64(y / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y / (x * (x * 0.16666666666666666));
	tmp = 0.0;
	if (y <= -160000.0)
		tmp = t_0;
	elseif (y <= 430.0)
		tmp = x / (x / y);
	elseif (y <= 4.8e+187)
		tmp = t_0;
	elseif (y <= 1.1e+248)
		tmp = (y * x) / x;
	else
		tmp = x * (y / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y / N[(x * N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -160000.0], t$95$0, If[LessEqual[y, 430.0], N[(x / N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.8e+187], t$95$0, If[LessEqual[y, 1.1e+248], N[(N[(y * x), $MachinePrecision] / x), $MachinePrecision], N[(x * N[(y / x), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -1.6e5 or 430 < y < 4.79999999999999971e187

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

      1. clear-num 100.0%

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

      2. associate-/r/ 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in y around 0 4.1%

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

   7. Taylor expanded in x around 0 3.3%

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

   8. Taylor expanded in x around inf 43.1%

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

      1. *-commutative 43.1%

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

   10. Simplified 43.1%

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

   if -1.6e5 < y < 430

   1. Initial program 72.7%

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

   2. Taylor expanded in y around 0 69.7%

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

   3. Taylor expanded in x around 0 33.1%

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

      1. div-inv 32.9%

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

      2. associate-*l* 72.2%

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

   5. Applied egg-rr 72.2%

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

      1. un-div-inv 72.3%

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

      2. associate-*r/ 33.1%

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

      3. associate-/l* 74.0%

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

   7. Applied egg-rr 74.0%

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

   if 4.79999999999999971e187 < y < 1.1e248

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 4.5%

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

   3. Taylor expanded in x around 0 49.0%

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

   if 1.1e248 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 6.6%

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

   3. Taylor expanded in x around 0 37.0%

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

      1. div-inv 37.0%

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

      2. associate-*l* 44.6%

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

   5. Applied egg-rr 44.6%

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

   6. Taylor expanded in y around 0 44.6%

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

4. Final simplification 57.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -160000:\\ \;\;\;\;\frac{y}{x \cdot \left(x \cdot 0.16666666666666666\right)}\\ \mathbf{elif}\;y \leq 430:\\ \;\;\;\;\frac{x}{\frac{x}{y}}\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{+187}:\\ \;\;\;\;\frac{y}{x \cdot \left(x \cdot 0.16666666666666666\right)}\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{+248}:\\ \;\;\;\;\frac{y \cdot x}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{x}\\ \end{array} \]

Alternative 8: 55.8% accurate, 15.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot 0.16666666666666666\right)\\ t_1 := \frac{y}{t_0}\\ \mathbf{if}\;y \leq -160000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 320:\\ \;\;\;\;\frac{y}{1 + t_0}\\ \mathbf{elif}\;y \leq 8 \cdot 10^{+187}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.95 \cdot 10^{+247}:\\ \;\;\;\;\frac{y \cdot x}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* x (* x 0.16666666666666666))) (t_1 (/ y t_0)))
   (if (<= y -160000.0)
     t_1
     (if (<= y 320.0)
       (/ y (+ 1.0 t_0))
       (if (<= y 8e+187)
         t_1
         (if (<= y 1.95e+247) (/ (* y x) x) (* x (/ y x))))))))

C

double code(double x, double y) {
	double t_0 = x * (x * 0.16666666666666666);
	double t_1 = y / t_0;
	double tmp;
	if (y <= -160000.0) {
		tmp = t_1;
	} else if (y <= 320.0) {
		tmp = y / (1.0 + t_0);
	} else if (y <= 8e+187) {
		tmp = t_1;
	} else if (y <= 1.95e+247) {
		tmp = (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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x * (x * 0.16666666666666666d0)
    t_1 = y / t_0
    if (y <= (-160000.0d0)) then
        tmp = t_1
    else if (y <= 320.0d0) then
        tmp = y / (1.0d0 + t_0)
    else if (y <= 8d+187) then
        tmp = t_1
    else if (y <= 1.95d+247) then
        tmp = (y * x) / x
    else
        tmp = x * (y / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x * (x * 0.16666666666666666);
	double t_1 = y / t_0;
	double tmp;
	if (y <= -160000.0) {
		tmp = t_1;
	} else if (y <= 320.0) {
		tmp = y / (1.0 + t_0);
	} else if (y <= 8e+187) {
		tmp = t_1;
	} else if (y <= 1.95e+247) {
		tmp = (y * x) / x;
	} else {
		tmp = x * (y / x);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x * (x * 0.16666666666666666)
	t_1 = y / t_0
	tmp = 0
	if y <= -160000.0:
		tmp = t_1
	elif y <= 320.0:
		tmp = y / (1.0 + t_0)
	elif y <= 8e+187:
		tmp = t_1
	elif y <= 1.95e+247:
		tmp = (y * x) / x
	else:
		tmp = x * (y / x)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x * Float64(x * 0.16666666666666666))
	t_1 = Float64(y / t_0)
	tmp = 0.0
	if (y <= -160000.0)
		tmp = t_1;
	elseif (y <= 320.0)
		tmp = Float64(y / Float64(1.0 + t_0));
	elseif (y <= 8e+187)
		tmp = t_1;
	elseif (y <= 1.95e+247)
		tmp = Float64(Float64(y * x) / x);
	else
		tmp = Float64(x * Float64(y / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x * (x * 0.16666666666666666);
	t_1 = y / t_0;
	tmp = 0.0;
	if (y <= -160000.0)
		tmp = t_1;
	elseif (y <= 320.0)
		tmp = y / (1.0 + t_0);
	elseif (y <= 8e+187)
		tmp = t_1;
	elseif (y <= 1.95e+247)
		tmp = (y * x) / x;
	else
		tmp = x * (y / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x * N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(y / t$95$0), $MachinePrecision]}, If[LessEqual[y, -160000.0], t$95$1, If[LessEqual[y, 320.0], N[(y / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 8e+187], t$95$1, If[LessEqual[y, 1.95e+247], N[(N[(y * x), $MachinePrecision] / x), $MachinePrecision], N[(x * N[(y / x), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -1.6e5 or 320 < y < 7.99999999999999926e187

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

      1. clear-num 100.0%

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

      2. associate-/r/ 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in y around 0 4.1%

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

   7. Taylor expanded in x around 0 3.3%

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

   8. Taylor expanded in x around inf 43.1%

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

      1. *-commutative 43.1%

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

   10. Simplified 43.1%

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

   if -1.6e5 < y < 320

   1. Initial program 72.7%

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

      1. *-commutative 72.7%

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

      2. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

      1. clear-num 99.8%

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

      2. associate-/r/ 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in y around 0 96.8%

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

   7. Taylor expanded in x around 0 74.9%

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

      1. *-commutative 74.9%

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

      2. distribute-rgt-in 74.9%

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

      3. *-commutative 74.9%

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

      4. lft-mult-inverse 75.0%

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

   9. Applied egg-rr 75.0%

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

   if 7.99999999999999926e187 < y < 1.95000000000000001e247

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 4.5%

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

   3. Taylor expanded in x around 0 49.0%

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

   if 1.95000000000000001e247 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 6.6%

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

   3. Taylor expanded in x around 0 37.0%

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

      1. div-inv 37.0%

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

      2. associate-*l* 44.6%

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

   5. Applied egg-rr 44.6%

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

   6. Taylor expanded in y around 0 44.6%

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

4. Final simplification 58.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -160000:\\ \;\;\;\;\frac{y}{x \cdot \left(x \cdot 0.16666666666666666\right)}\\ \mathbf{elif}\;y \leq 320:\\ \;\;\;\;\frac{y}{1 + x \cdot \left(x \cdot 0.16666666666666666\right)}\\ \mathbf{elif}\;y \leq 8 \cdot 10^{+187}:\\ \;\;\;\;\frac{y}{x \cdot \left(x \cdot 0.16666666666666666\right)}\\ \mathbf{elif}\;y \leq 1.95 \cdot 10^{+247}:\\ \;\;\;\;\frac{y \cdot x}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{x}\\ \end{array} \]

Alternative 9: 50.9% accurate, 29.3× speedup

Math

\[\begin{array}{l} \\ x \cdot \frac{1}{\frac{x}{y}} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* x (/ 1.0 (/ x y))))

C

double code(double x, double y) {
	return x * (1.0 / (x / y));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x * (1.0d0 / (x / y))
end function

Java

public static double code(double x, double y) {
	return x * (1.0 / (x / y));
}

Python

def code(x, y):
	return x * (1.0 / (x / y))

Julia

function code(x, y)
	return Float64(x * Float64(1.0 / Float64(x / y)))
end

MATLAB

function tmp = code(x, y)
	tmp = x * (1.0 / (x / y));
end

Wolfram

code[x_, y_] := N[(x * N[(1.0 / N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 87.7%

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

2. Taylor expanded in y around 0 33.7%

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

3. Taylor expanded in x around 0 24.7%

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

   1. associate-/l* 47.1%

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

   2. div-inv 48.5%

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

5. Applied egg-rr 48.5%

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

6. Final simplification 48.5%

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

Alternative 10: 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 87.7%

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

2. Taylor expanded in y around 0 33.7%

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

3. Taylor expanded in x around 0 24.7%

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

   1. div-inv 24.6%

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

   2. associate-*l* 47.7%

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

5. Applied egg-rr 47.7%

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

6. Taylor expanded in y around 0 47.8%

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

7. Final simplification 47.8%

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

Alternative 11: 27.2% accurate, 205.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 y)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return y

Julia

function code(x, y)
	return y
end

MATLAB

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

Wolfram

code[x_, y_] := y

Derivation

1. Initial program 87.7%

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

   1. *-commutative 87.7%

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

   2. associate-/l* 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in y around 0 33.7%

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

   1. associate-/l* 45.9%

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

6. Simplified 45.9%

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

7. Taylor expanded in x around 0 28.8%

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

8. Final simplification 28.8%

   \[\leadsto y \]

Developer target: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023310 
(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))