Linear.Quaternion:$ccosh from linear-1.19.1.3

Percentage Accurate: 88.5% → 99.9%

Time: 8.3s

Alternatives: 10

Speedup: 1.0×

• 48.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 88.5% 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 87.8%

   \[\frac{\sin x \cdot \sinh y}{x} \]
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. Add Preprocessing

5. Final simplification 99.9%

   \[\leadsto \frac{\sin x}{x} \cdot \sinh y \]
6. Add Preprocessing

Alternative 2: 75.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sinh y \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\sin x \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sinh y) 5e-9) (* (sin x) (/ y x)) (sinh y)))

C

double code(double x, double y) {
	double tmp;
	if (sinh(y) <= 5e-9) {
		tmp = sin(x) * (y / x);
	} else {
		tmp = sinh(y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (sinh(y) <= 5d-9) then
        tmp = sin(x) * (y / x)
    else
        tmp = sinh(y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (Math.sinh(y) <= 5e-9) {
		tmp = Math.sin(x) * (y / x);
	} else {
		tmp = Math.sinh(y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if math.sinh(y) <= 5e-9:
		tmp = math.sin(x) * (y / x)
	else:
		tmp = math.sinh(y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (sinh(y) <= 5e-9)
		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) <= 5e-9)
		tmp = sin(x) * (y / x);
	else
		tmp = sinh(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[Sinh[y], $MachinePrecision], 5e-9], 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) < 5.0000000000000001e-9

   1. Initial program 83.5%

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

      1. *-commutative 83.5%

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

      2. associate-*l/ 99.8%

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

      3. *-commutative 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. Add Preprocessing

   5. Taylor expanded in y around 0 51.4%

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

      1. associate-/l* 67.8%

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

      2. associate-/r/ 75.0%

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

   7. Simplified 75.0%

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

   if 5.0000000000000001e-9 < (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. Add Preprocessing

   5. Taylor expanded in x around 0 79.1%

      \[\leadsto \color{blue}{1} \cdot \sinh y \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sinh y \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\sin x \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 69.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sinh y \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{\sin x}{x} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sinh y) 5e-9) (* (/ (sin x) x) y) (sinh y)))

C

double code(double x, double y) {
	double tmp;
	if (sinh(y) <= 5e-9) {
		tmp = (sin(x) / x) * y;
	} else {
		tmp = sinh(y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (sinh(y) <= 5d-9) then
        tmp = (sin(x) / x) * y
    else
        tmp = sinh(y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (Math.sinh(y) <= 5e-9) {
		tmp = (Math.sin(x) / x) * y;
	} else {
		tmp = Math.sinh(y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if math.sinh(y) <= 5e-9:
		tmp = (math.sin(x) / x) * y
	else:
		tmp = math.sinh(y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (sinh(y) <= 5e-9)
		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) <= 5e-9)
		tmp = (sin(x) / x) * y;
	else
		tmp = sinh(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[Sinh[y], $MachinePrecision], 5e-9], 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) < 5.0000000000000001e-9

   1. Initial program 83.5%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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. Add Preprocessing

   5. Taylor expanded in y around 0 67.8%

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

   if 5.0000000000000001e-9 < (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. Add Preprocessing

   5. Taylor expanded in x around 0 79.1%

      \[\leadsto \color{blue}{1} \cdot \sinh y \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sinh y \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{\sin x}{x} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 69.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sinh y \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{y}{\frac{x}{\sin x}}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sinh y) 5e-9) (/ y (/ x (sin x))) (sinh y)))

C

double code(double x, double y) {
	double tmp;
	if (sinh(y) <= 5e-9) {
		tmp = y / (x / sin(x));
	} else {
		tmp = sinh(y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (sinh(y) <= 5d-9) then
        tmp = y / (x / sin(x))
    else
        tmp = sinh(y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (Math.sinh(y) <= 5e-9) {
		tmp = y / (x / Math.sin(x));
	} else {
		tmp = Math.sinh(y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if math.sinh(y) <= 5e-9:
		tmp = y / (x / math.sin(x))
	else:
		tmp = math.sinh(y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (sinh(y) <= 5e-9)
		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) <= 5e-9)
		tmp = y / (x / sin(x));
	else
		tmp = sinh(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[Sinh[y], $MachinePrecision], 5e-9], 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) < 5.0000000000000001e-9

   1. Initial program 83.5%

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

      1. *-commutative 83.5%

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

      2. associate-*l/ 99.8%

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

      3. *-commutative 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. Add Preprocessing

   5. Taylor expanded in y around 0 51.4%

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

      1. associate-/l* 67.8%

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

   7. Simplified 67.8%

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

   if 5.0000000000000001e-9 < (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. Add Preprocessing

   5. Taylor expanded in x around 0 79.1%

      \[\leadsto \color{blue}{1} \cdot \sinh y \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sinh y \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{y}{\frac{x}{\sin x}}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 62.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sinh y \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{\frac{y}{x}}{x \cdot 0.16666666666666666 + \frac{1}{x}}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sinh y) 5e-9)
   (/ (/ y x) (+ (* x 0.16666666666666666) (/ 1.0 x)))
   (sinh y)))

C

double code(double x, double y) {
	double tmp;
	if (sinh(y) <= 5e-9) {
		tmp = (y / x) / ((x * 0.16666666666666666) + (1.0 / x));
	} else {
		tmp = sinh(y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (sinh(y) <= 5d-9) then
        tmp = (y / x) / ((x * 0.16666666666666666d0) + (1.0d0 / x))
    else
        tmp = sinh(y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (Math.sinh(y) <= 5e-9) {
		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) <= 5e-9:
		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) <= 5e-9)
		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) <= 5e-9)
		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], 5e-9], 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) < 5.0000000000000001e-9

   1. Initial program 83.5%

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

      1. *-commutative 83.5%

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

      2. associate-*l/ 99.8%

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

      3. *-commutative 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. Add Preprocessing

   5. Taylor expanded in y around 0 51.4%

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

      1. associate-/l* 67.8%

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

      2. associate-/r/ 75.0%

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

   7. Simplified 75.0%

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

      1. clear-num 74.4%

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

      2. associate-/r/ 73.9%

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

      3. div-inv 73.8%

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

      4. associate-/r* 74.3%

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

      5. clear-num 74.9%

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

   9. Applied egg-rr 74.9%

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

   10. Taylor expanded in x around 0 56.1%

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

   if 5.0000000000000001e-9 < (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. Add Preprocessing

   5. Taylor expanded in x around 0 79.1%

      \[\leadsto \color{blue}{1} \cdot \sinh y \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sinh y \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{\frac{y}{x}}{x \cdot 0.16666666666666666 + \frac{1}{x}}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.8%

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

   1. *-commutative 87.8%

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

   2. associate-*l/ 99.9%

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

   3. *-commutative 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. Add Preprocessing

5. Final simplification 99.9%

   \[\leadsto \sin x \cdot \frac{\sinh y}{x} \]
6. Add Preprocessing

Alternative 7: 50.3% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.5 \cdot 10^{+38} \lor \neg \left(x \leq 1.2 \cdot 10^{+168}\right):\\ \;\;\;\;\frac{\frac{y}{x}}{x \cdot 0.16666666666666666 + \frac{1}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x 4.5e+38) (not (<= x 1.2e+168)))
   (/ (/ y x) (+ (* x 0.16666666666666666) (/ 1.0 x)))
   (/ (* x y) x)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= 4.5e+38) || !(x <= 1.2e+168)) {
		tmp = (y / x) / ((x * 0.16666666666666666) + (1.0 / 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 <= 4.5d+38) .or. (.not. (x <= 1.2d+168))) then
        tmp = (y / x) / ((x * 0.16666666666666666d0) + (1.0d0 / 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 <= 4.5e+38) || !(x <= 1.2e+168)) {
		tmp = (y / x) / ((x * 0.16666666666666666) + (1.0 / x));
	} else {
		tmp = (x * y) / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= 4.5e+38) or not (x <= 1.2e+168):
		tmp = (y / x) / ((x * 0.16666666666666666) + (1.0 / x))
	else:
		tmp = (x * y) / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= 4.5e+38) || !(x <= 1.2e+168))
		tmp = Float64(Float64(y / x) / Float64(Float64(x * 0.16666666666666666) + Float64(1.0 / x)));
	else
		tmp = Float64(Float64(x * y) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= 4.5e+38) || ~((x <= 1.2e+168)))
		tmp = (y / x) / ((x * 0.16666666666666666) + (1.0 / x));
	else
		tmp = (x * y) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, 4.5e+38], N[Not[LessEqual[x, 1.2e+168]], $MachinePrecision]], N[(N[(y / x), $MachinePrecision] / N[(N[(x * 0.16666666666666666), $MachinePrecision] + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * y), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.4999999999999998e38 or 1.20000000000000005e168 < x

   1. Initial program 86.1%

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

      1. *-commutative 86.1%

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

      2. associate-*l/ 99.9%

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

      3. *-commutative 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. Add Preprocessing

   5. Taylor expanded in y around 0 40.2%

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

      1. associate-/l* 54.0%

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

      2. associate-/r/ 67.8%

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

   7. Simplified 67.8%

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

      1. clear-num 67.3%

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

      2. associate-/r/ 65.6%

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

      3. div-inv 65.6%

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

      4. associate-/r* 67.2%

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

      5. clear-num 67.7%

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

   9. Applied egg-rr 67.7%

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

   10. Taylor expanded in x around 0 55.5%

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

   if 4.4999999999999998e38 < x < 1.20000000000000005e168

   1. Initial program 99.8%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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. Add Preprocessing

   5. Taylor expanded in y around 0 32.9%

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

      1. associate-*l/ 32.8%

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

   7. Applied egg-rr 32.8%

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

   8. Taylor expanded in x around 0 21.8%

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

      1. *-commutative 21.8%

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

   10. Simplified 21.8%

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

4. Final simplification 51.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.5 \cdot 10^{+38} \lor \neg \left(x \leq 1.2 \cdot 10^{+168}\right):\\ \;\;\;\;\frac{\frac{y}{x}}{x \cdot 0.16666666666666666 + \frac{1}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 49.7% accurate, 12.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{-5} \lor \neg \left(x \leq 1.15 \cdot 10^{+168}\right):\\ \;\;\;\;\frac{\frac{y}{x}}{\frac{1}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x 4e-5) (not (<= x 1.15e+168)))
   (/ (/ y x) (/ 1.0 x))
   (/ (* x y) x)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= 4e-5) || !(x <= 1.15e+168)) {
		tmp = (y / x) / (1.0 / 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 <= 4d-5) .or. (.not. (x <= 1.15d+168))) then
        tmp = (y / x) / (1.0d0 / 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 <= 4e-5) || !(x <= 1.15e+168)) {
		tmp = (y / x) / (1.0 / x);
	} else {
		tmp = (x * y) / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= 4e-5) or not (x <= 1.15e+168):
		tmp = (y / x) / (1.0 / x)
	else:
		tmp = (x * y) / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= 4e-5) || !(x <= 1.15e+168))
		tmp = Float64(Float64(y / x) / Float64(1.0 / x));
	else
		tmp = Float64(Float64(x * y) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= 4e-5) || ~((x <= 1.15e+168)))
		tmp = (y / x) / (1.0 / x);
	else
		tmp = (x * y) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, 4e-5], N[Not[LessEqual[x, 1.15e+168]], $MachinePrecision]], N[(N[(y / x), $MachinePrecision] / N[(1.0 / x), $MachinePrecision]), $MachinePrecision], N[(N[(x * y), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.00000000000000033e-5 or 1.15e168 < x

   1. Initial program 85.3%

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

      1. *-commutative 85.3%

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

      2. associate-*l/ 99.9%

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

      3. *-commutative 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. Add Preprocessing

   5. Taylor expanded in y around 0 38.6%

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

      1. associate-/l* 53.3%

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

      2. associate-/r/ 67.8%

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

   7. Simplified 67.8%

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

      1. clear-num 67.3%

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

      2. associate-/r/ 65.5%

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

      3. div-inv 65.5%

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

      4. associate-/r* 67.2%

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

      5. clear-num 67.7%

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

   9. Applied egg-rr 67.7%

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

   10. Taylor expanded in x around 0 57.7%

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

   if 4.00000000000000033e-5 < x < 1.15e168

   1. Initial program 99.7%

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

      1. associate-*l/ 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot \sinh y} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 42.1%

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

      1. associate-*l/ 42.0%

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

   7. Applied egg-rr 42.0%

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

   8. Taylor expanded in x around 0 17.7%

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

      1. *-commutative 17.7%

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

   10. Simplified 17.7%

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

4. Final simplification 50.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{-5} \lor \neg \left(x \leq 1.15 \cdot 10^{+168}\right):\\ \;\;\;\;\frac{\frac{y}{x}}{\frac{1}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 30.6% accurate, 20.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4 \cdot 10^{+110}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 4e+110) y (/ (* x y) x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 4e+110) {
		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 (y <= 4d+110) 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 (y <= 4e+110) {
		tmp = y;
	} else {
		tmp = (x * y) / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 4e+110:
		tmp = y
	else:
		tmp = (x * y) / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 4e+110)
		tmp = y;
	else
		tmp = Float64(Float64(x * y) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 4e+110)
		tmp = y;
	else
		tmp = (x * y) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 4e+110], y, N[(N[(x * y), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.0000000000000001e110

   1. Initial program 85.6%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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. Add Preprocessing

   5. Taylor expanded in y around 0 59.9%

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

   6. Taylor expanded in x around 0 35.5%

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

   if 4.0000000000000001e110 < 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. Add Preprocessing

   5. Taylor expanded in y around 0 4.9%

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

      1. associate-*l/ 4.9%

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

   7. Applied egg-rr 4.9%

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

   8. Taylor expanded in x around 0 18.7%

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

      1. *-commutative 18.7%

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

   10. Simplified 18.7%

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

4. Final simplification 32.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4 \cdot 10^{+110}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 28.2% 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 87.8%

   \[\frac{\sin x \cdot \sinh y}{x} \]
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. Add Preprocessing

5. Taylor expanded in y around 0 51.3%

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

6. Taylor expanded in x around 0 30.7%

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

7. Final simplification 30.7%

   \[\leadsto y \]
8. Add Preprocessing

Developer target: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024021 
(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))