Linear.Quaternion:$ccosh from linear-1.19.1.3

Percentage Accurate: 88.8% → 99.9%

Time: 9.1s

Alternatives: 10

Speedup: 1.0×

• 49.1% 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: 88.8% 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 90.1%

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

   1. associate-*l/ 99.9%

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

3. Simplified 99.9%

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

5. Final simplification 99.9%

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

Alternative 2: 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 90.1%

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

   1. *-commutative 90.1%

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

   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. Final simplification 99.9%

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

Alternative 3: 74.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1350:\\ \;\;\;\;\sin x \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1350.0) (* (sin x) (/ y x)) (sinh y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1350.0) {
		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 (y <= 1350.0d0) 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 (y <= 1350.0) {
		tmp = Math.sin(x) * (y / x);
	} else {
		tmp = Math.sinh(y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1350.0:
		tmp = math.sin(x) * (y / x)
	else:
		tmp = math.sinh(y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1350.0)
		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 (y <= 1350.0)
		tmp = sin(x) * (y / x);
	else
		tmp = sinh(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1350.0], N[(N[Sin[x], $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision], N[Sinh[y], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1350

   1. Initial program 87.0%

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

      1. *-commutative 87.0%

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

      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 54.5%

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

      1. associate-/l* 67.4%

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

   7. Simplified 67.4%

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

      1. associate-/r/ 76.4%

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

   9. Applied egg-rr 76.4%

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

   if 1350 < 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 62.3%

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1350:\\ \;\;\;\;\sin x \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 69.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1350:\\ \;\;\;\;\frac{\sin x}{x} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1350.0) (* (/ (sin x) x) y) (sinh y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1350.0) {
		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 (y <= 1350.0d0) 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 (y <= 1350.0) {
		tmp = (Math.sin(x) / x) * y;
	} else {
		tmp = Math.sinh(y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1350.0:
		tmp = (math.sin(x) / x) * y
	else:
		tmp = math.sinh(y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1350.0)
		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 (y <= 1350.0)
		tmp = (sin(x) / x) * y;
	else
		tmp = sinh(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1350.0], N[(N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision] * y), $MachinePrecision], N[Sinh[y], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1350

   1. Initial program 87.0%

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

      1. *-commutative 87.0%

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

      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 54.5%

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

      1. associate-/l* 67.4%

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

   7. Simplified 67.4%

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

      1. clear-num 66.8%

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

      2. associate-/r/ 67.3%

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

      3. clear-num 67.4%

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

   9. Applied egg-rr 67.4%

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

   if 1350 < 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 62.3%

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

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1350:\\ \;\;\;\;\frac{\sin x}{x} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 69.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1350:\\ \;\;\;\;\frac{y}{\frac{x}{\sin x}}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1350.0) (/ y (/ x (sin x))) (sinh y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1350.0) {
		tmp = y / (x / sin(x));
	} else {
		tmp = sinh(y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1350.0d0) then
        tmp = y / (x / sin(x))
    else
        tmp = sinh(y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1350.0) {
		tmp = y / (x / Math.sin(x));
	} else {
		tmp = Math.sinh(y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1350.0:
		tmp = y / (x / math.sin(x))
	else:
		tmp = math.sinh(y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1350.0)
		tmp = Float64(y / Float64(x / sin(x)));
	else
		tmp = sinh(y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1350.0)
		tmp = y / (x / sin(x));
	else
		tmp = sinh(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1350.0], N[(y / N[(x / N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sinh[y], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1350

   1. Initial program 87.0%

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

      1. *-commutative 87.0%

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

      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 54.5%

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

      1. associate-/l* 67.4%

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

   7. Simplified 67.4%

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

   if 1350 < 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 62.3%

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

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1350:\\ \;\;\;\;\frac{y}{\frac{x}{\sin x}}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 58.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.3 \cdot 10^{-8}:\\ \;\;\;\;\frac{y}{1 + x \cdot \left(x \cdot 0.16666666666666666\right)}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 4.3e-8) (/ y (+ 1.0 (* x (* x 0.16666666666666666)))) (sinh y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 4.3e-8) {
		tmp = y / (1.0 + (x * (x * 0.16666666666666666)));
	} else {
		tmp = sinh(y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 4.3d-8) then
        tmp = y / (1.0d0 + (x * (x * 0.16666666666666666d0)))
    else
        tmp = sinh(y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 4.3e-8) {
		tmp = y / (1.0 + (x * (x * 0.16666666666666666)));
	} else {
		tmp = Math.sinh(y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 4.3e-8:
		tmp = y / (1.0 + (x * (x * 0.16666666666666666)))
	else:
		tmp = math.sinh(y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 4.3e-8)
		tmp = Float64(y / Float64(1.0 + Float64(x * Float64(x * 0.16666666666666666))));
	else
		tmp = sinh(y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 4.3e-8)
		tmp = y / (1.0 + (x * (x * 0.16666666666666666)));
	else
		tmp = sinh(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 4.3e-8], N[(y / N[(1.0 + N[(x * N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sinh[y], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.3000000000000001e-8

   1. Initial program 86.9%

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

      1. *-commutative 86.9%

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

      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 54.8%

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

      1. associate-/l* 67.7%

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

   7. Simplified 67.7%

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

      1. clear-num 67.7%

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

      2. associate-/r/ 67.6%

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

   9. Applied egg-rr 67.6%

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

   10. Taylor expanded in x around 0 47.8%

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

       1. *-commutative 47.8%

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

       2. distribute-lft-in 47.8%

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

       3. *-commutative 47.8%

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

       4. div-inv 47.8%

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

       5. *-inverses 47.8%

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

   12. Applied egg-rr 47.8%

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

   if 4.3000000000000001e-8 < 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 61.3%

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

4. Final simplification 51.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.3 \cdot 10^{-8}:\\ \;\;\;\;\frac{y}{1 + x \cdot \left(x \cdot 0.16666666666666666\right)}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 48.6% accurate, 14.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot 0.16666666666666666\right)\\ \mathbf{if}\;y \leq 1.55 \cdot 10^{-5}:\\ \;\;\;\;\frac{y}{1 + t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* x (* x 0.16666666666666666))))
   (if (<= y 1.55e-5) (/ y (+ 1.0 t_0)) (/ y t_0))))

C

double code(double x, double y) {
	double t_0 = x * (x * 0.16666666666666666);
	double tmp;
	if (y <= 1.55e-5) {
		tmp = y / (1.0 + t_0);
	} else {
		tmp = y / t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * (x * 0.16666666666666666d0)
    if (y <= 1.55d-5) then
        tmp = y / (1.0d0 + t_0)
    else
        tmp = y / t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x * (x * 0.16666666666666666);
	double tmp;
	if (y <= 1.55e-5) {
		tmp = y / (1.0 + t_0);
	} else {
		tmp = y / t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x * (x * 0.16666666666666666)
	tmp = 0
	if y <= 1.55e-5:
		tmp = y / (1.0 + t_0)
	else:
		tmp = y / t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x * Float64(x * 0.16666666666666666))
	tmp = 0.0
	if (y <= 1.55e-5)
		tmp = Float64(y / Float64(1.0 + t_0));
	else
		tmp = Float64(y / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x * (x * 0.16666666666666666);
	tmp = 0.0;
	if (y <= 1.55e-5)
		tmp = y / (1.0 + t_0);
	else
		tmp = y / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x * N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 1.55e-5], N[(y / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision], N[(y / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < 1.55000000000000007e-5

   1. Initial program 86.9%

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

      1. *-commutative 86.9%

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

      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 54.8%

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

      1. associate-/l* 67.7%

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

   7. Simplified 67.7%

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

      1. clear-num 67.7%

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

      2. associate-/r/ 67.6%

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

   9. Applied egg-rr 67.6%

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

   10. Taylor expanded in x around 0 47.8%

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

       1. *-commutative 47.8%

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

       2. distribute-lft-in 47.8%

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

       3. *-commutative 47.8%

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

       4. div-inv 47.8%

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

       5. *-inverses 47.8%

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

   12. Applied egg-rr 47.8%

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

   if 1.55000000000000007e-5 < y

   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%

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

      3. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 4.2%

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

      1. associate-/l* 4.2%

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

   7. Simplified 4.2%

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

      1. clear-num 4.2%

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

      2. associate-/r/ 4.2%

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

   9. Applied egg-rr 4.2%

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

   10. Taylor expanded in x around 0 3.1%

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

   11. Taylor expanded in x around inf 30.7%

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

       1. *-commutative 30.7%

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

   13. Simplified 30.7%

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

4. Final simplification 43.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.55 \cdot 10^{-5}:\\ \;\;\;\;\frac{y}{1 + x \cdot \left(x \cdot 0.16666666666666666\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot \left(x \cdot 0.16666666666666666\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 52.9% accurate, 17.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.55 \cdot 10^{-5}:\\ \;\;\;\;x \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot \left(x \cdot 0.16666666666666666\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.55e-5) (* x (/ y x)) (/ y (* x (* x 0.16666666666666666)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.55e-5) {
		tmp = x * (y / x);
	} else {
		tmp = y / (x * (x * 0.16666666666666666));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.55d-5) then
        tmp = x * (y / x)
    else
        tmp = y / (x * (x * 0.16666666666666666d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.55e-5) {
		tmp = x * (y / x);
	} else {
		tmp = y / (x * (x * 0.16666666666666666));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.55e-5:
		tmp = x * (y / x)
	else:
		tmp = y / (x * (x * 0.16666666666666666))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.55e-5)
		tmp = Float64(x * Float64(y / x));
	else
		tmp = Float64(y / Float64(x * Float64(x * 0.16666666666666666)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.55e-5)
		tmp = x * (y / x);
	else
		tmp = y / (x * (x * 0.16666666666666666));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.55e-5], N[(x * N[(y / x), $MachinePrecision]), $MachinePrecision], N[(y / N[(x * N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.55000000000000007e-5

   1. Initial program 86.9%

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

      1. *-commutative 86.9%

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

      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 54.8%

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

   6. Taylor expanded in x around 0 22.9%

      \[\leadsto \frac{\color{blue}{-0.16666666666666666 \cdot \left({x}^{3} \cdot y\right) + x \cdot y}}{x} \]
   7. Step-by-step derivation

      1. associate-*r* 22.9%

         \[\leadsto \frac{\color{blue}{\left(-0.16666666666666666 \cdot {x}^{3}\right) \cdot y} + x \cdot y}{x} \]

      2. distribute-rgt-out 29.1%

         \[\leadsto \frac{\color{blue}{y \cdot \left(-0.16666666666666666 \cdot {x}^{3} + x\right)}}{x} \]

   8. Simplified 29.1%

      \[\leadsto \frac{\color{blue}{y \cdot \left(-0.16666666666666666 \cdot {x}^{3} + x\right)}}{x} \]

   9. Taylor expanded in x around 0 23.9%

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

       1. *-commutative 23.9%

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

   11. Simplified 23.9%

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

       1. associate-/l* 34.8%

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

       2. associate-/r/ 55.4%

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

   13. Applied egg-rr 55.4%

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

   if 1.55000000000000007e-5 < y

   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%

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

      3. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 4.2%

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

      1. associate-/l* 4.2%

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

   7. Simplified 4.2%

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

      1. clear-num 4.2%

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

      2. associate-/r/ 4.2%

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

   9. Applied egg-rr 4.2%

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

   10. Taylor expanded in x around 0 3.1%

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

   11. Taylor expanded in x around inf 30.7%

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

       1. *-commutative 30.7%

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

   13. Simplified 30.7%

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

4. Final simplification 49.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.55 \cdot 10^{-5}:\\ \;\;\;\;x \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot \left(x \cdot 0.16666666666666666\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 50.5% 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 90.1%

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

   1. *-commutative 90.1%

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

   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 42.5%

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

6. Taylor expanded in x around 0 22.2%

   \[\leadsto \frac{\color{blue}{-0.16666666666666666 \cdot \left({x}^{3} \cdot y\right) + x \cdot y}}{x} \]
7. Step-by-step derivation

   1. associate-*r* 22.2%

      \[\leadsto \frac{\color{blue}{\left(-0.16666666666666666 \cdot {x}^{3}\right) \cdot y} + x \cdot y}{x} \]

   2. distribute-rgt-out 30.8%

      \[\leadsto \frac{\color{blue}{y \cdot \left(-0.16666666666666666 \cdot {x}^{3} + x\right)}}{x} \]

8. Simplified 30.8%

   \[\leadsto \frac{\color{blue}{y \cdot \left(-0.16666666666666666 \cdot {x}^{3} + x\right)}}{x} \]

9. Taylor expanded in x around 0 20.7%

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

    1. *-commutative 20.7%

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

11. Simplified 20.7%

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

    1. associate-/l* 27.2%

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

    2. associate-/r/ 46.7%

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

13. Applied egg-rr 46.7%

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

14. Final simplification 46.7%

    \[\leadsto x \cdot \frac{y}{x} \]
15. Add Preprocessing

Alternative 10: 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 90.1%

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

   1. *-commutative 90.1%

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

   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 42.5%

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

   1. associate-/l* 52.3%

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

7. Simplified 52.3%

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

8. Taylor expanded in x around 0 27.2%

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

9. Final simplification 27.2%

   \[\leadsto y \]
10. Add Preprocessing

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 2024041 
(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))