Linear.Quaternion:$ccosh from linear-1.19.1.3

Percentage Accurate: 89.3% → 99.9%

Time: 9.3s

Alternatives: 12

Speedup: 1.0×

• 48.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 89.3% 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 88.2%

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

   1. associate-*l/ 99.9%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

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

Alternative 2: 86.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sinh y \leq -4 \cdot 10^{-8}:\\ \;\;\;\;\left(1 + \left(x \cdot x\right) \cdot -0.16666666666666666\right) \cdot \sinh y\\ \mathbf{elif}\;\sinh y \leq 2 \cdot 10^{-21}:\\ \;\;\;\;\frac{y}{\frac{x}{\sin x}}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sinh y) -4e-8)
   (* (+ 1.0 (* (* x x) -0.16666666666666666)) (sinh y))
   (if (<= (sinh y) 2e-21) (/ y (/ x (sin x))) (sinh y))))

C

double code(double x, double y) {
	double tmp;
	if (sinh(y) <= -4e-8) {
		tmp = (1.0 + ((x * x) * -0.16666666666666666)) * sinh(y);
	} else if (sinh(y) <= 2e-21) {
		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) <= (-4d-8)) then
        tmp = (1.0d0 + ((x * x) * (-0.16666666666666666d0))) * sinh(y)
    else if (sinh(y) <= 2d-21) 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) <= -4e-8) {
		tmp = (1.0 + ((x * x) * -0.16666666666666666)) * Math.sinh(y);
	} else if (Math.sinh(y) <= 2e-21) {
		tmp = y / (x / Math.sin(x));
	} else {
		tmp = Math.sinh(y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if math.sinh(y) <= -4e-8:
		tmp = (1.0 + ((x * x) * -0.16666666666666666)) * math.sinh(y)
	elif math.sinh(y) <= 2e-21:
		tmp = y / (x / math.sin(x))
	else:
		tmp = math.sinh(y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (sinh(y) <= -4e-8)
		tmp = Float64(Float64(1.0 + Float64(Float64(x * x) * -0.16666666666666666)) * sinh(y));
	elseif (sinh(y) <= 2e-21)
		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) <= -4e-8)
		tmp = (1.0 + ((x * x) * -0.16666666666666666)) * sinh(y);
	elseif (sinh(y) <= 2e-21)
		tmp = y / (x / sin(x));
	else
		tmp = sinh(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[Sinh[y], $MachinePrecision], -4e-8], N[(N[(1.0 + N[(N[(x * x), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] * N[Sinh[y], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sinh[y], $MachinePrecision], 2e-21], N[(y / N[(x / N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sinh[y], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sinh.f64 y) < -4.0000000000000001e-8

   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 83.1%

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

      1. *-commutative 83.1%

         \[\leadsto \left(1 + \color{blue}{{x}^{2} \cdot -0.16666666666666666}\right) \cdot \sinh y \]

      2. unpow2 83.1%

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

   6. Simplified 83.1%

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

   if -4.0000000000000001e-8 < (sinh.f64 y) < 1.99999999999999982e-21

   1. Initial program 77.5%

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

      1. associate-*r/ 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in y around 0 77.5%

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

      1. associate-/l* 100.0%

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

   6. Simplified 100.0%

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

   if 1.99999999999999982e-21 < (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 76.2%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sinh y \leq -4 \cdot 10^{-8}:\\ \;\;\;\;\left(1 + \left(x \cdot x\right) \cdot -0.16666666666666666\right) \cdot \sinh y\\ \mathbf{elif}\;\sinh y \leq 2 \cdot 10^{-21}:\\ \;\;\;\;\frac{y}{\frac{x}{\sin x}}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \]

Alternative 3: 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 88.2%

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

   1. associate-*r/ 99.9%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

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

Alternative 4: 84.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.9 \cdot 10^{+140}:\\ \;\;\;\;x \cdot \left(y \cdot \left(x \cdot -0.16666666666666666\right) - \frac{y}{-x}\right)\\ \mathbf{elif}\;y \leq -4 \cdot 10^{-8}:\\ \;\;\;\;\sinh y\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{-15}:\\ \;\;\;\;\sin x \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.9e+140)
   (* x (- (* y (* x -0.16666666666666666)) (/ y (- x))))
   (if (<= y -4e-8)
     (sinh y)
     (if (<= y 5.5e-15) (* (sin x) (/ y x)) (sinh y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.9e+140) {
		tmp = x * ((y * (x * -0.16666666666666666)) - (y / -x));
	} else if (y <= -4e-8) {
		tmp = sinh(y);
	} else if (y <= 5.5e-15) {
		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 <= (-1.9d+140)) then
        tmp = x * ((y * (x * (-0.16666666666666666d0))) - (y / -x))
    else if (y <= (-4d-8)) then
        tmp = sinh(y)
    else if (y <= 5.5d-15) 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 <= -1.9e+140) {
		tmp = x * ((y * (x * -0.16666666666666666)) - (y / -x));
	} else if (y <= -4e-8) {
		tmp = Math.sinh(y);
	} else if (y <= 5.5e-15) {
		tmp = Math.sin(x) * (y / x);
	} else {
		tmp = Math.sinh(y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.9e+140:
		tmp = x * ((y * (x * -0.16666666666666666)) - (y / -x))
	elif y <= -4e-8:
		tmp = math.sinh(y)
	elif y <= 5.5e-15:
		tmp = math.sin(x) * (y / x)
	else:
		tmp = math.sinh(y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.9e+140)
		tmp = Float64(x * Float64(Float64(y * Float64(x * -0.16666666666666666)) - Float64(y / Float64(-x))));
	elseif (y <= -4e-8)
		tmp = sinh(y);
	elseif (y <= 5.5e-15)
		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 <= -1.9e+140)
		tmp = x * ((y * (x * -0.16666666666666666)) - (y / -x));
	elseif (y <= -4e-8)
		tmp = sinh(y);
	elseif (y <= 5.5e-15)
		tmp = sin(x) * (y / x);
	else
		tmp = sinh(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.9e+140], N[(x * N[(N[(y * N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] - N[(y / (-x)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -4e-8], N[Sinh[y], $MachinePrecision], If[LessEqual[y, 5.5e-15], N[(N[Sin[x], $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision], N[Sinh[y], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.9e140

   1. Initial program 100.0%

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

      1. associate-*r/ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 5.3%

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

      1. associate-/l* 5.3%

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

      2. associate-/r/ 31.4%

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

   6. Simplified 31.4%

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

   7. Taylor expanded in x around 0 49.6%

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

      1. unpow2 49.6%

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

   9. Simplified 49.6%

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

       1. *-un-lft-identity 49.6%

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

       2. *-commutative 49.6%

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

       3. *-inverses 49.6%

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

       4. div-inv 49.6%

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

       5. associate-*l* 13.9%

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

       6. div-inv 13.9%

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

       7. frac-2neg 13.9%

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

       8. div-inv 13.9%

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

       9. fma-def 13.9%

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

       10. *-commutative 13.9%

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

       11. distribute-rgt-neg-in 13.9%

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

       12. associate-*r* 13.9%

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

       13. *-commutative 13.9%

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

   11. Applied egg-rr 13.9%

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

       1. fma-udef 13.9%

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

       2. associate-*l* 75.6%

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

       3. associate-*l* 75.6%

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

       4. *-commutative 75.6%

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

       5. distribute-lft-out 75.6%

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

       6. associate-*r/ 75.6%

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

       7. distribute-lft-neg-in 75.6%

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

       8. *-rgt-identity 75.6%

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

       9. *-commutative 75.6%

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

       10. associate-*l* 75.6%

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

       11. *-commutative 75.6%

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

   13. Simplified 75.6%

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

   if -1.9e140 < y < -4.0000000000000001e-8 or 5.5000000000000002e-15 < 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 78.7%

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

   if -4.0000000000000001e-8 < y < 5.5000000000000002e-15

   1. Initial program 77.5%

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

      1. associate-*r/ 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in y around 0 77.5%

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

      1. associate-/l* 100.0%

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

      2. associate-/r/ 99.8%

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

   6. Simplified 99.8%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.9 \cdot 10^{+140}:\\ \;\;\;\;x \cdot \left(y \cdot \left(x \cdot -0.16666666666666666\right) - \frac{y}{-x}\right)\\ \mathbf{elif}\;y \leq -4 \cdot 10^{-8}:\\ \;\;\;\;\sinh y\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{-15}:\\ \;\;\;\;\sin x \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \]

Alternative 5: 84.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.9 \cdot 10^{+140}:\\ \;\;\;\;x \cdot \left(y \cdot \left(x \cdot -0.16666666666666666\right) - \frac{y}{-x}\right)\\ \mathbf{elif}\;y \leq -4 \cdot 10^{-8}:\\ \;\;\;\;\sinh y\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{-15}:\\ \;\;\;\;\frac{y}{\frac{x}{\sin x}}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.9e+140)
   (* x (- (* y (* x -0.16666666666666666)) (/ y (- x))))
   (if (<= y -4e-8)
     (sinh y)
     (if (<= y 5.5e-15) (/ y (/ x (sin x))) (sinh y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.9e+140) {
		tmp = x * ((y * (x * -0.16666666666666666)) - (y / -x));
	} else if (y <= -4e-8) {
		tmp = sinh(y);
	} else if (y <= 5.5e-15) {
		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 <= (-1.9d+140)) then
        tmp = x * ((y * (x * (-0.16666666666666666d0))) - (y / -x))
    else if (y <= (-4d-8)) then
        tmp = sinh(y)
    else if (y <= 5.5d-15) 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 <= -1.9e+140) {
		tmp = x * ((y * (x * -0.16666666666666666)) - (y / -x));
	} else if (y <= -4e-8) {
		tmp = Math.sinh(y);
	} else if (y <= 5.5e-15) {
		tmp = y / (x / Math.sin(x));
	} else {
		tmp = Math.sinh(y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.9e+140:
		tmp = x * ((y * (x * -0.16666666666666666)) - (y / -x))
	elif y <= -4e-8:
		tmp = math.sinh(y)
	elif y <= 5.5e-15:
		tmp = y / (x / math.sin(x))
	else:
		tmp = math.sinh(y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.9e+140)
		tmp = Float64(x * Float64(Float64(y * Float64(x * -0.16666666666666666)) - Float64(y / Float64(-x))));
	elseif (y <= -4e-8)
		tmp = sinh(y);
	elseif (y <= 5.5e-15)
		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 <= -1.9e+140)
		tmp = x * ((y * (x * -0.16666666666666666)) - (y / -x));
	elseif (y <= -4e-8)
		tmp = sinh(y);
	elseif (y <= 5.5e-15)
		tmp = y / (x / sin(x));
	else
		tmp = sinh(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.9e+140], N[(x * N[(N[(y * N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] - N[(y / (-x)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -4e-8], N[Sinh[y], $MachinePrecision], If[LessEqual[y, 5.5e-15], N[(y / N[(x / N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sinh[y], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.9e140

   1. Initial program 100.0%

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

      1. associate-*r/ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 5.3%

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

      1. associate-/l* 5.3%

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

      2. associate-/r/ 31.4%

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

   6. Simplified 31.4%

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

   7. Taylor expanded in x around 0 49.6%

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

      1. unpow2 49.6%

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

   9. Simplified 49.6%

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

       1. *-un-lft-identity 49.6%

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

       2. *-commutative 49.6%

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

       3. *-inverses 49.6%

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

       4. div-inv 49.6%

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

       5. associate-*l* 13.9%

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

       6. div-inv 13.9%

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

       7. frac-2neg 13.9%

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

       8. div-inv 13.9%

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

       9. fma-def 13.9%

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

       10. *-commutative 13.9%

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

       11. distribute-rgt-neg-in 13.9%

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

       12. associate-*r* 13.9%

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

       13. *-commutative 13.9%

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

   11. Applied egg-rr 13.9%

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

       1. fma-udef 13.9%

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

       2. associate-*l* 75.6%

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

       3. associate-*l* 75.6%

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

       4. *-commutative 75.6%

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

       5. distribute-lft-out 75.6%

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

       6. associate-*r/ 75.6%

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

       7. distribute-lft-neg-in 75.6%

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

       8. *-rgt-identity 75.6%

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

       9. *-commutative 75.6%

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

       10. associate-*l* 75.6%

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

       11. *-commutative 75.6%

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

   13. Simplified 75.6%

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

   if -1.9e140 < y < -4.0000000000000001e-8 or 5.5000000000000002e-15 < 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 78.7%

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

   if -4.0000000000000001e-8 < y < 5.5000000000000002e-15

   1. Initial program 77.5%

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

      1. associate-*r/ 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in y around 0 77.5%

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

      1. associate-/l* 100.0%

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

   6. Simplified 100.0%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.9 \cdot 10^{+140}:\\ \;\;\;\;x \cdot \left(y \cdot \left(x \cdot -0.16666666666666666\right) - \frac{y}{-x}\right)\\ \mathbf{elif}\;y \leq -4 \cdot 10^{-8}:\\ \;\;\;\;\sinh y\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{-15}:\\ \;\;\;\;\frac{y}{\frac{x}{\sin x}}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \]

Alternative 6: 71.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{+140}:\\ \;\;\;\;x \cdot \left(y \cdot \left(x \cdot -0.16666666666666666\right) - \frac{y}{-x}\right)\\ \mathbf{elif}\;y \leq -4 \cdot 10^{-8}:\\ \;\;\;\;\sinh y\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{-72}:\\ \;\;\;\;\frac{x}{\frac{x}{y}}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1e+140)
   (* x (- (* y (* x -0.16666666666666666)) (/ y (- x))))
   (if (<= y -4e-8) (sinh y) (if (<= y 2.1e-72) (/ x (/ x y)) (sinh y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1e+140) {
		tmp = x * ((y * (x * -0.16666666666666666)) - (y / -x));
	} else if (y <= -4e-8) {
		tmp = sinh(y);
	} else if (y <= 2.1e-72) {
		tmp = 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 <= (-1d+140)) then
        tmp = x * ((y * (x * (-0.16666666666666666d0))) - (y / -x))
    else if (y <= (-4d-8)) then
        tmp = sinh(y)
    else if (y <= 2.1d-72) then
        tmp = 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 <= -1e+140) {
		tmp = x * ((y * (x * -0.16666666666666666)) - (y / -x));
	} else if (y <= -4e-8) {
		tmp = Math.sinh(y);
	} else if (y <= 2.1e-72) {
		tmp = x / (x / y);
	} else {
		tmp = Math.sinh(y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1e+140:
		tmp = x * ((y * (x * -0.16666666666666666)) - (y / -x))
	elif y <= -4e-8:
		tmp = math.sinh(y)
	elif y <= 2.1e-72:
		tmp = x / (x / y)
	else:
		tmp = math.sinh(y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1e+140)
		tmp = Float64(x * Float64(Float64(y * Float64(x * -0.16666666666666666)) - Float64(y / Float64(-x))));
	elseif (y <= -4e-8)
		tmp = sinh(y);
	elseif (y <= 2.1e-72)
		tmp = Float64(x / Float64(x / y));
	else
		tmp = sinh(y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1e+140)
		tmp = x * ((y * (x * -0.16666666666666666)) - (y / -x));
	elseif (y <= -4e-8)
		tmp = sinh(y);
	elseif (y <= 2.1e-72)
		tmp = x / (x / y);
	else
		tmp = sinh(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1e+140], N[(x * N[(N[(y * N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] - N[(y / (-x)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -4e-8], N[Sinh[y], $MachinePrecision], If[LessEqual[y, 2.1e-72], N[(x / N[(x / y), $MachinePrecision]), $MachinePrecision], N[Sinh[y], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.00000000000000006e140

   1. Initial program 100.0%

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

      1. associate-*r/ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 5.3%

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

      1. associate-/l* 5.3%

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

      2. associate-/r/ 31.4%

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

   6. Simplified 31.4%

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

   7. Taylor expanded in x around 0 49.6%

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

      1. unpow2 49.6%

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

   9. Simplified 49.6%

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

       1. *-un-lft-identity 49.6%

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

       2. *-commutative 49.6%

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

       3. *-inverses 49.6%

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

       4. div-inv 49.6%

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

       5. associate-*l* 13.9%

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

       6. div-inv 13.9%

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

       7. frac-2neg 13.9%

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

       8. div-inv 13.9%

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

       9. fma-def 13.9%

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

       10. *-commutative 13.9%

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

       11. distribute-rgt-neg-in 13.9%

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

       12. associate-*r* 13.9%

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

       13. *-commutative 13.9%

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

   11. Applied egg-rr 13.9%

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

       1. fma-udef 13.9%

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

       2. associate-*l* 75.6%

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

       3. associate-*l* 75.6%

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

       4. *-commutative 75.6%

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

       5. distribute-lft-out 75.6%

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

       6. associate-*r/ 75.6%

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

       7. distribute-lft-neg-in 75.6%

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

       8. *-rgt-identity 75.6%

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

       9. *-commutative 75.6%

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

       10. associate-*l* 75.6%

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

       11. *-commutative 75.6%

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

   13. Simplified 75.6%

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

   if -1.00000000000000006e140 < y < -4.0000000000000001e-8 or 2.1e-72 < y

   1. Initial program 99.0%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 75.7%

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

   if -4.0000000000000001e-8 < y < 2.1e-72

   1. Initial program 76.1%

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

      1. associate-*r/ 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in y around 0 76.1%

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

   5. Taylor expanded in x around 0 29.9%

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

      1. associate-/l* 53.7%

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

      2. associate-/r/ 76.5%

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

   7. Applied egg-rr 76.5%

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

      1. *-commutative 76.5%

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

      2. clear-num 79.2%

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

      3. un-div-inv 79.3%

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

   9. Applied egg-rr 79.3%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{+140}:\\ \;\;\;\;x \cdot \left(y \cdot \left(x \cdot -0.16666666666666666\right) - \frac{y}{-x}\right)\\ \mathbf{elif}\;y \leq -4 \cdot 10^{-8}:\\ \;\;\;\;\sinh y\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{-72}:\\ \;\;\;\;\frac{x}{\frac{x}{y}}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \]

Alternative 7: 58.7% accurate, 12.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.3 \cdot 10^{-10} \lor \neg \left(y \leq 1.8 \cdot 10^{+24}\right):\\ \;\;\;\;x \cdot \left(y \cdot \left(x \cdot -0.16666666666666666\right) - \frac{y}{-x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{x}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.3e-10) (not (<= y 1.8e+24)))
   (* x (- (* y (* x -0.16666666666666666)) (/ y (- x))))
   (/ x (/ x y))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.3e-10) || !(y <= 1.8e+24)) {
		tmp = x * ((y * (x * -0.16666666666666666)) - (y / -x));
	} else {
		tmp = 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 ((y <= (-1.3d-10)) .or. (.not. (y <= 1.8d+24))) then
        tmp = x * ((y * (x * (-0.16666666666666666d0))) - (y / -x))
    else
        tmp = x / (x / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.3e-10) || !(y <= 1.8e+24)) {
		tmp = x * ((y * (x * -0.16666666666666666)) - (y / -x));
	} else {
		tmp = x / (x / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.3e-10) or not (y <= 1.8e+24):
		tmp = x * ((y * (x * -0.16666666666666666)) - (y / -x))
	else:
		tmp = x / (x / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.3e-10) || !(y <= 1.8e+24))
		tmp = Float64(x * Float64(Float64(y * Float64(x * -0.16666666666666666)) - Float64(y / Float64(-x))));
	else
		tmp = Float64(x / Float64(x / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.3e-10) || ~((y <= 1.8e+24)))
		tmp = x * ((y * (x * -0.16666666666666666)) - (y / -x));
	else
		tmp = x / (x / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.3e-10], N[Not[LessEqual[y, 1.8e+24]], $MachinePrecision]], N[(x * N[(N[(y * N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] - N[(y / (-x)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(x / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.29999999999999991e-10 or 1.79999999999999992e24 < y

   1. Initial program 100.0%

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

      1. associate-*r/ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 6.2%

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

      1. associate-/l* 6.2%

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

      2. associate-/r/ 27.9%

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

   6. Simplified 27.9%

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

   7. Taylor expanded in x around 0 29.0%

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

      1. unpow2 29.0%

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

   9. Simplified 29.0%

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

       1. *-un-lft-identity 29.0%

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

       2. *-commutative 29.0%

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

       3. *-inverses 29.0%

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

       4. div-inv 29.0%

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

       5. associate-*l* 11.6%

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

       6. div-inv 11.6%

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

       7. frac-2neg 11.6%

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

       8. div-inv 11.6%

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

       9. fma-def 11.6%

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

       10. *-commutative 11.6%

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

       11. distribute-rgt-neg-in 11.6%

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

       12. associate-*r* 11.6%

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

       13. *-commutative 11.6%

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

   11. Applied egg-rr 11.6%

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

       1. fma-udef 11.6%

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

       2. associate-*l* 50.8%

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

       3. associate-*l* 50.8%

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

       4. *-commutative 50.8%

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

       5. distribute-lft-out 50.8%

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

       6. associate-*r/ 50.8%

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

       7. distribute-lft-neg-in 50.8%

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

       8. *-rgt-identity 50.8%

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

       9. *-commutative 50.8%

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

       10. associate-*l* 50.8%

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

       11. *-commutative 50.8%

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

   13. Simplified 50.8%

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

   if -1.29999999999999991e-10 < y < 1.79999999999999992e24

   1. Initial program 78.6%

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

      1. associate-*r/ 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in y around 0 75.4%

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

   5. Taylor expanded in x around 0 32.1%

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

      1. associate-/l* 53.4%

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

      2. associate-/r/ 73.1%

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

   7. Applied egg-rr 73.1%

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

      1. *-commutative 73.1%

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

      2. clear-num 75.4%

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

      3. un-div-inv 75.5%

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

   9. Applied egg-rr 75.5%

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

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.3 \cdot 10^{-10} \lor \neg \left(y \leq 1.8 \cdot 10^{+24}\right):\\ \;\;\;\;x \cdot \left(y \cdot \left(x \cdot -0.16666666666666666\right) - \frac{y}{-x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{x}{y}}\\ \end{array} \]

Alternative 8: 47.4% accurate, 20.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.3 \cdot 10^{+290}:\\ \;\;\;\;x \cdot \frac{y}{x}\\ \mathbf{elif}\;y \leq -1.25 \cdot 10^{+91}:\\ \;\;\;\;\frac{x \cdot \left(-y\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{x}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.3e+290)
   (* x (/ y x))
   (if (<= y -1.25e+91) (/ (* x (- y)) x) (/ x (/ x y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.3e+290) {
		tmp = x * (y / x);
	} else if (y <= -1.25e+91) {
		tmp = (x * -y) / x;
	} else {
		tmp = 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 (y <= (-1.3d+290)) then
        tmp = x * (y / x)
    else if (y <= (-1.25d+91)) then
        tmp = (x * -y) / x
    else
        tmp = x / (x / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.3e+290) {
		tmp = x * (y / x);
	} else if (y <= -1.25e+91) {
		tmp = (x * -y) / x;
	} else {
		tmp = x / (x / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.3e+290:
		tmp = x * (y / x)
	elif y <= -1.25e+91:
		tmp = (x * -y) / x
	else:
		tmp = x / (x / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.3e+290)
		tmp = Float64(x * Float64(y / x));
	elseif (y <= -1.25e+91)
		tmp = Float64(Float64(x * Float64(-y)) / x);
	else
		tmp = Float64(x / Float64(x / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.3e+290)
		tmp = x * (y / x);
	elseif (y <= -1.25e+91)
		tmp = (x * -y) / x;
	else
		tmp = x / (x / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.3e+290], N[(x * N[(y / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -1.25e+91], N[(N[(x * (-y)), $MachinePrecision] / x), $MachinePrecision], N[(x / N[(x / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.29999999999999999e290

   1. Initial program 100.0%

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

      1. associate-*r/ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 9.8%

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

   5. Taylor expanded in x around 0 28.8%

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

      1. associate-/l* 12.2%

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

      2. associate-/r/ 83.4%

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

   7. Applied egg-rr 83.4%

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

   if -1.29999999999999999e290 < y < -1.2500000000000001e91

   1. Initial program 100.0%

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

      1. associate-*r/ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 4.0%

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

   5. Taylor expanded in x around 0 11.4%

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

      1. associate-/l* 2.8%

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

      2. associate-/r/ 14.1%

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

   7. Applied egg-rr 14.1%

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

      1. associate-*l/ 11.4%

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

      2. frac-2neg 11.4%

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

      3. add-sqr-sqrt 4.3%

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

      4. sqrt-unprod 7.4%

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

      5. sqr-neg 7.4%

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

      6. sqrt-unprod 21.6%

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

      7. add-sqr-sqrt 37.1%

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

   9. Applied egg-rr 37.1%

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

   if -1.2500000000000001e91 < y

   1. Initial program 86.1%

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

      1. associate-*r/ 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around 0 51.2%

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

   5. Taylor expanded in x around 0 25.6%

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

      1. associate-/l* 36.4%

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

      2. associate-/r/ 57.3%

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

   7. Applied egg-rr 57.3%

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

      1. *-commutative 57.3%

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

      2. clear-num 58.8%

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

      3. un-div-inv 58.4%

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

   9. Applied egg-rr 58.4%

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

4. Final simplification 56.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.3 \cdot 10^{+290}:\\ \;\;\;\;x \cdot \frac{y}{x}\\ \mathbf{elif}\;y \leq -1.25 \cdot 10^{+91}:\\ \;\;\;\;\frac{x \cdot \left(-y\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{x}{y}}\\ \end{array} \]

Alternative 9: 51.4% accurate, 25.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.3 \cdot 10^{+18}:\\ \;\;\;\;x \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(-y\right)}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 4.3e+18) (* x (/ y x)) (/ (* y (- y)) y)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 4.3e+18) {
		tmp = x * (y / x);
	} else {
		tmp = (y * -y) / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 4.3d+18) then
        tmp = x * (y / x)
    else
        tmp = (y * -y) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 4.3e+18) {
		tmp = x * (y / x);
	} else {
		tmp = (y * -y) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 4.3e+18:
		tmp = x * (y / x)
	else:
		tmp = (y * -y) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 4.3e+18)
		tmp = Float64(x * Float64(y / x));
	else
		tmp = Float64(Float64(y * Float64(-y)) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 4.3e+18)
		tmp = x * (y / x);
	else
		tmp = (y * -y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 4.3e+18], N[(x * N[(y / x), $MachinePrecision]), $MachinePrecision], N[(N[(y * (-y)), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.3e18

   1. Initial program 84.9%

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

      1. associate-*r/ 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around 0 42.8%

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

   5. Taylor expanded in x around 0 26.6%

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

      1. associate-/l* 39.4%

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

      2. associate-/r/ 60.3%

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

   7. Applied egg-rr 60.3%

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

   if 4.3e18 < x

   1. Initial program 99.9%

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

      1. associate-*r/ 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in y around 0 49.7%

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

   5. Taylor expanded in x around 0 13.8%

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

      1. clear-num 13.8%

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

      2. inv-pow 13.8%

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

      3. *-commutative 13.8%

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

      4. associate-/r* 3.6%

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

      5. *-inverses 3.6%

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

   7. Applied egg-rr 3.6%

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

      1. unpow-1 3.6%

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

      2. remove-double-div 3.6%

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

      3. add-sqr-sqrt 2.3%

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

      4. sqrt-unprod 37.7%

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

      5. sqr-neg 37.7%

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

      6. sqrt-unprod 2.1%

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

      7. add-sqr-sqrt 4.9%

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

      8. neg-sub0 4.9%

         \[\leadsto \color{blue}{0 - y} \]

      9. metadata-eval 4.9%

         \[\leadsto \color{blue}{\log 1} - y \]

      10. flip-- 43.0%

          \[\leadsto \color{blue}{\frac{\log 1 \cdot \log 1 - y \cdot y}{\log 1 + y}} \]

      11. metadata-eval 43.0%

          \[\leadsto \frac{\color{blue}{0} \cdot \log 1 - y \cdot y}{\log 1 + y} \]

      12. metadata-eval 43.0%

          \[\leadsto \frac{0 \cdot \color{blue}{0} - y \cdot y}{\log 1 + y} \]

      13. metadata-eval 43.0%

          \[\leadsto \frac{\color{blue}{0} - y \cdot y}{\log 1 + y} \]

      14. add-sqr-sqrt 24.2%

          \[\leadsto \frac{0 - y \cdot y}{\log 1 + \color{blue}{\sqrt{y} \cdot \sqrt{y}}} \]

      15. sqrt-unprod 1.4%

          \[\leadsto \frac{0 - y \cdot y}{\log 1 + \color{blue}{\sqrt{y \cdot y}}} \]

      16. sqr-neg 1.4%

          \[\leadsto \frac{0 - y \cdot y}{\log 1 + \sqrt{\color{blue}{\left(-y\right) \cdot \left(-y\right)}}} \]

      17. sqrt-unprod 9.7%

          \[\leadsto \frac{0 - y \cdot y}{\log 1 + \color{blue}{\sqrt{-y} \cdot \sqrt{-y}}} \]

      18. add-sqr-sqrt 28.6%

          \[\leadsto \frac{0 - y \cdot y}{\log 1 + \color{blue}{\left(-y\right)}} \]

      19. sub-neg 28.6%

          \[\leadsto \frac{0 - y \cdot y}{\color{blue}{\log 1 - y}} \]

      20. metadata-eval 28.6%

          \[\leadsto \frac{0 - y \cdot y}{\color{blue}{0} - y} \]

      21. neg-sub0 28.6%

          \[\leadsto \frac{0 - y \cdot y}{\color{blue}{-y}} \]

      22. add-sqr-sqrt 9.7%

          \[\leadsto \frac{0 - y \cdot y}{\color{blue}{\sqrt{-y} \cdot \sqrt{-y}}} \]

      23. sqrt-unprod 1.4%

          \[\leadsto \frac{0 - y \cdot y}{\color{blue}{\sqrt{\left(-y\right) \cdot \left(-y\right)}}} \]

      24. sqr-neg 1.4%

          \[\leadsto \frac{0 - y \cdot y}{\sqrt{\color{blue}{y \cdot y}}} \]

      25. sqrt-unprod 24.2%

          \[\leadsto \frac{0 - y \cdot y}{\color{blue}{\sqrt{y} \cdot \sqrt{y}}} \]

      26. add-sqr-sqrt 43.0%

          \[\leadsto \frac{0 - y \cdot y}{\color{blue}{y}} \]

   9. Applied egg-rr 43.0%

      \[\leadsto \color{blue}{\frac{0 - y \cdot y}{y}} \]
   10. Step-by-step derivation

       1. sub0-neg 43.0%

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

   11. Simplified 43.0%

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

4. Final simplification 56.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.3 \cdot 10^{+18}:\\ \;\;\;\;x \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(-y\right)}{y}\\ \end{array} \]

Alternative 10: 50.0% 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 88.2%

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

   1. associate-*r/ 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in y around 0 44.3%

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

5. Taylor expanded in x around 0 23.8%

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

   1. associate-/l* 31.6%

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

   2. associate-/r/ 52.2%

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

7. Applied egg-rr 52.2%

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

8. Final simplification 52.2%

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

Alternative 11: 49.7% accurate, 41.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.2%

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

   1. associate-*r/ 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in y around 0 44.3%

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

5. Taylor expanded in x around 0 23.8%

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

   1. associate-/l* 31.6%

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

   2. associate-/r/ 52.2%

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

7. Applied egg-rr 52.2%

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

   1. *-commutative 52.2%

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

   2. clear-num 53.5%

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

   3. un-div-inv 53.2%

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

9. Applied egg-rr 53.2%

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

10. Final simplification 53.2%

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

Alternative 12: 27.6% 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 88.2%

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

   1. associate-*r/ 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in y around 0 44.3%

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

   1. associate-/l* 56.1%

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

   2. associate-/r/ 65.8%

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

6. Simplified 65.8%

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

7. Taylor expanded in x around 0 31.6%

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

8. Final simplification 31.6%

   \[\leadsto y \]

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 2023200 
(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))