Linear.Quaternion:$ccosh from linear-1.19.1.3

Percentage Accurate: 88.7% → 99.9%

Time: 7.0s

Alternatives: 8

Speedup: 1.0×

• 49.0% 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: 88.7% 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 87.0%

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

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

Alternative 2: 86.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sinh y \leq -10000000 \lor \neg \left(\sinh y \leq 2 \cdot 10^{-6}\right):\\ \;\;\;\;\sinh y\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin x}{-0.16666666666666666 \cdot \left(x \cdot y\right) + \frac{x}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= (sinh y) -10000000.0) (not (<= (sinh y) 2e-6)))
   (sinh y)
   (/ (sin x) (+ (* -0.16666666666666666 (* x y)) (/ x y)))))

C

double code(double x, double y) {
	double tmp;
	if ((sinh(y) <= -10000000.0) || !(sinh(y) <= 2e-6)) {
		tmp = sinh(y);
	} else {
		tmp = sin(x) / ((-0.16666666666666666 * (x * y)) + (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 ((sinh(y) <= (-10000000.0d0)) .or. (.not. (sinh(y) <= 2d-6))) then
        tmp = sinh(y)
    else
        tmp = sin(x) / (((-0.16666666666666666d0) * (x * y)) + (x / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((Math.sinh(y) <= -10000000.0) || !(Math.sinh(y) <= 2e-6)) {
		tmp = Math.sinh(y);
	} else {
		tmp = Math.sin(x) / ((-0.16666666666666666 * (x * y)) + (x / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (math.sinh(y) <= -10000000.0) or not (math.sinh(y) <= 2e-6):
		tmp = math.sinh(y)
	else:
		tmp = math.sin(x) / ((-0.16666666666666666 * (x * y)) + (x / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((sinh(y) <= -10000000.0) || !(sinh(y) <= 2e-6))
		tmp = sinh(y);
	else
		tmp = Float64(sin(x) / Float64(Float64(-0.16666666666666666 * Float64(x * y)) + Float64(x / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((sinh(y) <= -10000000.0) || ~((sinh(y) <= 2e-6)))
		tmp = sinh(y);
	else
		tmp = sin(x) / ((-0.16666666666666666 * (x * y)) + (x / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[N[Sinh[y], $MachinePrecision], -10000000.0], N[Not[LessEqual[N[Sinh[y], $MachinePrecision], 2e-6]], $MachinePrecision]], N[Sinh[y], $MachinePrecision], N[(N[Sin[x], $MachinePrecision] / N[(N[(-0.16666666666666666 * N[(x * y), $MachinePrecision]), $MachinePrecision] + N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sinh.f64 y) < -1e7 or 1.99999999999999991e-6 < (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 77.5%

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

   if -1e7 < (sinh.f64 y) < 1.99999999999999991e-6

   1. Initial program 73.8%

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

      1. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in y around 0 99.7%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sinh y \leq -10000000 \lor \neg \left(\sinh y \leq 2 \cdot 10^{-6}\right):\\ \;\;\;\;\sinh y\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin x}{-0.16666666666666666 \cdot \left(x \cdot y\right) + \frac{x}{y}}\\ \end{array} \]

Alternative 3: 87.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sinh y \leq -10000000 \lor \neg \left(\sinh y \leq 2 \cdot 10^{-6}\right):\\ \;\;\;\;\sinh y\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \frac{y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= (sinh y) -10000000.0) (not (<= (sinh y) 2e-6)))
   (sinh y)
   (* (sin x) (/ y x))))

C

double code(double x, double y) {
	double tmp;
	if ((sinh(y) <= -10000000.0) || !(sinh(y) <= 2e-6)) {
		tmp = sinh(y);
	} else {
		tmp = sin(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 ((sinh(y) <= (-10000000.0d0)) .or. (.not. (sinh(y) <= 2d-6))) then
        tmp = sinh(y)
    else
        tmp = sin(x) * (y / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((Math.sinh(y) <= -10000000.0) || !(Math.sinh(y) <= 2e-6)) {
		tmp = Math.sinh(y);
	} else {
		tmp = Math.sin(x) * (y / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (math.sinh(y) <= -10000000.0) or not (math.sinh(y) <= 2e-6):
		tmp = math.sinh(y)
	else:
		tmp = math.sin(x) * (y / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((sinh(y) <= -10000000.0) || !(sinh(y) <= 2e-6))
		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) <= -10000000.0) || ~((sinh(y) <= 2e-6)))
		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], -10000000.0], N[Not[LessEqual[N[Sinh[y], $MachinePrecision], 2e-6]], $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) < -1e7 or 1.99999999999999991e-6 < (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 77.5%

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

   if -1e7 < (sinh.f64 y) < 1.99999999999999991e-6

   1. Initial program 73.8%

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

      1. associate-*l/ 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around 0 72.9%

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

      1. associate-/l* 98.9%

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

      2. associate-/r/ 98.9%

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

   6. Simplified 98.9%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sinh y \leq -10000000 \lor \neg \left(\sinh y \leq 2 \cdot 10^{-6}\right):\\ \;\;\;\;\sinh y\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \frac{y}{x}\\ \end{array} \]

Alternative 4: 87.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sinh y \leq -10000000 \lor \neg \left(\sinh y \leq 2 \cdot 10^{-6}\right):\\ \;\;\;\;\sinh y\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin x}{x} \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= (sinh y) -10000000.0) (not (<= (sinh y) 2e-6)))
   (sinh y)
   (* (/ (sin x) x) y)))

C

double code(double x, double y) {
	double tmp;
	if ((sinh(y) <= -10000000.0) || !(sinh(y) <= 2e-6)) {
		tmp = sinh(y);
	} else {
		tmp = (sin(x) / 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 ((sinh(y) <= (-10000000.0d0)) .or. (.not. (sinh(y) <= 2d-6))) then
        tmp = sinh(y)
    else
        tmp = (sin(x) / x) * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((Math.sinh(y) <= -10000000.0) || !(Math.sinh(y) <= 2e-6)) {
		tmp = Math.sinh(y);
	} else {
		tmp = (Math.sin(x) / x) * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (math.sinh(y) <= -10000000.0) or not (math.sinh(y) <= 2e-6):
		tmp = math.sinh(y)
	else:
		tmp = (math.sin(x) / x) * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((sinh(y) <= -10000000.0) || !(sinh(y) <= 2e-6))
		tmp = sinh(y);
	else
		tmp = Float64(Float64(sin(x) / x) * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((sinh(y) <= -10000000.0) || ~((sinh(y) <= 2e-6)))
		tmp = sinh(y);
	else
		tmp = (sin(x) / x) * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[N[Sinh[y], $MachinePrecision], -10000000.0], N[Not[LessEqual[N[Sinh[y], $MachinePrecision], 2e-6]], $MachinePrecision]], N[Sinh[y], $MachinePrecision], N[(N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sinh.f64 y) < -1e7 or 1.99999999999999991e-6 < (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 77.5%

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

   if -1e7 < (sinh.f64 y) < 1.99999999999999991e-6

   1. Initial program 73.8%

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

      1. associate-*l/ 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around 0 72.9%

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

      1. associate-/l* 98.9%

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

   6. Simplified 98.9%

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

      1. associate-/l* 72.9%

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

      2. *-commutative 72.9%

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

      3. associate-*l/ 99.0%

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

   8. Applied egg-rr 99.0%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sinh y \leq -10000000 \lor \neg \left(\sinh y \leq 2 \cdot 10^{-6}\right):\\ \;\;\;\;\sinh y\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin x}{x} \cdot y\\ \end{array} \]

Alternative 5: 74.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sinh y \leq -5 \cdot 10^{-6} \lor \neg \left(\sinh y \leq 5 \cdot 10^{-16}\right):\\ \;\;\;\;\sinh y\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot \left(x \cdot 0.16666666666666666 + \frac{1}{x}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= (sinh y) -5e-6) (not (<= (sinh y) 5e-16)))
   (sinh y)
   (/ y (* x (+ (* x 0.16666666666666666) (/ 1.0 x))))))

C

double code(double x, double y) {
	double tmp;
	if ((sinh(y) <= -5e-6) || !(sinh(y) <= 5e-16)) {
		tmp = sinh(y);
	} else {
		tmp = y / (x * ((x * 0.16666666666666666) + (1.0 / x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((sinh(y) <= (-5d-6)) .or. (.not. (sinh(y) <= 5d-16))) then
        tmp = sinh(y)
    else
        tmp = y / (x * ((x * 0.16666666666666666d0) + (1.0d0 / x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((Math.sinh(y) <= -5e-6) || !(Math.sinh(y) <= 5e-16)) {
		tmp = Math.sinh(y);
	} else {
		tmp = y / (x * ((x * 0.16666666666666666) + (1.0 / x)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (math.sinh(y) <= -5e-6) or not (math.sinh(y) <= 5e-16):
		tmp = math.sinh(y)
	else:
		tmp = y / (x * ((x * 0.16666666666666666) + (1.0 / x)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((sinh(y) <= -5e-6) || !(sinh(y) <= 5e-16))
		tmp = sinh(y);
	else
		tmp = Float64(y / Float64(x * Float64(Float64(x * 0.16666666666666666) + Float64(1.0 / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((sinh(y) <= -5e-6) || ~((sinh(y) <= 5e-16)))
		tmp = sinh(y);
	else
		tmp = y / (x * ((x * 0.16666666666666666) + (1.0 / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[N[Sinh[y], $MachinePrecision], -5e-6], N[Not[LessEqual[N[Sinh[y], $MachinePrecision], 5e-16]], $MachinePrecision]], N[Sinh[y], $MachinePrecision], N[(y / N[(x * N[(N[(x * 0.16666666666666666), $MachinePrecision] + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sinh.f64 y) < -5.00000000000000041e-6 or 5.0000000000000004e-16 < (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 77.1%

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

   if -5.00000000000000041e-6 < (sinh.f64 y) < 5.0000000000000004e-16

   1. Initial program 72.6%

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

      1. associate-*l/ 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around 0 72.6%

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

      1. associate-/l* 99.9%

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

   6. Simplified 99.9%

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

      1. clear-num 99.8%

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

      2. associate-/r/ 99.7%

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

   8. Applied egg-rr 99.7%

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

   9. Taylor expanded in x around 0 75.2%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sinh y \leq -5 \cdot 10^{-6} \lor \neg \left(\sinh y \leq 5 \cdot 10^{-16}\right):\\ \;\;\;\;\sinh y\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot \left(x \cdot 0.16666666666666666 + \frac{1}{x}\right)}\\ \end{array} \]

Alternative 6: 45.2% accurate, 13.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -12200 \lor \neg \left(y \leq 7.5 \cdot 10^{+103}\right):\\ \;\;\;\;\frac{x \cdot y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot \left(x \cdot 0.16666666666666666 + \frac{1}{x}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -12200.0) (not (<= y 7.5e+103)))
   (/ (* x y) x)
   (/ y (* x (+ (* x 0.16666666666666666) (/ 1.0 x))))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -12200.0) || !(y <= 7.5e+103)) {
		tmp = (x * y) / x;
	} else {
		tmp = y / (x * ((x * 0.16666666666666666) + (1.0 / x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-12200.0d0)) .or. (.not. (y <= 7.5d+103))) then
        tmp = (x * y) / x
    else
        tmp = y / (x * ((x * 0.16666666666666666d0) + (1.0d0 / x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -12200.0) || !(y <= 7.5e+103)) {
		tmp = (x * y) / x;
	} else {
		tmp = y / (x * ((x * 0.16666666666666666) + (1.0 / x)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -12200.0) or not (y <= 7.5e+103):
		tmp = (x * y) / x
	else:
		tmp = y / (x * ((x * 0.16666666666666666) + (1.0 / x)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -12200.0) || !(y <= 7.5e+103))
		tmp = Float64(Float64(x * y) / x);
	else
		tmp = Float64(y / Float64(x * Float64(Float64(x * 0.16666666666666666) + Float64(1.0 / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -12200.0) || ~((y <= 7.5e+103)))
		tmp = (x * y) / x;
	else
		tmp = y / (x * ((x * 0.16666666666666666) + (1.0 / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -12200.0], N[Not[LessEqual[y, 7.5e+103]], $MachinePrecision]], N[(N[(x * y), $MachinePrecision] / x), $MachinePrecision], N[(y / N[(x * N[(N[(x * 0.16666666666666666), $MachinePrecision] + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -12200 or 7.49999999999999922e103 < 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 y around 0 4.8%

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

      1. associate-/l* 4.8%

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

      2. associate-/r/ 31.6%

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

   6. Simplified 31.6%

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

      1. *-commutative 31.6%

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

      2. associate-*r/ 4.8%

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

   8. Applied egg-rr 4.8%

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

   9. Taylor expanded in x around 0 20.2%

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

       1. *-commutative 20.2%

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

   11. Simplified 20.2%

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

   if -12200 < y < 7.49999999999999922e103

   1. Initial program 77.6%

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

      1. associate-*l/ 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around 0 63.3%

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

      1. associate-/l* 85.6%

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

   6. Simplified 85.6%

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

      1. clear-num 85.6%

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

      2. associate-/r/ 85.4%

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

   8. Applied egg-rr 85.4%

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

   9. Taylor expanded in x around 0 64.5%

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

4. Final simplification 45.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -12200 \lor \neg \left(y \leq 7.5 \cdot 10^{+103}\right):\\ \;\;\;\;\frac{x \cdot y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot \left(x \cdot 0.16666666666666666 + \frac{1}{x}\right)}\\ \end{array} \]

Alternative 7: 31.5% accurate, 29.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{-25}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x 5e-25) y (/ (* x y) x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 5e-25) {
		tmp = y;
	} 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 <= 5d-25) then
        tmp = y
    else
        tmp = (x * y) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 5e-25) {
		tmp = y;
	} else {
		tmp = (x * y) / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 5e-25:
		tmp = y
	else:
		tmp = (x * y) / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 5e-25)
		tmp = y;
	else
		tmp = Float64(Float64(x * y) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 5e-25)
		tmp = y;
	else
		tmp = (x * y) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 5e-25], y, N[(N[(x * y), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.99999999999999962e-25

   1. Initial program 82.6%

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

      1. associate-*l/ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 33.2%

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

      1. associate-/l* 50.6%

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

      2. associate-/r/ 65.7%

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

   6. Simplified 65.7%

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

   7. Taylor expanded in x around 0 37.7%

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

   if 4.99999999999999962e-25 < x

   1. Initial program 99.9%

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

      1. associate-*l/ 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around 0 54.2%

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

      1. associate-/l* 54.1%

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

      2. associate-/r/ 54.3%

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

   6. Simplified 54.3%

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

      1. *-commutative 54.3%

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

      2. associate-*r/ 54.2%

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

   8. Applied egg-rr 54.2%

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

   9. Taylor expanded in x around 0 22.8%

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

       1. *-commutative 22.8%

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

   11. Simplified 22.8%

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

4. Final simplification 33.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{-25}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{x}\\ \end{array} \]

Alternative 8: 28.5% 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 87.0%

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

   1. associate-*l/ 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in y around 0 38.6%

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

   1. associate-/l* 51.5%

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

   2. associate-/r/ 62.8%

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

6. Simplified 62.8%

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

7. Taylor expanded in x around 0 29.8%

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

8. Final simplification 29.8%

   \[\leadsto y \]

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 2023311 
(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))