Linear.Quaternion:$ccosh from linear-1.19.1.3

Percentage Accurate: 88.5% → 99.8%

Time: 15.3s

Alternatives: 20

Speedup: 1.7×

• 49.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 88.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% 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 89.8%

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

   1. associate-/l* N/A

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

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\sin x, \color{blue}{\left(\frac{\sinh y}{x}\right)}\right) \]

   3. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\frac{\color{blue}{\sinh y}}{x}\right)\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\sinh y, \color{blue}{x}\right)\right) \]

   5. sinh-lowering-sinh.f64 99.9%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]

3. Simplified 99.9%

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

Alternative 2: 61.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sinh y \leq 4 \cdot 10^{-52}:\\ \;\;\;\;\frac{x}{\frac{x}{y}}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sinh y) 4e-52) (/ x (/ x y)) (sinh y)))

C

double code(double x, double y) {
	double tmp;
	if (sinh(y) <= 4e-52) {
		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 (sinh(y) <= 4d-52) 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 (Math.sinh(y) <= 4e-52) {
		tmp = x / (x / y);
	} else {
		tmp = Math.sinh(y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if math.sinh(y) <= 4e-52:
		tmp = x / (x / y)
	else:
		tmp = math.sinh(y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (sinh(y) <= 4e-52)
		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 (sinh(y) <= 4e-52)
		tmp = x / (x / y);
	else
		tmp = sinh(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[Sinh[y], $MachinePrecision], 4e-52], N[(x / N[(x / y), $MachinePrecision]), $MachinePrecision], N[Sinh[y], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sinh.f64 y) < 4e-52

   1. Initial program 86.8%

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

      1. associate-/l* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin x, \color{blue}{\left(\frac{\sinh y}{x}\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\frac{\color{blue}{\sinh y}}{x}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\sinh y, \color{blue}{x}\right)\right) \]

      5. sinh-lowering-sinh.f64 99.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{x}, \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 76.6%

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

   8. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{y}{x}\right)}\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 59.7%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \color{blue}{x}\right)\right) \]

   10. Simplified 59.7%

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

       1. clear-num N/A

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

       2. un-div-inv N/A

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

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(\frac{x}{y}\right)}\right) \]

       4. /-lowering-/.f64 59.5%

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(x, \color{blue}{y}\right)\right) \]

   12. Applied egg-rr 59.5%

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

   if 4e-52 < (sinh.f64 y)

   1. Initial program 98.5%

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

      1. associate-/l* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin x, \color{blue}{\left(\frac{\sinh y}{x}\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\frac{\color{blue}{\sinh y}}{x}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\sinh y, \color{blue}{x}\right)\right) \]

      5. sinh-lowering-sinh.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{x}, \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 66.9%

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

      1. *-commutative N/A

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

      2. div-inv N/A

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

      3. associate-*l* N/A

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

      4. lft-mult-inverse N/A

         \[\leadsto \sinh y \cdot 1 \]

      5. *-rgt-identity N/A

         \[\leadsto \sinh y \]

      6. sinh-lowering-sinh.f64 66.9%

         \[\leadsto \mathsf{sinh.f64}\left(y\right) \]

   9. Applied egg-rr 66.9%

      \[\leadsto \color{blue}{\sinh y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 89.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.0039:\\ \;\;\;\;y \cdot \left(\left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \cdot \frac{\sin x}{x}\right)\\ \mathbf{elif}\;y \leq 9 \cdot 10^{+61}:\\ \;\;\;\;\frac{\sinh y \cdot \left(x \cdot \left(1 + x \cdot \left(x \cdot -0.16666666666666666\right)\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(\sin x \cdot \left(1 + y \cdot \left(y \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right)\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 0.0039)
   (* y (* (+ 1.0 (* 0.16666666666666666 (* y y))) (/ (sin x) x)))
   (if (<= y 9e+61)
     (/ (* (sinh y) (* x (+ 1.0 (* x (* x -0.16666666666666666))))) x)
     (/
      (*
       y
       (*
        (sin x)
        (+
         1.0
         (*
          y
          (* y (+ 0.16666666666666666 (* (* y y) 0.008333333333333333)))))))
      x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 0.0039) {
		tmp = y * ((1.0 + (0.16666666666666666 * (y * y))) * (sin(x) / x));
	} else if (y <= 9e+61) {
		tmp = (sinh(y) * (x * (1.0 + (x * (x * -0.16666666666666666))))) / x;
	} else {
		tmp = (y * (sin(x) * (1.0 + (y * (y * (0.16666666666666666 + ((y * y) * 0.008333333333333333))))))) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 0.0039d0) then
        tmp = y * ((1.0d0 + (0.16666666666666666d0 * (y * y))) * (sin(x) / x))
    else if (y <= 9d+61) then
        tmp = (sinh(y) * (x * (1.0d0 + (x * (x * (-0.16666666666666666d0)))))) / x
    else
        tmp = (y * (sin(x) * (1.0d0 + (y * (y * (0.16666666666666666d0 + ((y * y) * 0.008333333333333333d0))))))) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 0.0039) {
		tmp = y * ((1.0 + (0.16666666666666666 * (y * y))) * (Math.sin(x) / x));
	} else if (y <= 9e+61) {
		tmp = (Math.sinh(y) * (x * (1.0 + (x * (x * -0.16666666666666666))))) / x;
	} else {
		tmp = (y * (Math.sin(x) * (1.0 + (y * (y * (0.16666666666666666 + ((y * y) * 0.008333333333333333))))))) / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 0.0039:
		tmp = y * ((1.0 + (0.16666666666666666 * (y * y))) * (math.sin(x) / x))
	elif y <= 9e+61:
		tmp = (math.sinh(y) * (x * (1.0 + (x * (x * -0.16666666666666666))))) / x
	else:
		tmp = (y * (math.sin(x) * (1.0 + (y * (y * (0.16666666666666666 + ((y * y) * 0.008333333333333333))))))) / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 0.0039)
		tmp = Float64(y * Float64(Float64(1.0 + Float64(0.16666666666666666 * Float64(y * y))) * Float64(sin(x) / x)));
	elseif (y <= 9e+61)
		tmp = Float64(Float64(sinh(y) * Float64(x * Float64(1.0 + Float64(x * Float64(x * -0.16666666666666666))))) / x);
	else
		tmp = Float64(Float64(y * Float64(sin(x) * Float64(1.0 + Float64(y * Float64(y * Float64(0.16666666666666666 + Float64(Float64(y * y) * 0.008333333333333333))))))) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 0.0039)
		tmp = y * ((1.0 + (0.16666666666666666 * (y * y))) * (sin(x) / x));
	elseif (y <= 9e+61)
		tmp = (sinh(y) * (x * (1.0 + (x * (x * -0.16666666666666666))))) / x;
	else
		tmp = (y * (sin(x) * (1.0 + (y * (y * (0.16666666666666666 + ((y * y) * 0.008333333333333333))))))) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 0.0039], N[(y * N[(N[(1.0 + N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 9e+61], N[(N[(N[Sinh[y], $MachinePrecision] * N[(x * N[(1.0 + N[(x * N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(y * N[(N[Sin[x], $MachinePrecision] * N[(1.0 + N[(y * N[(y * N[(0.16666666666666666 + N[(N[(y * y), $MachinePrecision] * 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 0.0038999999999999998

   1. Initial program 87.0%

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

      1. associate-/l* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin x, \color{blue}{\left(\frac{\sinh y}{x}\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\frac{\color{blue}{\sinh y}}{x}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\sinh y, \color{blue}{x}\right)\right) \]

      5. sinh-lowering-sinh.f64 99.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]

   3. Simplified 99.9%

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

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left(\frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{\sin x}{x}\right)} \]
   6. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto y \cdot \left(\frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x}\right) + \color{blue}{y \cdot \frac{\sin x}{x}} \]

      2. fma-define N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x}}, y \cdot \frac{\sin x}{x}\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{1}{6} \cdot \left({y}^{2} \cdot \color{blue}{\frac{\sin x}{x}}\right), y \cdot \frac{\sin x}{x}\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(y, \left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \color{blue}{\frac{\sin x}{x}}, y \cdot \frac{\sin x}{x}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, \left({y}^{2} \cdot \frac{1}{6}\right) \cdot \frac{\color{blue}{\sin x}}{x}, y \cdot \frac{\sin x}{x}\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(y, {y}^{2} \cdot \color{blue}{\left(\frac{1}{6} \cdot \frac{\sin x}{x}\right)}, y \cdot \frac{\sin x}{x}\right) \]

      7. fma-define N/A

         \[\leadsto y \cdot \left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right)\right) + \color{blue}{y \cdot \frac{\sin x}{x}} \]

      8. distribute-lft-in N/A

         \[\leadsto y \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right)} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right)}\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left({y}^{2} \cdot \frac{1}{6}\right) \cdot \frac{\sin x}{x} + \frac{\color{blue}{\sin x}}{x}\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \frac{\sin x}{x} + \frac{\sin \color{blue}{x}}{x}\right)\right) \]

      12. distribute-lft1-in N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left(\frac{1}{6} \cdot {y}^{2} + 1\right) \cdot \color{blue}{\frac{\sin x}{x}}\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left(1 + \frac{1}{6} \cdot {y}^{2}\right) \cdot \frac{\color{blue}{\sin x}}{x}\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left(1 + \frac{1}{6} \cdot {y}^{2}\right), \color{blue}{\left(\frac{\sin x}{x}\right)}\right)\right) \]

   7. Simplified 86.9%

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

   if 0.0038999999999999998 < y < 9e61

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)}, \mathsf{sinh.f64}\left(y\right)\right), x\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right), \mathsf{sinh.f64}\left(y\right)\right), x\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot {x}^{2}\right)\right)\right), \mathsf{sinh.f64}\left(y\right)\right), x\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left({x}^{2} \cdot \frac{-1}{6}\right)\right)\right), \mathsf{sinh.f64}\left(y\right)\right), x\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \frac{-1}{6}\right)\right)\right), \mathsf{sinh.f64}\left(y\right)\right), x\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(x \cdot \left(x \cdot \frac{-1}{6}\right)\right)\right)\right), \mathsf{sinh.f64}\left(y\right)\right), x\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \frac{-1}{6}\right)\right)\right)\right), \mathsf{sinh.f64}\left(y\right)\right), x\right) \]

      7. *-lowering-*.f64 68.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{-1}{6}\right)\right)\right)\right), \mathsf{sinh.f64}\left(y\right)\right), x\right) \]

   5. Simplified 68.8%

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

   if 9e61 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(y \cdot \left(\sin x + {y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{6} \cdot \sin x\right)\right)\right)}, x\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\sin x + {y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{6} \cdot \sin x\right)\right)\right), x\right) \]

      2. *-rgt-identity N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\sin x \cdot 1 + {y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{6} \cdot \sin x\right)\right)\right), x\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\sin x \cdot 1 + {y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x + \frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right)\right)\right)\right), x\right) \]

      4. distribute-lft-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\sin x \cdot 1 + \left({y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x\right) + {y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right)\right)\right)\right)\right), x\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\sin x \cdot 1 + \left(\left({y}^{2} \cdot \frac{1}{6}\right) \cdot \sin x + {y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right)\right)\right)\right)\right), x\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\sin x \cdot 1 + \left(\left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \sin x + {y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right)\right)\right)\right)\right), x\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\sin x \cdot 1 + \left(\left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \sin x + {y}^{2} \cdot \left(\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot \sin x\right)\right)\right)\right), x\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\sin x \cdot 1 + \left(\left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \sin x + \left({y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2}\right)\right) \cdot \sin x\right)\right)\right), x\right) \]

      9. distribute-rgt-out N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\sin x \cdot 1 + \sin x \cdot \left(\frac{1}{6} \cdot {y}^{2} + {y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right), x\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\sin x \cdot 1 + \sin x \cdot \left({y}^{2} \cdot \frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right), x\right) \]

      11. distribute-lft-in N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\sin x \cdot 1 + \sin x \cdot \left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right), x\right) \]

   5. Simplified 97.6%

      \[\leadsto \frac{\color{blue}{y \cdot \left(\sin x \cdot \left(1 + y \cdot \left(y \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right)\right)}}{x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.0039:\\ \;\;\;\;y \cdot \left(\left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \cdot \frac{\sin x}{x}\right)\\ \mathbf{elif}\;y \leq 9 \cdot 10^{+61}:\\ \;\;\;\;\frac{\sinh y \cdot \left(x \cdot \left(1 + x \cdot \left(x \cdot -0.16666666666666666\right)\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(\sin x \cdot \left(1 + y \cdot \left(y \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right)\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 87.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\\ t_1 := y \cdot \left(t\_0 \cdot \frac{\sin x}{x}\right)\\ \mathbf{if}\;y \leq 0.0039:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{+102}:\\ \;\;\;\;\sinh y\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{+132}:\\ \;\;\;\;y \cdot \left(t\_0 \cdot \left(1 + -0.16666666666666666 \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* 0.16666666666666666 (* y y))))
        (t_1 (* y (* t_0 (/ (sin x) x)))))
   (if (<= y 0.0039)
     t_1
     (if (<= y 9.5e+102)
       (sinh y)
       (if (<= y 2.8e+132)
         (* y (* t_0 (+ 1.0 (* -0.16666666666666666 (* x x)))))
         t_1)))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (0.16666666666666666 * (y * y));
	double t_1 = y * (t_0 * (sin(x) / x));
	double tmp;
	if (y <= 0.0039) {
		tmp = t_1;
	} else if (y <= 9.5e+102) {
		tmp = sinh(y);
	} else if (y <= 2.8e+132) {
		tmp = y * (t_0 * (1.0 + (-0.16666666666666666 * (x * x))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 + (0.16666666666666666d0 * (y * y))
    t_1 = y * (t_0 * (sin(x) / x))
    if (y <= 0.0039d0) then
        tmp = t_1
    else if (y <= 9.5d+102) then
        tmp = sinh(y)
    else if (y <= 2.8d+132) then
        tmp = y * (t_0 * (1.0d0 + ((-0.16666666666666666d0) * (x * x))))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 + (0.16666666666666666 * (y * y));
	double t_1 = y * (t_0 * (Math.sin(x) / x));
	double tmp;
	if (y <= 0.0039) {
		tmp = t_1;
	} else if (y <= 9.5e+102) {
		tmp = Math.sinh(y);
	} else if (y <= 2.8e+132) {
		tmp = y * (t_0 * (1.0 + (-0.16666666666666666 * (x * x))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + (0.16666666666666666 * (y * y))
	t_1 = y * (t_0 * (math.sin(x) / x))
	tmp = 0
	if y <= 0.0039:
		tmp = t_1
	elif y <= 9.5e+102:
		tmp = math.sinh(y)
	elif y <= 2.8e+132:
		tmp = y * (t_0 * (1.0 + (-0.16666666666666666 * (x * x))))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(0.16666666666666666 * Float64(y * y)))
	t_1 = Float64(y * Float64(t_0 * Float64(sin(x) / x)))
	tmp = 0.0
	if (y <= 0.0039)
		tmp = t_1;
	elseif (y <= 9.5e+102)
		tmp = sinh(y);
	elseif (y <= 2.8e+132)
		tmp = Float64(y * Float64(t_0 * Float64(1.0 + Float64(-0.16666666666666666 * Float64(x * x)))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + (0.16666666666666666 * (y * y));
	t_1 = y * (t_0 * (sin(x) / x));
	tmp = 0.0;
	if (y <= 0.0039)
		tmp = t_1;
	elseif (y <= 9.5e+102)
		tmp = sinh(y);
	elseif (y <= 2.8e+132)
		tmp = y * (t_0 * (1.0 + (-0.16666666666666666 * (x * x))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(y * N[(t$95$0 * N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 0.0039], t$95$1, If[LessEqual[y, 9.5e+102], N[Sinh[y], $MachinePrecision], If[LessEqual[y, 2.8e+132], N[(y * N[(t$95$0 * N[(1.0 + N[(-0.16666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if y < 0.0038999999999999998 or 2.7999999999999999e132 < y

   1. Initial program 88.7%

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

      1. associate-/l* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin x, \color{blue}{\left(\frac{\sinh y}{x}\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\frac{\color{blue}{\sinh y}}{x}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\sinh y, \color{blue}{x}\right)\right) \]

      5. sinh-lowering-sinh.f64 99.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]

   3. Simplified 99.9%

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

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left(\frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{\sin x}{x}\right)} \]
   6. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto y \cdot \left(\frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x}\right) + \color{blue}{y \cdot \frac{\sin x}{x}} \]

      2. fma-define N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x}}, y \cdot \frac{\sin x}{x}\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{1}{6} \cdot \left({y}^{2} \cdot \color{blue}{\frac{\sin x}{x}}\right), y \cdot \frac{\sin x}{x}\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(y, \left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \color{blue}{\frac{\sin x}{x}}, y \cdot \frac{\sin x}{x}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, \left({y}^{2} \cdot \frac{1}{6}\right) \cdot \frac{\color{blue}{\sin x}}{x}, y \cdot \frac{\sin x}{x}\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(y, {y}^{2} \cdot \color{blue}{\left(\frac{1}{6} \cdot \frac{\sin x}{x}\right)}, y \cdot \frac{\sin x}{x}\right) \]

      7. fma-define N/A

         \[\leadsto y \cdot \left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right)\right) + \color{blue}{y \cdot \frac{\sin x}{x}} \]

      8. distribute-lft-in N/A

         \[\leadsto y \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right)} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right)}\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left({y}^{2} \cdot \frac{1}{6}\right) \cdot \frac{\sin x}{x} + \frac{\color{blue}{\sin x}}{x}\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \frac{\sin x}{x} + \frac{\sin \color{blue}{x}}{x}\right)\right) \]

      12. distribute-lft1-in N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left(\frac{1}{6} \cdot {y}^{2} + 1\right) \cdot \color{blue}{\frac{\sin x}{x}}\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left(1 + \frac{1}{6} \cdot {y}^{2}\right) \cdot \frac{\color{blue}{\sin x}}{x}\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left(1 + \frac{1}{6} \cdot {y}^{2}\right), \color{blue}{\left(\frac{\sin x}{x}\right)}\right)\right) \]

   7. Simplified 88.2%

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

   if 0.0038999999999999998 < y < 9.4999999999999992e102

   1. Initial program 100.0%

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

      1. associate-/l* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin x, \color{blue}{\left(\frac{\sinh y}{x}\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\frac{\color{blue}{\sinh y}}{x}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\sinh y, \color{blue}{x}\right)\right) \]

      5. sinh-lowering-sinh.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{x}, \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 75.0%

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

      1. *-commutative N/A

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

      2. div-inv N/A

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

      3. associate-*l* N/A

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

      4. lft-mult-inverse N/A

         \[\leadsto \sinh y \cdot 1 \]

      5. *-rgt-identity N/A

         \[\leadsto \sinh y \]

      6. sinh-lowering-sinh.f64 75.0%

         \[\leadsto \mathsf{sinh.f64}\left(y\right) \]

   9. Applied egg-rr 75.0%

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

   if 9.4999999999999992e102 < y < 2.7999999999999999e132

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)}, \mathsf{sinh.f64}\left(y\right)\right), x\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right), \mathsf{sinh.f64}\left(y\right)\right), x\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot {x}^{2}\right)\right)\right), \mathsf{sinh.f64}\left(y\right)\right), x\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left({x}^{2} \cdot \frac{-1}{6}\right)\right)\right), \mathsf{sinh.f64}\left(y\right)\right), x\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \frac{-1}{6}\right)\right)\right), \mathsf{sinh.f64}\left(y\right)\right), x\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(x \cdot \left(x \cdot \frac{-1}{6}\right)\right)\right)\right), \mathsf{sinh.f64}\left(y\right)\right), x\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \frac{-1}{6}\right)\right)\right)\right), \mathsf{sinh.f64}\left(y\right)\right), x\right) \]

      7. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{-1}{6}\right)\right)\right)\right), \mathsf{sinh.f64}\left(y\right)\right), x\right) \]

   5. Simplified 100.0%

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

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left(1 + \left(\frac{-1}{6} \cdot {x}^{2} + \frac{1}{6} \cdot \left({y}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right)\right)} \]
   7. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto y \cdot \left(\left(1 + \frac{-1}{6} \cdot {x}^{2}\right) + \color{blue}{\frac{1}{6} \cdot \left({y}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)}\right) \]

      2. distribute-lft-in N/A

         \[\leadsto y \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right) + \color{blue}{y \cdot \left(\frac{1}{6} \cdot \left({y}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto y \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right) + y \cdot \left(\frac{1}{6} \cdot \left(\left(1 + \frac{-1}{6} \cdot {x}^{2}\right) \cdot \color{blue}{{y}^{2}}\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto y \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right) + y \cdot \left(\left(\frac{1}{6} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right) \cdot \color{blue}{{y}^{2}}\right) \]

      5. distribute-lft-out N/A

         \[\leadsto y \cdot \color{blue}{\left(\left(1 + \frac{-1}{6} \cdot {x}^{2}\right) + \left(\frac{1}{6} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right) \cdot {y}^{2}\right)} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(\left(1 + \frac{-1}{6} \cdot {x}^{2}\right) + \left(\frac{1}{6} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right) \cdot {y}^{2}\right)}\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(\left(1 + \frac{-1}{6} \cdot {x}^{2}\right) + \frac{1}{6} \cdot \color{blue}{\left(\left(1 + \frac{-1}{6} \cdot {x}^{2}\right) \cdot {y}^{2}\right)}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(\left(1 + \frac{-1}{6} \cdot {x}^{2}\right) + \frac{1}{6} \cdot \left({y}^{2} \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {x}^{2}\right)}\right)\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(\left(1 + \frac{-1}{6} \cdot {x}^{2}\right) + \left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {x}^{2}\right)}\right)\right) \]

      10. distribute-rgt1-in N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left(\frac{1}{6} \cdot {y}^{2} + 1\right) \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {x}^{2}\right)}\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left(1 + \frac{1}{6} \cdot {y}^{2}\right) \cdot \left(\color{blue}{1} + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right) \]

   8. Simplified 100.0%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.0039:\\ \;\;\;\;y \cdot \left(\left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \cdot \frac{\sin x}{x}\right)\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{+102}:\\ \;\;\;\;\sinh y\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{+132}:\\ \;\;\;\;y \cdot \left(\left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \cdot \left(1 + -0.16666666666666666 \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \cdot \frac{\sin x}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 88.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\\ t_1 := \frac{\sin x}{x}\\ \mathbf{if}\;y \leq 0.0039:\\ \;\;\;\;y \cdot \left(t\_0 \cdot t\_1\right)\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{+103}:\\ \;\;\;\;\frac{\sinh y \cdot \left(x \cdot \left(1 + x \cdot \left(x \cdot -0.16666666666666666\right)\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left(y \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* 0.16666666666666666 (* y y)))) (t_1 (/ (sin x) x)))
   (if (<= y 0.0039)
     (* y (* t_0 t_1))
     (if (<= y 1.05e+103)
       (/ (* (sinh y) (* x (+ 1.0 (* x (* x -0.16666666666666666))))) x)
       (* t_1 (* y t_0))))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (0.16666666666666666 * (y * y));
	double t_1 = sin(x) / x;
	double tmp;
	if (y <= 0.0039) {
		tmp = y * (t_0 * t_1);
	} else if (y <= 1.05e+103) {
		tmp = (sinh(y) * (x * (1.0 + (x * (x * -0.16666666666666666))))) / x;
	} else {
		tmp = t_1 * (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) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 + (0.16666666666666666d0 * (y * y))
    t_1 = sin(x) / x
    if (y <= 0.0039d0) then
        tmp = y * (t_0 * t_1)
    else if (y <= 1.05d+103) then
        tmp = (sinh(y) * (x * (1.0d0 + (x * (x * (-0.16666666666666666d0)))))) / x
    else
        tmp = t_1 * (y * t_0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 + (0.16666666666666666 * (y * y));
	double t_1 = Math.sin(x) / x;
	double tmp;
	if (y <= 0.0039) {
		tmp = y * (t_0 * t_1);
	} else if (y <= 1.05e+103) {
		tmp = (Math.sinh(y) * (x * (1.0 + (x * (x * -0.16666666666666666))))) / x;
	} else {
		tmp = t_1 * (y * t_0);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + (0.16666666666666666 * (y * y))
	t_1 = math.sin(x) / x
	tmp = 0
	if y <= 0.0039:
		tmp = y * (t_0 * t_1)
	elif y <= 1.05e+103:
		tmp = (math.sinh(y) * (x * (1.0 + (x * (x * -0.16666666666666666))))) / x
	else:
		tmp = t_1 * (y * t_0)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(0.16666666666666666 * Float64(y * y)))
	t_1 = Float64(sin(x) / x)
	tmp = 0.0
	if (y <= 0.0039)
		tmp = Float64(y * Float64(t_0 * t_1));
	elseif (y <= 1.05e+103)
		tmp = Float64(Float64(sinh(y) * Float64(x * Float64(1.0 + Float64(x * Float64(x * -0.16666666666666666))))) / x);
	else
		tmp = Float64(t_1 * Float64(y * t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + (0.16666666666666666 * (y * y));
	t_1 = sin(x) / x;
	tmp = 0.0;
	if (y <= 0.0039)
		tmp = y * (t_0 * t_1);
	elseif (y <= 1.05e+103)
		tmp = (sinh(y) * (x * (1.0 + (x * (x * -0.16666666666666666))))) / x;
	else
		tmp = t_1 * (y * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[y, 0.0039], N[(y * N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.05e+103], N[(N[(N[Sinh[y], $MachinePrecision] * N[(x * N[(1.0 + N[(x * N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(t$95$1 * N[(y * t$95$0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < 0.0038999999999999998

   1. Initial program 87.0%

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

      1. associate-/l* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin x, \color{blue}{\left(\frac{\sinh y}{x}\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\frac{\color{blue}{\sinh y}}{x}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\sinh y, \color{blue}{x}\right)\right) \]

      5. sinh-lowering-sinh.f64 99.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]

   3. Simplified 99.9%

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

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left(\frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{\sin x}{x}\right)} \]
   6. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto y \cdot \left(\frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x}\right) + \color{blue}{y \cdot \frac{\sin x}{x}} \]

      2. fma-define N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x}}, y \cdot \frac{\sin x}{x}\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{1}{6} \cdot \left({y}^{2} \cdot \color{blue}{\frac{\sin x}{x}}\right), y \cdot \frac{\sin x}{x}\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(y, \left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \color{blue}{\frac{\sin x}{x}}, y \cdot \frac{\sin x}{x}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, \left({y}^{2} \cdot \frac{1}{6}\right) \cdot \frac{\color{blue}{\sin x}}{x}, y \cdot \frac{\sin x}{x}\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(y, {y}^{2} \cdot \color{blue}{\left(\frac{1}{6} \cdot \frac{\sin x}{x}\right)}, y \cdot \frac{\sin x}{x}\right) \]

      7. fma-define N/A

         \[\leadsto y \cdot \left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right)\right) + \color{blue}{y \cdot \frac{\sin x}{x}} \]

      8. distribute-lft-in N/A

         \[\leadsto y \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right)} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right)}\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left({y}^{2} \cdot \frac{1}{6}\right) \cdot \frac{\sin x}{x} + \frac{\color{blue}{\sin x}}{x}\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \frac{\sin x}{x} + \frac{\sin \color{blue}{x}}{x}\right)\right) \]

      12. distribute-lft1-in N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left(\frac{1}{6} \cdot {y}^{2} + 1\right) \cdot \color{blue}{\frac{\sin x}{x}}\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left(1 + \frac{1}{6} \cdot {y}^{2}\right) \cdot \frac{\color{blue}{\sin x}}{x}\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left(1 + \frac{1}{6} \cdot {y}^{2}\right), \color{blue}{\left(\frac{\sin x}{x}\right)}\right)\right) \]

   7. Simplified 86.9%

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

   if 0.0038999999999999998 < y < 1.0500000000000001e103

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)}, \mathsf{sinh.f64}\left(y\right)\right), x\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right), \mathsf{sinh.f64}\left(y\right)\right), x\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot {x}^{2}\right)\right)\right), \mathsf{sinh.f64}\left(y\right)\right), x\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left({x}^{2} \cdot \frac{-1}{6}\right)\right)\right), \mathsf{sinh.f64}\left(y\right)\right), x\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \frac{-1}{6}\right)\right)\right), \mathsf{sinh.f64}\left(y\right)\right), x\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(x \cdot \left(x \cdot \frac{-1}{6}\right)\right)\right)\right), \mathsf{sinh.f64}\left(y\right)\right), x\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \frac{-1}{6}\right)\right)\right)\right), \mathsf{sinh.f64}\left(y\right)\right), x\right) \]

      7. *-lowering-*.f64 70.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{-1}{6}\right)\right)\right)\right), \mathsf{sinh.f64}\left(y\right)\right), x\right) \]

   5. Simplified 70.0%

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

   if 1.0500000000000001e103 < y

   1. Initial program 100.0%

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

      1. associate-/l* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin x, \color{blue}{\left(\frac{\sinh y}{x}\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\frac{\color{blue}{\sinh y}}{x}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\sinh y, \color{blue}{x}\right)\right) \]

      5. sinh-lowering-sinh.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left(\frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{\sin x}{x}\right)} \]
   6. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto y \cdot \left(\frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x}\right) + \color{blue}{y \cdot \frac{\sin x}{x}} \]

      2. fma-define N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x}}, y \cdot \frac{\sin x}{x}\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{1}{6} \cdot \left({y}^{2} \cdot \color{blue}{\frac{\sin x}{x}}\right), y \cdot \frac{\sin x}{x}\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(y, \left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \color{blue}{\frac{\sin x}{x}}, y \cdot \frac{\sin x}{x}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, \left({y}^{2} \cdot \frac{1}{6}\right) \cdot \frac{\color{blue}{\sin x}}{x}, y \cdot \frac{\sin x}{x}\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(y, {y}^{2} \cdot \color{blue}{\left(\frac{1}{6} \cdot \frac{\sin x}{x}\right)}, y \cdot \frac{\sin x}{x}\right) \]

      7. fma-define N/A

         \[\leadsto y \cdot \left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right)\right) + \color{blue}{y \cdot \frac{\sin x}{x}} \]

      8. distribute-lft-in N/A

         \[\leadsto y \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right)} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right)}\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left({y}^{2} \cdot \frac{1}{6}\right) \cdot \frac{\sin x}{x} + \frac{\color{blue}{\sin x}}{x}\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \frac{\sin x}{x} + \frac{\sin \color{blue}{x}}{x}\right)\right) \]

      12. distribute-lft1-in N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left(\frac{1}{6} \cdot {y}^{2} + 1\right) \cdot \color{blue}{\frac{\sin x}{x}}\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left(1 + \frac{1}{6} \cdot {y}^{2}\right) \cdot \frac{\color{blue}{\sin x}}{x}\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left(1 + \frac{1}{6} \cdot {y}^{2}\right), \color{blue}{\left(\frac{\sin x}{x}\right)}\right)\right) \]

   7. Simplified 92.2%

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

      1. associate-*r* N/A

         \[\leadsto \left(y \cdot \left(1 + \frac{1}{6} \cdot \left(y \cdot y\right)\right)\right) \cdot \color{blue}{\frac{\sin x}{x}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\sin x}{x} \cdot \color{blue}{\left(y \cdot \left(1 + \frac{1}{6} \cdot \left(y \cdot y\right)\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\sin x}{x}\right), \color{blue}{\left(y \cdot \left(1 + \frac{1}{6} \cdot \left(y \cdot y\right)\right)\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\sin x, x\right), \left(\color{blue}{y} \cdot \left(1 + \frac{1}{6} \cdot \left(y \cdot y\right)\right)\right)\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(x\right), x\right), \left(y \cdot \left(1 + \frac{1}{6} \cdot \left(y \cdot y\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(x\right), x\right), \mathsf{*.f64}\left(y, \color{blue}{\left(1 + \frac{1}{6} \cdot \left(y \cdot y\right)\right)}\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(x\right), x\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{6} \cdot \left(y \cdot y\right)\right)}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(x\right), x\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \color{blue}{\left(y \cdot y\right)}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(x\right), x\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right)\right)\right)\right) \]

   9. Applied egg-rr 100.0%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.0039:\\ \;\;\;\;y \cdot \left(\left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \cdot \frac{\sin x}{x}\right)\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{+103}:\\ \;\;\;\;\frac{\sinh y \cdot \left(x \cdot \left(1 + x \cdot \left(x \cdot -0.16666666666666666\right)\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin x}{x} \cdot \left(y \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 88.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\\ t_1 := \frac{\sin x}{x}\\ \mathbf{if}\;y \leq 0.0039:\\ \;\;\;\;y \cdot \left(t\_0 \cdot t\_1\right)\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{+102}:\\ \;\;\;\;\sinh y\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left(y \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* 0.16666666666666666 (* y y)))) (t_1 (/ (sin x) x)))
   (if (<= y 0.0039)
     (* y (* t_0 t_1))
     (if (<= y 9.5e+102) (sinh y) (* t_1 (* y t_0))))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (0.16666666666666666 * (y * y));
	double t_1 = sin(x) / x;
	double tmp;
	if (y <= 0.0039) {
		tmp = y * (t_0 * t_1);
	} else if (y <= 9.5e+102) {
		tmp = sinh(y);
	} else {
		tmp = t_1 * (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) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 + (0.16666666666666666d0 * (y * y))
    t_1 = sin(x) / x
    if (y <= 0.0039d0) then
        tmp = y * (t_0 * t_1)
    else if (y <= 9.5d+102) then
        tmp = sinh(y)
    else
        tmp = t_1 * (y * t_0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 + (0.16666666666666666 * (y * y));
	double t_1 = Math.sin(x) / x;
	double tmp;
	if (y <= 0.0039) {
		tmp = y * (t_0 * t_1);
	} else if (y <= 9.5e+102) {
		tmp = Math.sinh(y);
	} else {
		tmp = t_1 * (y * t_0);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + (0.16666666666666666 * (y * y))
	t_1 = math.sin(x) / x
	tmp = 0
	if y <= 0.0039:
		tmp = y * (t_0 * t_1)
	elif y <= 9.5e+102:
		tmp = math.sinh(y)
	else:
		tmp = t_1 * (y * t_0)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(0.16666666666666666 * Float64(y * y)))
	t_1 = Float64(sin(x) / x)
	tmp = 0.0
	if (y <= 0.0039)
		tmp = Float64(y * Float64(t_0 * t_1));
	elseif (y <= 9.5e+102)
		tmp = sinh(y);
	else
		tmp = Float64(t_1 * Float64(y * t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + (0.16666666666666666 * (y * y));
	t_1 = sin(x) / x;
	tmp = 0.0;
	if (y <= 0.0039)
		tmp = y * (t_0 * t_1);
	elseif (y <= 9.5e+102)
		tmp = sinh(y);
	else
		tmp = t_1 * (y * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[y, 0.0039], N[(y * N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 9.5e+102], N[Sinh[y], $MachinePrecision], N[(t$95$1 * N[(y * t$95$0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < 0.0038999999999999998

   1. Initial program 87.0%

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

      1. associate-/l* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin x, \color{blue}{\left(\frac{\sinh y}{x}\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\frac{\color{blue}{\sinh y}}{x}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\sinh y, \color{blue}{x}\right)\right) \]

      5. sinh-lowering-sinh.f64 99.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]

   3. Simplified 99.9%

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

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left(\frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{\sin x}{x}\right)} \]
   6. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto y \cdot \left(\frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x}\right) + \color{blue}{y \cdot \frac{\sin x}{x}} \]

      2. fma-define N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x}}, y \cdot \frac{\sin x}{x}\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{1}{6} \cdot \left({y}^{2} \cdot \color{blue}{\frac{\sin x}{x}}\right), y \cdot \frac{\sin x}{x}\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(y, \left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \color{blue}{\frac{\sin x}{x}}, y \cdot \frac{\sin x}{x}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, \left({y}^{2} \cdot \frac{1}{6}\right) \cdot \frac{\color{blue}{\sin x}}{x}, y \cdot \frac{\sin x}{x}\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(y, {y}^{2} \cdot \color{blue}{\left(\frac{1}{6} \cdot \frac{\sin x}{x}\right)}, y \cdot \frac{\sin x}{x}\right) \]

      7. fma-define N/A

         \[\leadsto y \cdot \left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right)\right) + \color{blue}{y \cdot \frac{\sin x}{x}} \]

      8. distribute-lft-in N/A

         \[\leadsto y \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right)} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right)}\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left({y}^{2} \cdot \frac{1}{6}\right) \cdot \frac{\sin x}{x} + \frac{\color{blue}{\sin x}}{x}\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \frac{\sin x}{x} + \frac{\sin \color{blue}{x}}{x}\right)\right) \]

      12. distribute-lft1-in N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left(\frac{1}{6} \cdot {y}^{2} + 1\right) \cdot \color{blue}{\frac{\sin x}{x}}\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left(1 + \frac{1}{6} \cdot {y}^{2}\right) \cdot \frac{\color{blue}{\sin x}}{x}\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left(1 + \frac{1}{6} \cdot {y}^{2}\right), \color{blue}{\left(\frac{\sin x}{x}\right)}\right)\right) \]

   7. Simplified 86.9%

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

   if 0.0038999999999999998 < y < 9.4999999999999992e102

   1. Initial program 100.0%

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

      1. associate-/l* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin x, \color{blue}{\left(\frac{\sinh y}{x}\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\frac{\color{blue}{\sinh y}}{x}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\sinh y, \color{blue}{x}\right)\right) \]

      5. sinh-lowering-sinh.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{x}, \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 75.0%

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

      1. *-commutative N/A

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

      2. div-inv N/A

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

      3. associate-*l* N/A

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

      4. lft-mult-inverse N/A

         \[\leadsto \sinh y \cdot 1 \]

      5. *-rgt-identity N/A

         \[\leadsto \sinh y \]

      6. sinh-lowering-sinh.f64 75.0%

         \[\leadsto \mathsf{sinh.f64}\left(y\right) \]

   9. Applied egg-rr 75.0%

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

   if 9.4999999999999992e102 < y

   1. Initial program 100.0%

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

      1. associate-/l* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin x, \color{blue}{\left(\frac{\sinh y}{x}\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\frac{\color{blue}{\sinh y}}{x}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\sinh y, \color{blue}{x}\right)\right) \]

      5. sinh-lowering-sinh.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left(\frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{\sin x}{x}\right)} \]
   6. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto y \cdot \left(\frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x}\right) + \color{blue}{y \cdot \frac{\sin x}{x}} \]

      2. fma-define N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x}}, y \cdot \frac{\sin x}{x}\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{1}{6} \cdot \left({y}^{2} \cdot \color{blue}{\frac{\sin x}{x}}\right), y \cdot \frac{\sin x}{x}\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(y, \left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \color{blue}{\frac{\sin x}{x}}, y \cdot \frac{\sin x}{x}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, \left({y}^{2} \cdot \frac{1}{6}\right) \cdot \frac{\color{blue}{\sin x}}{x}, y \cdot \frac{\sin x}{x}\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(y, {y}^{2} \cdot \color{blue}{\left(\frac{1}{6} \cdot \frac{\sin x}{x}\right)}, y \cdot \frac{\sin x}{x}\right) \]

      7. fma-define N/A

         \[\leadsto y \cdot \left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right)\right) + \color{blue}{y \cdot \frac{\sin x}{x}} \]

      8. distribute-lft-in N/A

         \[\leadsto y \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right)} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right)}\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left({y}^{2} \cdot \frac{1}{6}\right) \cdot \frac{\sin x}{x} + \frac{\color{blue}{\sin x}}{x}\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \frac{\sin x}{x} + \frac{\sin \color{blue}{x}}{x}\right)\right) \]

      12. distribute-lft1-in N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left(\frac{1}{6} \cdot {y}^{2} + 1\right) \cdot \color{blue}{\frac{\sin x}{x}}\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left(1 + \frac{1}{6} \cdot {y}^{2}\right) \cdot \frac{\color{blue}{\sin x}}{x}\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left(1 + \frac{1}{6} \cdot {y}^{2}\right), \color{blue}{\left(\frac{\sin x}{x}\right)}\right)\right) \]

   7. Simplified 92.2%

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

      1. associate-*r* N/A

         \[\leadsto \left(y \cdot \left(1 + \frac{1}{6} \cdot \left(y \cdot y\right)\right)\right) \cdot \color{blue}{\frac{\sin x}{x}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\sin x}{x} \cdot \color{blue}{\left(y \cdot \left(1 + \frac{1}{6} \cdot \left(y \cdot y\right)\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\sin x}{x}\right), \color{blue}{\left(y \cdot \left(1 + \frac{1}{6} \cdot \left(y \cdot y\right)\right)\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\sin x, x\right), \left(\color{blue}{y} \cdot \left(1 + \frac{1}{6} \cdot \left(y \cdot y\right)\right)\right)\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(x\right), x\right), \left(y \cdot \left(1 + \frac{1}{6} \cdot \left(y \cdot y\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(x\right), x\right), \mathsf{*.f64}\left(y, \color{blue}{\left(1 + \frac{1}{6} \cdot \left(y \cdot y\right)\right)}\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(x\right), x\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{6} \cdot \left(y \cdot y\right)\right)}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(x\right), x\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \color{blue}{\left(y \cdot y\right)}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(x\right), x\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right)\right)\right)\right) \]

   9. Applied egg-rr 100.0%

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

Alternative 7: 73.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{\sinh y}{x}\\ \mathbf{if}\;x \leq 1.66 \cdot 10^{+97}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 4.3 \cdot 10^{+189}:\\ \;\;\;\;y \cdot \left(\left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \cdot \left(1 + -0.16666666666666666 \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* x (/ (sinh y) x))))
   (if (<= x 1.66e+97)
     t_0
     (if (<= x 4.3e+189)
       (*
        y
        (*
         (+ 1.0 (* 0.16666666666666666 (* y y)))
         (+ 1.0 (* -0.16666666666666666 (* x x)))))
       t_0))))

C

double code(double x, double y) {
	double t_0 = x * (sinh(y) / x);
	double tmp;
	if (x <= 1.66e+97) {
		tmp = t_0;
	} else if (x <= 4.3e+189) {
		tmp = y * ((1.0 + (0.16666666666666666 * (y * y))) * (1.0 + (-0.16666666666666666 * (x * x))));
	} else {
		tmp = 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 * (sinh(y) / x)
    if (x <= 1.66d+97) then
        tmp = t_0
    else if (x <= 4.3d+189) then
        tmp = y * ((1.0d0 + (0.16666666666666666d0 * (y * y))) * (1.0d0 + ((-0.16666666666666666d0) * (x * x))))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x * (Math.sinh(y) / x);
	double tmp;
	if (x <= 1.66e+97) {
		tmp = t_0;
	} else if (x <= 4.3e+189) {
		tmp = y * ((1.0 + (0.16666666666666666 * (y * y))) * (1.0 + (-0.16666666666666666 * (x * x))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x * (math.sinh(y) / x)
	tmp = 0
	if x <= 1.66e+97:
		tmp = t_0
	elif x <= 4.3e+189:
		tmp = y * ((1.0 + (0.16666666666666666 * (y * y))) * (1.0 + (-0.16666666666666666 * (x * x))))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x * Float64(sinh(y) / x))
	tmp = 0.0
	if (x <= 1.66e+97)
		tmp = t_0;
	elseif (x <= 4.3e+189)
		tmp = Float64(y * Float64(Float64(1.0 + Float64(0.16666666666666666 * Float64(y * y))) * Float64(1.0 + Float64(-0.16666666666666666 * Float64(x * x)))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x * (sinh(y) / x);
	tmp = 0.0;
	if (x <= 1.66e+97)
		tmp = t_0;
	elseif (x <= 4.3e+189)
		tmp = y * ((1.0 + (0.16666666666666666 * (y * y))) * (1.0 + (-0.16666666666666666 * (x * x))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x * N[(N[Sinh[y], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 1.66e+97], t$95$0, If[LessEqual[x, 4.3e+189], N[(y * N[(N[(1.0 + N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(-0.16666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.6599999999999999e97 or 4.29999999999999998e189 < x

   1. Initial program 89.1%

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

      1. associate-/l* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin x, \color{blue}{\left(\frac{\sinh y}{x}\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\frac{\color{blue}{\sinh y}}{x}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\sinh y, \color{blue}{x}\right)\right) \]

      5. sinh-lowering-sinh.f64 99.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{x}, \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 78.8%

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

   if 1.6599999999999999e97 < x < 4.29999999999999998e189

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)}, \mathsf{sinh.f64}\left(y\right)\right), x\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right), \mathsf{sinh.f64}\left(y\right)\right), x\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot {x}^{2}\right)\right)\right), \mathsf{sinh.f64}\left(y\right)\right), x\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left({x}^{2} \cdot \frac{-1}{6}\right)\right)\right), \mathsf{sinh.f64}\left(y\right)\right), x\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \frac{-1}{6}\right)\right)\right), \mathsf{sinh.f64}\left(y\right)\right), x\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(x \cdot \left(x \cdot \frac{-1}{6}\right)\right)\right)\right), \mathsf{sinh.f64}\left(y\right)\right), x\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \frac{-1}{6}\right)\right)\right)\right), \mathsf{sinh.f64}\left(y\right)\right), x\right) \]

      7. *-lowering-*.f64 50.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{-1}{6}\right)\right)\right)\right), \mathsf{sinh.f64}\left(y\right)\right), x\right) \]

   5. Simplified 50.7%

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

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left(1 + \left(\frac{-1}{6} \cdot {x}^{2} + \frac{1}{6} \cdot \left({y}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right)\right)} \]
   7. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto y \cdot \left(\left(1 + \frac{-1}{6} \cdot {x}^{2}\right) + \color{blue}{\frac{1}{6} \cdot \left({y}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)}\right) \]

      2. distribute-lft-in N/A

         \[\leadsto y \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right) + \color{blue}{y \cdot \left(\frac{1}{6} \cdot \left({y}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto y \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right) + y \cdot \left(\frac{1}{6} \cdot \left(\left(1 + \frac{-1}{6} \cdot {x}^{2}\right) \cdot \color{blue}{{y}^{2}}\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto y \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right) + y \cdot \left(\left(\frac{1}{6} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right) \cdot \color{blue}{{y}^{2}}\right) \]

      5. distribute-lft-out N/A

         \[\leadsto y \cdot \color{blue}{\left(\left(1 + \frac{-1}{6} \cdot {x}^{2}\right) + \left(\frac{1}{6} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right) \cdot {y}^{2}\right)} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(\left(1 + \frac{-1}{6} \cdot {x}^{2}\right) + \left(\frac{1}{6} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right) \cdot {y}^{2}\right)}\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(\left(1 + \frac{-1}{6} \cdot {x}^{2}\right) + \frac{1}{6} \cdot \color{blue}{\left(\left(1 + \frac{-1}{6} \cdot {x}^{2}\right) \cdot {y}^{2}\right)}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(\left(1 + \frac{-1}{6} \cdot {x}^{2}\right) + \frac{1}{6} \cdot \left({y}^{2} \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {x}^{2}\right)}\right)\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(\left(1 + \frac{-1}{6} \cdot {x}^{2}\right) + \left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {x}^{2}\right)}\right)\right) \]

      10. distribute-rgt1-in N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left(\frac{1}{6} \cdot {y}^{2} + 1\right) \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {x}^{2}\right)}\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left(1 + \frac{1}{6} \cdot {y}^{2}\right) \cdot \left(\color{blue}{1} + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right) \]

   8. Simplified 50.9%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.66 \cdot 10^{+97}:\\ \;\;\;\;x \cdot \frac{\sinh y}{x}\\ \mathbf{elif}\;x \leq 4.3 \cdot 10^{+189}:\\ \;\;\;\;y \cdot \left(\left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \cdot \left(1 + -0.16666666666666666 \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\sinh y}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.0026:\\ \;\;\;\;\sin x \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\sinh y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 0.0026) (* (sin x) (/ y x)) (* x (/ (sinh y) x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 0.0026) {
		tmp = sin(x) * (y / x);
	} else {
		tmp = x * (sinh(y) / x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 0.0026d0) then
        tmp = sin(x) * (y / x)
    else
        tmp = x * (sinh(y) / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 0.0026) {
		tmp = Math.sin(x) * (y / x);
	} else {
		tmp = x * (Math.sinh(y) / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 0.0026:
		tmp = math.sin(x) * (y / x)
	else:
		tmp = x * (math.sinh(y) / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 0.0026)
		tmp = Float64(sin(x) * Float64(y / x));
	else
		tmp = Float64(x * Float64(sinh(y) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 0.0026)
		tmp = sin(x) * (y / x);
	else
		tmp = x * (sinh(y) / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 0.0026], N[(N[Sin[x], $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[Sinh[y], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 0.0025999999999999999

   1. Initial program 87.0%

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

      1. associate-/l* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin x, \color{blue}{\left(\frac{\sinh y}{x}\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\frac{\color{blue}{\sinh y}}{x}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\sinh y, \color{blue}{x}\right)\right) \]

      5. sinh-lowering-sinh.f64 99.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]

   3. Simplified 99.9%

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

   5. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \color{blue}{\left(\frac{y}{x}\right)}\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 75.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(y, \color{blue}{x}\right)\right) \]

   7. Simplified 75.7%

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

   if 0.0025999999999999999 < y

   1. Initial program 100.0%

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

      1. associate-/l* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin x, \color{blue}{\left(\frac{\sinh y}{x}\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\frac{\color{blue}{\sinh y}}{x}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\sinh y, \color{blue}{x}\right)\right) \]

      5. sinh-lowering-sinh.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{x}, \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 69.6%

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

Alternative 9: 70.3% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{y \cdot \left(1 + \left(y \cdot y\right) \cdot \left(0.16666666666666666 + y \cdot \left(y \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right)\right)\right)}{x}\\ \mathbf{if}\;x \leq 1.66 \cdot 10^{+97}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 4.4 \cdot 10^{+189}:\\ \;\;\;\;y \cdot \left(\left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \cdot \left(1 + -0.16666666666666666 \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (*
          x
          (/
           (*
            y
            (+
             1.0
             (*
              (* y y)
              (+
               0.16666666666666666
               (*
                y
                (*
                 y
                 (+
                  0.008333333333333333
                  (* (* y y) 0.0001984126984126984))))))))
           x))))
   (if (<= x 1.66e+97)
     t_0
     (if (<= x 4.4e+189)
       (*
        y
        (*
         (+ 1.0 (* 0.16666666666666666 (* y y)))
         (+ 1.0 (* -0.16666666666666666 (* x x)))))
       t_0))))

C

double code(double x, double y) {
	double t_0 = x * ((y * (1.0 + ((y * y) * (0.16666666666666666 + (y * (y * (0.008333333333333333 + ((y * y) * 0.0001984126984126984)))))))) / x);
	double tmp;
	if (x <= 1.66e+97) {
		tmp = t_0;
	} else if (x <= 4.4e+189) {
		tmp = y * ((1.0 + (0.16666666666666666 * (y * y))) * (1.0 + (-0.16666666666666666 * (x * x))));
	} else {
		tmp = 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 * ((y * (1.0d0 + ((y * y) * (0.16666666666666666d0 + (y * (y * (0.008333333333333333d0 + ((y * y) * 0.0001984126984126984d0)))))))) / x)
    if (x <= 1.66d+97) then
        tmp = t_0
    else if (x <= 4.4d+189) then
        tmp = y * ((1.0d0 + (0.16666666666666666d0 * (y * y))) * (1.0d0 + ((-0.16666666666666666d0) * (x * x))))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x * ((y * (1.0 + ((y * y) * (0.16666666666666666 + (y * (y * (0.008333333333333333 + ((y * y) * 0.0001984126984126984)))))))) / x);
	double tmp;
	if (x <= 1.66e+97) {
		tmp = t_0;
	} else if (x <= 4.4e+189) {
		tmp = y * ((1.0 + (0.16666666666666666 * (y * y))) * (1.0 + (-0.16666666666666666 * (x * x))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x * ((y * (1.0 + ((y * y) * (0.16666666666666666 + (y * (y * (0.008333333333333333 + ((y * y) * 0.0001984126984126984)))))))) / x)
	tmp = 0
	if x <= 1.66e+97:
		tmp = t_0
	elif x <= 4.4e+189:
		tmp = y * ((1.0 + (0.16666666666666666 * (y * y))) * (1.0 + (-0.16666666666666666 * (x * x))))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x * Float64(Float64(y * Float64(1.0 + Float64(Float64(y * y) * Float64(0.16666666666666666 + Float64(y * Float64(y * Float64(0.008333333333333333 + Float64(Float64(y * y) * 0.0001984126984126984)))))))) / x))
	tmp = 0.0
	if (x <= 1.66e+97)
		tmp = t_0;
	elseif (x <= 4.4e+189)
		tmp = Float64(y * Float64(Float64(1.0 + Float64(0.16666666666666666 * Float64(y * y))) * Float64(1.0 + Float64(-0.16666666666666666 * Float64(x * x)))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x * ((y * (1.0 + ((y * y) * (0.16666666666666666 + (y * (y * (0.008333333333333333 + ((y * y) * 0.0001984126984126984)))))))) / x);
	tmp = 0.0;
	if (x <= 1.66e+97)
		tmp = t_0;
	elseif (x <= 4.4e+189)
		tmp = y * ((1.0 + (0.16666666666666666 * (y * y))) * (1.0 + (-0.16666666666666666 * (x * x))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x * N[(N[(y * N[(1.0 + N[(N[(y * y), $MachinePrecision] * N[(0.16666666666666666 + N[(y * N[(y * N[(0.008333333333333333 + N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 1.66e+97], t$95$0, If[LessEqual[x, 4.4e+189], N[(y * N[(N[(1.0 + N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(-0.16666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.6599999999999999e97 or 4.4000000000000001e189 < x

   1. Initial program 89.1%

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

      1. associate-/l* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin x, \color{blue}{\left(\frac{\sinh y}{x}\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\frac{\color{blue}{\sinh y}}{x}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\sinh y, \color{blue}{x}\right)\right) \]

      5. sinh-lowering-sinh.f64 99.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{x}, \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 78.8%

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

   8. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\color{blue}{\left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)}, x\right)\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right), x\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \left({y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right), x\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({y}^{2}\right), \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right), x\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(y \cdot y\right), \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right), x\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right), x\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \left({y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right)\right), x\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \left(\left(y \cdot y\right) \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right)\right), x\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \left(y \cdot \left(y \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right)\right)\right), x\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \left(y \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right)\right)\right), x\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right)\right)\right), x\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{120}, \left(\frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right)\right)\right)\right), x\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{120}, \left({y}^{2} \cdot \frac{1}{5040}\right)\right)\right)\right)\right)\right)\right)\right), x\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left({y}^{2}\right), \frac{1}{5040}\right)\right)\right)\right)\right)\right)\right)\right), x\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left(y \cdot y\right), \frac{1}{5040}\right)\right)\right)\right)\right)\right)\right)\right), x\right)\right) \]

      15. *-lowering-*.f64 74.4%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{5040}\right)\right)\right)\right)\right)\right)\right)\right), x\right)\right) \]

   10. Simplified 74.4%

       \[\leadsto x \cdot \frac{\color{blue}{y \cdot \left(1 + \left(y \cdot y\right) \cdot \left(0.16666666666666666 + y \cdot \left(y \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right)\right)\right)}}{x} \]

   if 1.6599999999999999e97 < x < 4.4000000000000001e189

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)}, \mathsf{sinh.f64}\left(y\right)\right), x\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right), \mathsf{sinh.f64}\left(y\right)\right), x\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot {x}^{2}\right)\right)\right), \mathsf{sinh.f64}\left(y\right)\right), x\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left({x}^{2} \cdot \frac{-1}{6}\right)\right)\right), \mathsf{sinh.f64}\left(y\right)\right), x\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \frac{-1}{6}\right)\right)\right), \mathsf{sinh.f64}\left(y\right)\right), x\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(x \cdot \left(x \cdot \frac{-1}{6}\right)\right)\right)\right), \mathsf{sinh.f64}\left(y\right)\right), x\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \frac{-1}{6}\right)\right)\right)\right), \mathsf{sinh.f64}\left(y\right)\right), x\right) \]

      7. *-lowering-*.f64 50.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{-1}{6}\right)\right)\right)\right), \mathsf{sinh.f64}\left(y\right)\right), x\right) \]

   5. Simplified 50.7%

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

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left(1 + \left(\frac{-1}{6} \cdot {x}^{2} + \frac{1}{6} \cdot \left({y}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right)\right)} \]
   7. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto y \cdot \left(\left(1 + \frac{-1}{6} \cdot {x}^{2}\right) + \color{blue}{\frac{1}{6} \cdot \left({y}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)}\right) \]

      2. distribute-lft-in N/A

         \[\leadsto y \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right) + \color{blue}{y \cdot \left(\frac{1}{6} \cdot \left({y}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto y \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right) + y \cdot \left(\frac{1}{6} \cdot \left(\left(1 + \frac{-1}{6} \cdot {x}^{2}\right) \cdot \color{blue}{{y}^{2}}\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto y \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right) + y \cdot \left(\left(\frac{1}{6} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right) \cdot \color{blue}{{y}^{2}}\right) \]

      5. distribute-lft-out N/A

         \[\leadsto y \cdot \color{blue}{\left(\left(1 + \frac{-1}{6} \cdot {x}^{2}\right) + \left(\frac{1}{6} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right) \cdot {y}^{2}\right)} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(\left(1 + \frac{-1}{6} \cdot {x}^{2}\right) + \left(\frac{1}{6} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right) \cdot {y}^{2}\right)}\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(\left(1 + \frac{-1}{6} \cdot {x}^{2}\right) + \frac{1}{6} \cdot \color{blue}{\left(\left(1 + \frac{-1}{6} \cdot {x}^{2}\right) \cdot {y}^{2}\right)}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(\left(1 + \frac{-1}{6} \cdot {x}^{2}\right) + \frac{1}{6} \cdot \left({y}^{2} \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {x}^{2}\right)}\right)\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(\left(1 + \frac{-1}{6} \cdot {x}^{2}\right) + \left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {x}^{2}\right)}\right)\right) \]

      10. distribute-rgt1-in N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left(\frac{1}{6} \cdot {y}^{2} + 1\right) \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {x}^{2}\right)}\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left(1 + \frac{1}{6} \cdot {y}^{2}\right) \cdot \left(\color{blue}{1} + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right) \]

   8. Simplified 50.9%

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

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.66 \cdot 10^{+97}:\\ \;\;\;\;x \cdot \frac{y \cdot \left(1 + \left(y \cdot y\right) \cdot \left(0.16666666666666666 + y \cdot \left(y \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right)\right)\right)}{x}\\ \mathbf{elif}\;x \leq 4.4 \cdot 10^{+189}:\\ \;\;\;\;y \cdot \left(\left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \cdot \left(1 + -0.16666666666666666 \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y \cdot \left(1 + \left(y \cdot y\right) \cdot \left(0.16666666666666666 + y \cdot \left(y \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right)\right)\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 69.5% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y \cdot y\right) \cdot 0.008333333333333333\\ \mathbf{if}\;x \leq 1.66 \cdot 10^{+97}:\\ \;\;\;\;x \cdot \frac{y \cdot \left(1 + y \cdot \left(y \cdot \left(0.16666666666666666 + t\_0\right)\right)\right)}{x}\\ \mathbf{elif}\;x \leq 4.1 \cdot 10^{+189}:\\ \;\;\;\;y \cdot \left(\left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \cdot \left(1 + -0.16666666666666666 \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(x \cdot \left(y \cdot \left(y \cdot t\_0\right)\right)\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* y y) 0.008333333333333333)))
   (if (<= x 1.66e+97)
     (* x (/ (* y (+ 1.0 (* y (* y (+ 0.16666666666666666 t_0))))) x))
     (if (<= x 4.1e+189)
       (*
        y
        (*
         (+ 1.0 (* 0.16666666666666666 (* y y)))
         (+ 1.0 (* -0.16666666666666666 (* x x)))))
       (/ (* y (* x (* y (* y t_0)))) x)))))

C

double code(double x, double y) {
	double t_0 = (y * y) * 0.008333333333333333;
	double tmp;
	if (x <= 1.66e+97) {
		tmp = x * ((y * (1.0 + (y * (y * (0.16666666666666666 + t_0))))) / x);
	} else if (x <= 4.1e+189) {
		tmp = y * ((1.0 + (0.16666666666666666 * (y * y))) * (1.0 + (-0.16666666666666666 * (x * x))));
	} else {
		tmp = (y * (x * (y * (y * t_0)))) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (y * y) * 0.008333333333333333d0
    if (x <= 1.66d+97) then
        tmp = x * ((y * (1.0d0 + (y * (y * (0.16666666666666666d0 + t_0))))) / x)
    else if (x <= 4.1d+189) then
        tmp = y * ((1.0d0 + (0.16666666666666666d0 * (y * y))) * (1.0d0 + ((-0.16666666666666666d0) * (x * x))))
    else
        tmp = (y * (x * (y * (y * t_0)))) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (y * y) * 0.008333333333333333;
	double tmp;
	if (x <= 1.66e+97) {
		tmp = x * ((y * (1.0 + (y * (y * (0.16666666666666666 + t_0))))) / x);
	} else if (x <= 4.1e+189) {
		tmp = y * ((1.0 + (0.16666666666666666 * (y * y))) * (1.0 + (-0.16666666666666666 * (x * x))));
	} else {
		tmp = (y * (x * (y * (y * t_0)))) / x;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (y * y) * 0.008333333333333333
	tmp = 0
	if x <= 1.66e+97:
		tmp = x * ((y * (1.0 + (y * (y * (0.16666666666666666 + t_0))))) / x)
	elif x <= 4.1e+189:
		tmp = y * ((1.0 + (0.16666666666666666 * (y * y))) * (1.0 + (-0.16666666666666666 * (x * x))))
	else:
		tmp = (y * (x * (y * (y * t_0)))) / x
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(y * y) * 0.008333333333333333)
	tmp = 0.0
	if (x <= 1.66e+97)
		tmp = Float64(x * Float64(Float64(y * Float64(1.0 + Float64(y * Float64(y * Float64(0.16666666666666666 + t_0))))) / x));
	elseif (x <= 4.1e+189)
		tmp = Float64(y * Float64(Float64(1.0 + Float64(0.16666666666666666 * Float64(y * y))) * Float64(1.0 + Float64(-0.16666666666666666 * Float64(x * x)))));
	else
		tmp = Float64(Float64(y * Float64(x * Float64(y * Float64(y * t_0)))) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (y * y) * 0.008333333333333333;
	tmp = 0.0;
	if (x <= 1.66e+97)
		tmp = x * ((y * (1.0 + (y * (y * (0.16666666666666666 + t_0))))) / x);
	elseif (x <= 4.1e+189)
		tmp = y * ((1.0 + (0.16666666666666666 * (y * y))) * (1.0 + (-0.16666666666666666 * (x * x))));
	else
		tmp = (y * (x * (y * (y * t_0)))) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * y), $MachinePrecision] * 0.008333333333333333), $MachinePrecision]}, If[LessEqual[x, 1.66e+97], N[(x * N[(N[(y * N[(1.0 + N[(y * N[(y * N[(0.16666666666666666 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.1e+189], N[(y * N[(N[(1.0 + N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(-0.16666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * N[(x * N[(y * N[(y * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.6599999999999999e97

   1. Initial program 88.0%

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

      1. associate-/l* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin x, \color{blue}{\left(\frac{\sinh y}{x}\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\frac{\color{blue}{\sinh y}}{x}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\sinh y, \color{blue}{x}\right)\right) \]

      5. sinh-lowering-sinh.f64 99.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{x}, \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 78.9%

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

   8. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\color{blue}{\left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)}, x\right)\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right), x\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right), x\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \left(\left(y \cdot y\right) \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right), x\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \left(y \cdot \left(y \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right)\right), x\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(y \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right)\right), x\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right)\right), x\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{6}, \left(\frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right)\right)\right), x\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{6}, \left({y}^{2} \cdot \frac{1}{120}\right)\right)\right)\right)\right)\right), x\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left({y}^{2}\right), \frac{1}{120}\right)\right)\right)\right)\right)\right), x\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left(y \cdot y\right), \frac{1}{120}\right)\right)\right)\right)\right)\right), x\right)\right) \]

      11. *-lowering-*.f64 72.3%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right)\right), x\right)\right) \]

   10. Simplified 72.3%

       \[\leadsto x \cdot \frac{\color{blue}{y \cdot \left(1 + y \cdot \left(y \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right)}}{x} \]

   if 1.6599999999999999e97 < x < 4.1000000000000002e189

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)}, \mathsf{sinh.f64}\left(y\right)\right), x\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right), \mathsf{sinh.f64}\left(y\right)\right), x\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot {x}^{2}\right)\right)\right), \mathsf{sinh.f64}\left(y\right)\right), x\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left({x}^{2} \cdot \frac{-1}{6}\right)\right)\right), \mathsf{sinh.f64}\left(y\right)\right), x\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \frac{-1}{6}\right)\right)\right), \mathsf{sinh.f64}\left(y\right)\right), x\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(x \cdot \left(x \cdot \frac{-1}{6}\right)\right)\right)\right), \mathsf{sinh.f64}\left(y\right)\right), x\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \frac{-1}{6}\right)\right)\right)\right), \mathsf{sinh.f64}\left(y\right)\right), x\right) \]

      7. *-lowering-*.f64 50.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{-1}{6}\right)\right)\right)\right), \mathsf{sinh.f64}\left(y\right)\right), x\right) \]

   5. Simplified 50.7%

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

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left(1 + \left(\frac{-1}{6} \cdot {x}^{2} + \frac{1}{6} \cdot \left({y}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right)\right)} \]
   7. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto y \cdot \left(\left(1 + \frac{-1}{6} \cdot {x}^{2}\right) + \color{blue}{\frac{1}{6} \cdot \left({y}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)}\right) \]

      2. distribute-lft-in N/A

         \[\leadsto y \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right) + \color{blue}{y \cdot \left(\frac{1}{6} \cdot \left({y}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto y \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right) + y \cdot \left(\frac{1}{6} \cdot \left(\left(1 + \frac{-1}{6} \cdot {x}^{2}\right) \cdot \color{blue}{{y}^{2}}\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto y \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right) + y \cdot \left(\left(\frac{1}{6} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right) \cdot \color{blue}{{y}^{2}}\right) \]

      5. distribute-lft-out N/A

         \[\leadsto y \cdot \color{blue}{\left(\left(1 + \frac{-1}{6} \cdot {x}^{2}\right) + \left(\frac{1}{6} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right) \cdot {y}^{2}\right)} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(\left(1 + \frac{-1}{6} \cdot {x}^{2}\right) + \left(\frac{1}{6} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right) \cdot {y}^{2}\right)}\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(\left(1 + \frac{-1}{6} \cdot {x}^{2}\right) + \frac{1}{6} \cdot \color{blue}{\left(\left(1 + \frac{-1}{6} \cdot {x}^{2}\right) \cdot {y}^{2}\right)}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(\left(1 + \frac{-1}{6} \cdot {x}^{2}\right) + \frac{1}{6} \cdot \left({y}^{2} \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {x}^{2}\right)}\right)\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(\left(1 + \frac{-1}{6} \cdot {x}^{2}\right) + \left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {x}^{2}\right)}\right)\right) \]

      10. distribute-rgt1-in N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left(\frac{1}{6} \cdot {y}^{2} + 1\right) \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {x}^{2}\right)}\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left(1 + \frac{1}{6} \cdot {y}^{2}\right) \cdot \left(\color{blue}{1} + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right) \]

   8. Simplified 50.9%

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

   if 4.1000000000000002e189 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(y \cdot \left(\sin x + {y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{6} \cdot \sin x\right)\right)\right)}, x\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\sin x + {y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{6} \cdot \sin x\right)\right)\right), x\right) \]

      2. *-rgt-identity N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\sin x \cdot 1 + {y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{6} \cdot \sin x\right)\right)\right), x\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\sin x \cdot 1 + {y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x + \frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right)\right)\right)\right), x\right) \]

      4. distribute-lft-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\sin x \cdot 1 + \left({y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x\right) + {y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right)\right)\right)\right)\right), x\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\sin x \cdot 1 + \left(\left({y}^{2} \cdot \frac{1}{6}\right) \cdot \sin x + {y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right)\right)\right)\right)\right), x\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\sin x \cdot 1 + \left(\left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \sin x + {y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right)\right)\right)\right)\right), x\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\sin x \cdot 1 + \left(\left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \sin x + {y}^{2} \cdot \left(\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot \sin x\right)\right)\right)\right), x\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\sin x \cdot 1 + \left(\left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \sin x + \left({y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2}\right)\right) \cdot \sin x\right)\right)\right), x\right) \]

      9. distribute-rgt-out N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\sin x \cdot 1 + \sin x \cdot \left(\frac{1}{6} \cdot {y}^{2} + {y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right), x\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\sin x \cdot 1 + \sin x \cdot \left({y}^{2} \cdot \frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right), x\right) \]

      11. distribute-lft-in N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\sin x \cdot 1 + \sin x \cdot \left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right), x\right) \]

   5. Simplified 86.7%

      \[\leadsto \frac{\color{blue}{y \cdot \left(\sin x \cdot \left(1 + y \cdot \left(y \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right)\right)}}{x} \]

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\color{blue}{x}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right)\right)\right), x\right) \]
   7. Step-by-step derivation

   8. Simplified 38.9%

      \[\leadsto \frac{y \cdot \left(\color{blue}{x} \cdot \left(1 + y \cdot \left(y \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right)\right)}{x} \]

   9. Taylor expanded in y around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{120} \cdot {y}^{4}\right)}\right)\right), x\right) \]
   10. Step-by-step derivation

       1. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \left(\frac{1}{120} \cdot {y}^{\left(2 \cdot 2\right)}\right)\right)\right), x\right) \]

       2. pow-sqr N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \left(\frac{1}{120} \cdot \left({y}^{2} \cdot {y}^{2}\right)\right)\right)\right), x\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \left(\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot {y}^{2}\right)\right)\right), x\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \left({y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right), x\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \left(\left(y \cdot y\right) \cdot \left(\frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right), x\right) \]

       6. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \left(y \cdot \left(y \cdot \left(\frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right)\right), x\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \left(y \cdot \left(\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot y\right)\right)\right)\right), x\right) \]

       8. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \left(y \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot y\right)\right)\right)\right)\right), x\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \left(y \cdot \left(\frac{1}{120} \cdot \left(\left(y \cdot y\right) \cdot y\right)\right)\right)\right)\right), x\right) \]

       10. unpow3 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \left(y \cdot \left(\frac{1}{120} \cdot {y}^{3}\right)\right)\right)\right), x\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \left(\frac{1}{120} \cdot {y}^{3}\right)\right)\right)\right), x\right) \]

       12. unpow3 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \left(\frac{1}{120} \cdot \left(\left(y \cdot y\right) \cdot y\right)\right)\right)\right)\right), x\right) \]

       13. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \left(\frac{1}{120} \cdot \left({y}^{2} \cdot y\right)\right)\right)\right)\right), x\right) \]

       14. associate-*r* N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \left(\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot y\right)\right)\right)\right), x\right) \]

       15. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \left(y \cdot \left(\frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right)\right), x\right) \]

       16. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left(\frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right)\right), x\right) \]

       17. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left({y}^{2} \cdot \frac{1}{120}\right)\right)\right)\right)\right), x\right) \]

       18. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left({y}^{2}\right), \frac{1}{120}\right)\right)\right)\right)\right), x\right) \]

       19. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left(y \cdot y\right), \frac{1}{120}\right)\right)\right)\right)\right), x\right) \]

       20. *-lowering-*.f64 77.6%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right), x\right) \]

   11. Simplified 77.6%

       \[\leadsto \frac{y \cdot \left(x \cdot \color{blue}{\left(y \cdot \left(y \cdot \left(\left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right)}\right)}{x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.66 \cdot 10^{+97}:\\ \;\;\;\;x \cdot \frac{y \cdot \left(1 + y \cdot \left(y \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right)}{x}\\ \mathbf{elif}\;x \leq 4.1 \cdot 10^{+189}:\\ \;\;\;\;y \cdot \left(\left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \cdot \left(1 + -0.16666666666666666 \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(x \cdot \left(y \cdot \left(y \cdot \left(\left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right)\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 66.9% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\\ \mathbf{if}\;x \leq 1.66 \cdot 10^{+97}:\\ \;\;\;\;x \cdot \frac{y \cdot t\_0}{x}\\ \mathbf{elif}\;x \leq 4.35 \cdot 10^{+189}:\\ \;\;\;\;y \cdot \left(t\_0 \cdot \left(1 + -0.16666666666666666 \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(x \cdot \left(y \cdot \left(y \cdot \left(\left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right)\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* 0.16666666666666666 (* y y)))))
   (if (<= x 1.66e+97)
     (* x (/ (* y t_0) x))
     (if (<= x 4.35e+189)
       (* y (* t_0 (+ 1.0 (* -0.16666666666666666 (* x x)))))
       (/ (* y (* x (* y (* y (* (* y y) 0.008333333333333333))))) x)))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (0.16666666666666666 * (y * y));
	double tmp;
	if (x <= 1.66e+97) {
		tmp = x * ((y * t_0) / x);
	} else if (x <= 4.35e+189) {
		tmp = y * (t_0 * (1.0 + (-0.16666666666666666 * (x * x))));
	} else {
		tmp = (y * (x * (y * (y * ((y * y) * 0.008333333333333333))))) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + (0.16666666666666666d0 * (y * y))
    if (x <= 1.66d+97) then
        tmp = x * ((y * t_0) / x)
    else if (x <= 4.35d+189) then
        tmp = y * (t_0 * (1.0d0 + ((-0.16666666666666666d0) * (x * x))))
    else
        tmp = (y * (x * (y * (y * ((y * y) * 0.008333333333333333d0))))) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 + (0.16666666666666666 * (y * y));
	double tmp;
	if (x <= 1.66e+97) {
		tmp = x * ((y * t_0) / x);
	} else if (x <= 4.35e+189) {
		tmp = y * (t_0 * (1.0 + (-0.16666666666666666 * (x * x))));
	} else {
		tmp = (y * (x * (y * (y * ((y * y) * 0.008333333333333333))))) / x;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + (0.16666666666666666 * (y * y))
	tmp = 0
	if x <= 1.66e+97:
		tmp = x * ((y * t_0) / x)
	elif x <= 4.35e+189:
		tmp = y * (t_0 * (1.0 + (-0.16666666666666666 * (x * x))))
	else:
		tmp = (y * (x * (y * (y * ((y * y) * 0.008333333333333333))))) / x
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(0.16666666666666666 * Float64(y * y)))
	tmp = 0.0
	if (x <= 1.66e+97)
		tmp = Float64(x * Float64(Float64(y * t_0) / x));
	elseif (x <= 4.35e+189)
		tmp = Float64(y * Float64(t_0 * Float64(1.0 + Float64(-0.16666666666666666 * Float64(x * x)))));
	else
		tmp = Float64(Float64(y * Float64(x * Float64(y * Float64(y * Float64(Float64(y * y) * 0.008333333333333333))))) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + (0.16666666666666666 * (y * y));
	tmp = 0.0;
	if (x <= 1.66e+97)
		tmp = x * ((y * t_0) / x);
	elseif (x <= 4.35e+189)
		tmp = y * (t_0 * (1.0 + (-0.16666666666666666 * (x * x))));
	else
		tmp = (y * (x * (y * (y * ((y * y) * 0.008333333333333333))))) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 1.66e+97], N[(x * N[(N[(y * t$95$0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.35e+189], N[(y * N[(t$95$0 * N[(1.0 + N[(-0.16666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * N[(x * N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.6599999999999999e97

   1. Initial program 88.0%

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

      1. associate-/l* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin x, \color{blue}{\left(\frac{\sinh y}{x}\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\frac{\color{blue}{\sinh y}}{x}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\sinh y, \color{blue}{x}\right)\right) \]

      5. sinh-lowering-sinh.f64 99.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{x}, \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 78.9%

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

   8. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\color{blue}{\left(y \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)}, x\right)\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right), x\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \left(\frac{1}{6} \cdot {y}^{2}\right)\right)\right), x\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \left({y}^{2}\right)\right)\right)\right), x\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \left(y \cdot y\right)\right)\right)\right), x\right)\right) \]

      5. *-lowering-*.f64 69.6%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, y\right)\right)\right)\right), x\right)\right) \]

   10. Simplified 69.6%

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

   if 1.6599999999999999e97 < x < 4.35e189

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)}, \mathsf{sinh.f64}\left(y\right)\right), x\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right), \mathsf{sinh.f64}\left(y\right)\right), x\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot {x}^{2}\right)\right)\right), \mathsf{sinh.f64}\left(y\right)\right), x\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left({x}^{2} \cdot \frac{-1}{6}\right)\right)\right), \mathsf{sinh.f64}\left(y\right)\right), x\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \frac{-1}{6}\right)\right)\right), \mathsf{sinh.f64}\left(y\right)\right), x\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(x \cdot \left(x \cdot \frac{-1}{6}\right)\right)\right)\right), \mathsf{sinh.f64}\left(y\right)\right), x\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \frac{-1}{6}\right)\right)\right)\right), \mathsf{sinh.f64}\left(y\right)\right), x\right) \]

      7. *-lowering-*.f64 50.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{-1}{6}\right)\right)\right)\right), \mathsf{sinh.f64}\left(y\right)\right), x\right) \]

   5. Simplified 50.7%

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

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left(1 + \left(\frac{-1}{6} \cdot {x}^{2} + \frac{1}{6} \cdot \left({y}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right)\right)} \]
   7. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto y \cdot \left(\left(1 + \frac{-1}{6} \cdot {x}^{2}\right) + \color{blue}{\frac{1}{6} \cdot \left({y}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)}\right) \]

      2. distribute-lft-in N/A

         \[\leadsto y \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right) + \color{blue}{y \cdot \left(\frac{1}{6} \cdot \left({y}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto y \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right) + y \cdot \left(\frac{1}{6} \cdot \left(\left(1 + \frac{-1}{6} \cdot {x}^{2}\right) \cdot \color{blue}{{y}^{2}}\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto y \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right) + y \cdot \left(\left(\frac{1}{6} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right) \cdot \color{blue}{{y}^{2}}\right) \]

      5. distribute-lft-out N/A

         \[\leadsto y \cdot \color{blue}{\left(\left(1 + \frac{-1}{6} \cdot {x}^{2}\right) + \left(\frac{1}{6} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right) \cdot {y}^{2}\right)} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(\left(1 + \frac{-1}{6} \cdot {x}^{2}\right) + \left(\frac{1}{6} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right) \cdot {y}^{2}\right)}\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(\left(1 + \frac{-1}{6} \cdot {x}^{2}\right) + \frac{1}{6} \cdot \color{blue}{\left(\left(1 + \frac{-1}{6} \cdot {x}^{2}\right) \cdot {y}^{2}\right)}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(\left(1 + \frac{-1}{6} \cdot {x}^{2}\right) + \frac{1}{6} \cdot \left({y}^{2} \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {x}^{2}\right)}\right)\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(\left(1 + \frac{-1}{6} \cdot {x}^{2}\right) + \left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {x}^{2}\right)}\right)\right) \]

      10. distribute-rgt1-in N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left(\frac{1}{6} \cdot {y}^{2} + 1\right) \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {x}^{2}\right)}\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left(1 + \frac{1}{6} \cdot {y}^{2}\right) \cdot \left(\color{blue}{1} + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right) \]

   8. Simplified 50.9%

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

   if 4.35e189 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(y \cdot \left(\sin x + {y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{6} \cdot \sin x\right)\right)\right)}, x\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\sin x + {y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{6} \cdot \sin x\right)\right)\right), x\right) \]

      2. *-rgt-identity N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\sin x \cdot 1 + {y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{6} \cdot \sin x\right)\right)\right), x\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\sin x \cdot 1 + {y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x + \frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right)\right)\right)\right), x\right) \]

      4. distribute-lft-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\sin x \cdot 1 + \left({y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x\right) + {y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right)\right)\right)\right)\right), x\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\sin x \cdot 1 + \left(\left({y}^{2} \cdot \frac{1}{6}\right) \cdot \sin x + {y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right)\right)\right)\right)\right), x\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\sin x \cdot 1 + \left(\left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \sin x + {y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right)\right)\right)\right)\right), x\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\sin x \cdot 1 + \left(\left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \sin x + {y}^{2} \cdot \left(\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot \sin x\right)\right)\right)\right), x\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\sin x \cdot 1 + \left(\left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \sin x + \left({y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2}\right)\right) \cdot \sin x\right)\right)\right), x\right) \]

      9. distribute-rgt-out N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\sin x \cdot 1 + \sin x \cdot \left(\frac{1}{6} \cdot {y}^{2} + {y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right), x\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\sin x \cdot 1 + \sin x \cdot \left({y}^{2} \cdot \frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right), x\right) \]

      11. distribute-lft-in N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\sin x \cdot 1 + \sin x \cdot \left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right), x\right) \]

   5. Simplified 86.7%

      \[\leadsto \frac{\color{blue}{y \cdot \left(\sin x \cdot \left(1 + y \cdot \left(y \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right)\right)}}{x} \]

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\color{blue}{x}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right)\right)\right), x\right) \]
   7. Step-by-step derivation

   8. Simplified 38.9%

      \[\leadsto \frac{y \cdot \left(\color{blue}{x} \cdot \left(1 + y \cdot \left(y \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right)\right)}{x} \]

   9. Taylor expanded in y around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{120} \cdot {y}^{4}\right)}\right)\right), x\right) \]
   10. Step-by-step derivation

       1. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \left(\frac{1}{120} \cdot {y}^{\left(2 \cdot 2\right)}\right)\right)\right), x\right) \]

       2. pow-sqr N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \left(\frac{1}{120} \cdot \left({y}^{2} \cdot {y}^{2}\right)\right)\right)\right), x\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \left(\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot {y}^{2}\right)\right)\right), x\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \left({y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right), x\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \left(\left(y \cdot y\right) \cdot \left(\frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right), x\right) \]

       6. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \left(y \cdot \left(y \cdot \left(\frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right)\right), x\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \left(y \cdot \left(\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot y\right)\right)\right)\right), x\right) \]

       8. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \left(y \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot y\right)\right)\right)\right)\right), x\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \left(y \cdot \left(\frac{1}{120} \cdot \left(\left(y \cdot y\right) \cdot y\right)\right)\right)\right)\right), x\right) \]

       10. unpow3 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \left(y \cdot \left(\frac{1}{120} \cdot {y}^{3}\right)\right)\right)\right), x\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \left(\frac{1}{120} \cdot {y}^{3}\right)\right)\right)\right), x\right) \]

       12. unpow3 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \left(\frac{1}{120} \cdot \left(\left(y \cdot y\right) \cdot y\right)\right)\right)\right)\right), x\right) \]

       13. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \left(\frac{1}{120} \cdot \left({y}^{2} \cdot y\right)\right)\right)\right)\right), x\right) \]

       14. associate-*r* N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \left(\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot y\right)\right)\right)\right), x\right) \]

       15. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \left(y \cdot \left(\frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right)\right), x\right) \]

       16. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left(\frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right)\right), x\right) \]

       17. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left({y}^{2} \cdot \frac{1}{120}\right)\right)\right)\right)\right), x\right) \]

       18. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left({y}^{2}\right), \frac{1}{120}\right)\right)\right)\right)\right), x\right) \]

       19. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left(y \cdot y\right), \frac{1}{120}\right)\right)\right)\right)\right), x\right) \]

       20. *-lowering-*.f64 77.6%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right), x\right) \]

   11. Simplified 77.6%

       \[\leadsto \frac{y \cdot \left(x \cdot \color{blue}{\left(y \cdot \left(y \cdot \left(\left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right)}\right)}{x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.66 \cdot 10^{+97}:\\ \;\;\;\;x \cdot \frac{y \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right)}{x}\\ \mathbf{elif}\;x \leq 4.35 \cdot 10^{+189}:\\ \;\;\;\;y \cdot \left(\left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \cdot \left(1 + -0.16666666666666666 \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(x \cdot \left(y \cdot \left(y \cdot \left(\left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right)\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 67.0% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.66 \cdot 10^{+97}:\\ \;\;\;\;x \cdot \frac{y \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right)}{x}\\ \mathbf{elif}\;x \leq 4.4 \cdot 10^{+189}:\\ \;\;\;\;\frac{\left(1 + x \cdot \left(x \cdot -0.16666666666666666\right)\right) \cdot \left(x \cdot y\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(x \cdot \left(y \cdot \left(y \cdot \left(\left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right)\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1.66e+97)
   (* x (/ (* y (+ 1.0 (* 0.16666666666666666 (* y y)))) x))
   (if (<= x 4.4e+189)
     (/ (* (+ 1.0 (* x (* x -0.16666666666666666))) (* x y)) x)
     (/ (* y (* x (* y (* y (* (* y y) 0.008333333333333333))))) x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.66e+97) {
		tmp = x * ((y * (1.0 + (0.16666666666666666 * (y * y)))) / x);
	} else if (x <= 4.4e+189) {
		tmp = ((1.0 + (x * (x * -0.16666666666666666))) * (x * y)) / x;
	} else {
		tmp = (y * (x * (y * (y * ((y * y) * 0.008333333333333333))))) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 1.66d+97) then
        tmp = x * ((y * (1.0d0 + (0.16666666666666666d0 * (y * y)))) / x)
    else if (x <= 4.4d+189) then
        tmp = ((1.0d0 + (x * (x * (-0.16666666666666666d0)))) * (x * y)) / x
    else
        tmp = (y * (x * (y * (y * ((y * y) * 0.008333333333333333d0))))) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 1.66e+97) {
		tmp = x * ((y * (1.0 + (0.16666666666666666 * (y * y)))) / x);
	} else if (x <= 4.4e+189) {
		tmp = ((1.0 + (x * (x * -0.16666666666666666))) * (x * y)) / x;
	} else {
		tmp = (y * (x * (y * (y * ((y * y) * 0.008333333333333333))))) / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 1.66e+97:
		tmp = x * ((y * (1.0 + (0.16666666666666666 * (y * y)))) / x)
	elif x <= 4.4e+189:
		tmp = ((1.0 + (x * (x * -0.16666666666666666))) * (x * y)) / x
	else:
		tmp = (y * (x * (y * (y * ((y * y) * 0.008333333333333333))))) / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.66e+97)
		tmp = Float64(x * Float64(Float64(y * Float64(1.0 + Float64(0.16666666666666666 * Float64(y * y)))) / x));
	elseif (x <= 4.4e+189)
		tmp = Float64(Float64(Float64(1.0 + Float64(x * Float64(x * -0.16666666666666666))) * Float64(x * y)) / x);
	else
		tmp = Float64(Float64(y * Float64(x * Float64(y * Float64(y * Float64(Float64(y * y) * 0.008333333333333333))))) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 1.66e+97)
		tmp = x * ((y * (1.0 + (0.16666666666666666 * (y * y)))) / x);
	elseif (x <= 4.4e+189)
		tmp = ((1.0 + (x * (x * -0.16666666666666666))) * (x * y)) / x;
	else
		tmp = (y * (x * (y * (y * ((y * y) * 0.008333333333333333))))) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.66e+97], N[(x * N[(N[(y * N[(1.0 + N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.4e+189], N[(N[(N[(1.0 + N[(x * N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x * y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(y * N[(x * N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.6599999999999999e97

   1. Initial program 88.0%

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

      1. associate-/l* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin x, \color{blue}{\left(\frac{\sinh y}{x}\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\frac{\color{blue}{\sinh y}}{x}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\sinh y, \color{blue}{x}\right)\right) \]

      5. sinh-lowering-sinh.f64 99.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{x}, \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 78.9%

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

   8. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\color{blue}{\left(y \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)}, x\right)\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right), x\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \left(\frac{1}{6} \cdot {y}^{2}\right)\right)\right), x\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \left({y}^{2}\right)\right)\right)\right), x\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \left(y \cdot y\right)\right)\right)\right), x\right)\right) \]

      5. *-lowering-*.f64 69.6%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, y\right)\right)\right)\right), x\right)\right) \]

   10. Simplified 69.6%

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

   if 1.6599999999999999e97 < x < 4.4000000000000001e189

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \color{blue}{y}\right), x\right) \]
   4. Step-by-step derivation

   5. Simplified 36.1%

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

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)}, y\right), x\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right), y\right), x\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot {x}^{2}\right)\right)\right), y\right), x\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left({x}^{2} \cdot \frac{-1}{6}\right)\right)\right), y\right), x\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \frac{-1}{6}\right)\right)\right), y\right), x\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-1}{6}\right)\right)\right), y\right), x\right) \]

      6. *-lowering-*.f64 50.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{6}\right)\right)\right), y\right), x\right) \]

   8. Simplified 50.7%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\left(1 + \left(x \cdot x\right) \cdot \frac{-1}{6}\right) \cdot x\right) \cdot y\right), x\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + \left(x \cdot x\right) \cdot \frac{-1}{6}\right) \cdot \left(x \cdot y\right)\right), x\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + \left(x \cdot x\right) \cdot \frac{-1}{6}\right) \cdot \left(y \cdot x\right)\right), x\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(1 + \left(x \cdot x\right) \cdot \frac{-1}{6}\right), \left(y \cdot x\right)\right), x\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \frac{-1}{6}\right)\right), \left(y \cdot x\right)\right), x\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-1}{6}\right)\right), \left(y \cdot x\right)\right), x\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{6}\right)\right), \left(y \cdot x\right)\right), x\right) \]

      8. *-lowering-*.f64 50.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{6}\right)\right), \mathsf{*.f64}\left(y, x\right)\right), x\right) \]

   10. Applied egg-rr 50.8%

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

       1. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(x \cdot \left(x \cdot \frac{-1}{6}\right)\right)\right), \mathsf{*.f64}\left(y, x\right)\right), x\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\left(x \cdot \frac{-1}{6}\right) \cdot x\right)\right), \mathsf{*.f64}\left(y, x\right)\right), x\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot \frac{-1}{6}\right), x\right)\right), \mathsf{*.f64}\left(y, x\right)\right), x\right) \]

       4. *-lowering-*.f64 50.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \frac{-1}{6}\right), x\right)\right), \mathsf{*.f64}\left(y, x\right)\right), x\right) \]

   12. Applied egg-rr 50.8%

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

   if 4.4000000000000001e189 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(y \cdot \left(\sin x + {y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{6} \cdot \sin x\right)\right)\right)}, x\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\sin x + {y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{6} \cdot \sin x\right)\right)\right), x\right) \]

      2. *-rgt-identity N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\sin x \cdot 1 + {y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{6} \cdot \sin x\right)\right)\right), x\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\sin x \cdot 1 + {y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x + \frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right)\right)\right)\right), x\right) \]

      4. distribute-lft-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\sin x \cdot 1 + \left({y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x\right) + {y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right)\right)\right)\right)\right), x\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\sin x \cdot 1 + \left(\left({y}^{2} \cdot \frac{1}{6}\right) \cdot \sin x + {y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right)\right)\right)\right)\right), x\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\sin x \cdot 1 + \left(\left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \sin x + {y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right)\right)\right)\right)\right), x\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\sin x \cdot 1 + \left(\left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \sin x + {y}^{2} \cdot \left(\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot \sin x\right)\right)\right)\right), x\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\sin x \cdot 1 + \left(\left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \sin x + \left({y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2}\right)\right) \cdot \sin x\right)\right)\right), x\right) \]

      9. distribute-rgt-out N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\sin x \cdot 1 + \sin x \cdot \left(\frac{1}{6} \cdot {y}^{2} + {y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right), x\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\sin x \cdot 1 + \sin x \cdot \left({y}^{2} \cdot \frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right), x\right) \]

      11. distribute-lft-in N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\sin x \cdot 1 + \sin x \cdot \left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right), x\right) \]

   5. Simplified 86.7%

      \[\leadsto \frac{\color{blue}{y \cdot \left(\sin x \cdot \left(1 + y \cdot \left(y \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right)\right)}}{x} \]

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\color{blue}{x}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right)\right)\right), x\right) \]
   7. Step-by-step derivation

   8. Simplified 38.9%

      \[\leadsto \frac{y \cdot \left(\color{blue}{x} \cdot \left(1 + y \cdot \left(y \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right)\right)}{x} \]

   9. Taylor expanded in y around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{120} \cdot {y}^{4}\right)}\right)\right), x\right) \]
   10. Step-by-step derivation

       1. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \left(\frac{1}{120} \cdot {y}^{\left(2 \cdot 2\right)}\right)\right)\right), x\right) \]

       2. pow-sqr N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \left(\frac{1}{120} \cdot \left({y}^{2} \cdot {y}^{2}\right)\right)\right)\right), x\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \left(\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot {y}^{2}\right)\right)\right), x\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \left({y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right), x\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \left(\left(y \cdot y\right) \cdot \left(\frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right), x\right) \]

       6. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \left(y \cdot \left(y \cdot \left(\frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right)\right), x\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \left(y \cdot \left(\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot y\right)\right)\right)\right), x\right) \]

       8. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \left(y \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot y\right)\right)\right)\right)\right), x\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \left(y \cdot \left(\frac{1}{120} \cdot \left(\left(y \cdot y\right) \cdot y\right)\right)\right)\right)\right), x\right) \]

       10. unpow3 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \left(y \cdot \left(\frac{1}{120} \cdot {y}^{3}\right)\right)\right)\right), x\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \left(\frac{1}{120} \cdot {y}^{3}\right)\right)\right)\right), x\right) \]

       12. unpow3 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \left(\frac{1}{120} \cdot \left(\left(y \cdot y\right) \cdot y\right)\right)\right)\right)\right), x\right) \]

       13. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \left(\frac{1}{120} \cdot \left({y}^{2} \cdot y\right)\right)\right)\right)\right), x\right) \]

       14. associate-*r* N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \left(\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot y\right)\right)\right)\right), x\right) \]

       15. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \left(y \cdot \left(\frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right)\right), x\right) \]

       16. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left(\frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right)\right), x\right) \]

       17. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left({y}^{2} \cdot \frac{1}{120}\right)\right)\right)\right)\right), x\right) \]

       18. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left({y}^{2}\right), \frac{1}{120}\right)\right)\right)\right)\right), x\right) \]

       19. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left(y \cdot y\right), \frac{1}{120}\right)\right)\right)\right)\right), x\right) \]

       20. *-lowering-*.f64 77.6%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right), x\right) \]

   11. Simplified 77.6%

       \[\leadsto \frac{y \cdot \left(x \cdot \color{blue}{\left(y \cdot \left(y \cdot \left(\left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right)}\right)}{x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.66 \cdot 10^{+97}:\\ \;\;\;\;x \cdot \frac{y \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right)}{x}\\ \mathbf{elif}\;x \leq 4.4 \cdot 10^{+189}:\\ \;\;\;\;\frac{\left(1 + x \cdot \left(x \cdot -0.16666666666666666\right)\right) \cdot \left(x \cdot y\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(x \cdot \left(y \cdot \left(y \cdot \left(\left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right)\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 66.3% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{y \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right)}{x}\\ \mathbf{if}\;x \leq 1.66 \cdot 10^{+97}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 4.4 \cdot 10^{+189}:\\ \;\;\;\;\frac{\left(1 + x \cdot \left(x \cdot -0.16666666666666666\right)\right) \cdot \left(x \cdot y\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* x (/ (* y (+ 1.0 (* 0.16666666666666666 (* y y)))) x))))
   (if (<= x 1.66e+97)
     t_0
     (if (<= x 4.4e+189)
       (/ (* (+ 1.0 (* x (* x -0.16666666666666666))) (* x y)) x)
       t_0))))

C

double code(double x, double y) {
	double t_0 = x * ((y * (1.0 + (0.16666666666666666 * (y * y)))) / x);
	double tmp;
	if (x <= 1.66e+97) {
		tmp = t_0;
	} else if (x <= 4.4e+189) {
		tmp = ((1.0 + (x * (x * -0.16666666666666666))) * (x * y)) / x;
	} else {
		tmp = 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 * ((y * (1.0d0 + (0.16666666666666666d0 * (y * y)))) / x)
    if (x <= 1.66d+97) then
        tmp = t_0
    else if (x <= 4.4d+189) then
        tmp = ((1.0d0 + (x * (x * (-0.16666666666666666d0)))) * (x * y)) / x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x * ((y * (1.0 + (0.16666666666666666 * (y * y)))) / x);
	double tmp;
	if (x <= 1.66e+97) {
		tmp = t_0;
	} else if (x <= 4.4e+189) {
		tmp = ((1.0 + (x * (x * -0.16666666666666666))) * (x * y)) / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x * ((y * (1.0 + (0.16666666666666666 * (y * y)))) / x)
	tmp = 0
	if x <= 1.66e+97:
		tmp = t_0
	elif x <= 4.4e+189:
		tmp = ((1.0 + (x * (x * -0.16666666666666666))) * (x * y)) / x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x * Float64(Float64(y * Float64(1.0 + Float64(0.16666666666666666 * Float64(y * y)))) / x))
	tmp = 0.0
	if (x <= 1.66e+97)
		tmp = t_0;
	elseif (x <= 4.4e+189)
		tmp = Float64(Float64(Float64(1.0 + Float64(x * Float64(x * -0.16666666666666666))) * Float64(x * y)) / x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x * ((y * (1.0 + (0.16666666666666666 * (y * y)))) / x);
	tmp = 0.0;
	if (x <= 1.66e+97)
		tmp = t_0;
	elseif (x <= 4.4e+189)
		tmp = ((1.0 + (x * (x * -0.16666666666666666))) * (x * y)) / x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x * N[(N[(y * N[(1.0 + N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 1.66e+97], t$95$0, If[LessEqual[x, 4.4e+189], N[(N[(N[(1.0 + N[(x * N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x * y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.6599999999999999e97 or 4.4000000000000001e189 < x

   1. Initial program 89.1%

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

      1. associate-/l* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin x, \color{blue}{\left(\frac{\sinh y}{x}\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\frac{\color{blue}{\sinh y}}{x}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\sinh y, \color{blue}{x}\right)\right) \]

      5. sinh-lowering-sinh.f64 99.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{x}, \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 78.8%

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

   8. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\color{blue}{\left(y \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)}, x\right)\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right), x\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \left(\frac{1}{6} \cdot {y}^{2}\right)\right)\right), x\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \left({y}^{2}\right)\right)\right)\right), x\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \left(y \cdot y\right)\right)\right)\right), x\right)\right) \]

      5. *-lowering-*.f64 69.5%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, y\right)\right)\right)\right), x\right)\right) \]

   10. Simplified 69.5%

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

   if 1.6599999999999999e97 < x < 4.4000000000000001e189

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \color{blue}{y}\right), x\right) \]
   4. Step-by-step derivation

   5. Simplified 36.1%

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

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)}, y\right), x\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right), y\right), x\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot {x}^{2}\right)\right)\right), y\right), x\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left({x}^{2} \cdot \frac{-1}{6}\right)\right)\right), y\right), x\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \frac{-1}{6}\right)\right)\right), y\right), x\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-1}{6}\right)\right)\right), y\right), x\right) \]

      6. *-lowering-*.f64 50.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{6}\right)\right)\right), y\right), x\right) \]

   8. Simplified 50.7%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\left(1 + \left(x \cdot x\right) \cdot \frac{-1}{6}\right) \cdot x\right) \cdot y\right), x\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + \left(x \cdot x\right) \cdot \frac{-1}{6}\right) \cdot \left(x \cdot y\right)\right), x\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + \left(x \cdot x\right) \cdot \frac{-1}{6}\right) \cdot \left(y \cdot x\right)\right), x\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(1 + \left(x \cdot x\right) \cdot \frac{-1}{6}\right), \left(y \cdot x\right)\right), x\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \frac{-1}{6}\right)\right), \left(y \cdot x\right)\right), x\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-1}{6}\right)\right), \left(y \cdot x\right)\right), x\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{6}\right)\right), \left(y \cdot x\right)\right), x\right) \]

      8. *-lowering-*.f64 50.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{6}\right)\right), \mathsf{*.f64}\left(y, x\right)\right), x\right) \]

   10. Applied egg-rr 50.8%

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

       1. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(x \cdot \left(x \cdot \frac{-1}{6}\right)\right)\right), \mathsf{*.f64}\left(y, x\right)\right), x\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\left(x \cdot \frac{-1}{6}\right) \cdot x\right)\right), \mathsf{*.f64}\left(y, x\right)\right), x\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot \frac{-1}{6}\right), x\right)\right), \mathsf{*.f64}\left(y, x\right)\right), x\right) \]

       4. *-lowering-*.f64 50.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \frac{-1}{6}\right), x\right)\right), \mathsf{*.f64}\left(y, x\right)\right), x\right) \]

   12. Applied egg-rr 50.8%

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

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.66 \cdot 10^{+97}:\\ \;\;\;\;x \cdot \frac{y \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right)}{x}\\ \mathbf{elif}\;x \leq 4.4 \cdot 10^{+189}:\\ \;\;\;\;\frac{\left(1 + x \cdot \left(x \cdot -0.16666666666666666\right)\right) \cdot \left(x \cdot y\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 66.3% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{y \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right)}{x}\\ \mathbf{if}\;x \leq 1.66 \cdot 10^{+97}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 4.3 \cdot 10^{+189}:\\ \;\;\;\;\frac{y \cdot \left(x \cdot \left(-0.16666666666666666 \cdot \left(x \cdot x\right)\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* x (/ (* y (+ 1.0 (* 0.16666666666666666 (* y y)))) x))))
   (if (<= x 1.66e+97)
     t_0
     (if (<= x 4.3e+189)
       (/ (* y (* x (* -0.16666666666666666 (* x x)))) x)
       t_0))))

C

double code(double x, double y) {
	double t_0 = x * ((y * (1.0 + (0.16666666666666666 * (y * y)))) / x);
	double tmp;
	if (x <= 1.66e+97) {
		tmp = t_0;
	} else if (x <= 4.3e+189) {
		tmp = (y * (x * (-0.16666666666666666 * (x * x)))) / x;
	} else {
		tmp = 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 * ((y * (1.0d0 + (0.16666666666666666d0 * (y * y)))) / x)
    if (x <= 1.66d+97) then
        tmp = t_0
    else if (x <= 4.3d+189) then
        tmp = (y * (x * ((-0.16666666666666666d0) * (x * x)))) / x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x * ((y * (1.0 + (0.16666666666666666 * (y * y)))) / x);
	double tmp;
	if (x <= 1.66e+97) {
		tmp = t_0;
	} else if (x <= 4.3e+189) {
		tmp = (y * (x * (-0.16666666666666666 * (x * x)))) / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x * ((y * (1.0 + (0.16666666666666666 * (y * y)))) / x)
	tmp = 0
	if x <= 1.66e+97:
		tmp = t_0
	elif x <= 4.3e+189:
		tmp = (y * (x * (-0.16666666666666666 * (x * x)))) / x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x * Float64(Float64(y * Float64(1.0 + Float64(0.16666666666666666 * Float64(y * y)))) / x))
	tmp = 0.0
	if (x <= 1.66e+97)
		tmp = t_0;
	elseif (x <= 4.3e+189)
		tmp = Float64(Float64(y * Float64(x * Float64(-0.16666666666666666 * Float64(x * x)))) / x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x * ((y * (1.0 + (0.16666666666666666 * (y * y)))) / x);
	tmp = 0.0;
	if (x <= 1.66e+97)
		tmp = t_0;
	elseif (x <= 4.3e+189)
		tmp = (y * (x * (-0.16666666666666666 * (x * x)))) / x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x * N[(N[(y * N[(1.0 + N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 1.66e+97], t$95$0, If[LessEqual[x, 4.3e+189], N[(N[(y * N[(x * N[(-0.16666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.6599999999999999e97 or 4.29999999999999998e189 < x

   1. Initial program 89.1%

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

      1. associate-/l* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin x, \color{blue}{\left(\frac{\sinh y}{x}\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\frac{\color{blue}{\sinh y}}{x}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\sinh y, \color{blue}{x}\right)\right) \]

      5. sinh-lowering-sinh.f64 99.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{x}, \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 78.8%

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

   8. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\color{blue}{\left(y \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)}, x\right)\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right), x\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \left(\frac{1}{6} \cdot {y}^{2}\right)\right)\right), x\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \left({y}^{2}\right)\right)\right)\right), x\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \left(y \cdot y\right)\right)\right)\right), x\right)\right) \]

      5. *-lowering-*.f64 69.5%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, y\right)\right)\right)\right), x\right)\right) \]

   10. Simplified 69.5%

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

   if 1.6599999999999999e97 < x < 4.29999999999999998e189

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \color{blue}{y}\right), x\right) \]
   4. Step-by-step derivation

   5. Simplified 36.1%

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

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)}, y\right), x\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right), y\right), x\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot {x}^{2}\right)\right)\right), y\right), x\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left({x}^{2} \cdot \frac{-1}{6}\right)\right)\right), y\right), x\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \frac{-1}{6}\right)\right)\right), y\right), x\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-1}{6}\right)\right)\right), y\right), x\right) \]

      6. *-lowering-*.f64 50.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{6}\right)\right)\right), y\right), x\right) \]

   8. Simplified 50.7%

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

   9. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(\frac{-1}{6} \cdot {x}^{3}\right)}, y\right), x\right) \]
   10. Step-by-step derivation

       1. unpow3 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{-1}{6} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), y\right), x\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{-1}{6} \cdot \left({x}^{2} \cdot x\right)\right), y\right), x\right) \]

       3. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\left(\frac{-1}{6} \cdot {x}^{2}\right) \cdot x\right), y\right), x\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(x \cdot \left(\frac{-1}{6} \cdot {x}^{2}\right)\right), y\right), x\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{-1}{6} \cdot {x}^{2}\right)\right), y\right), x\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{-1}{6}, \left({x}^{2}\right)\right)\right), y\right), x\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{-1}{6}, \left(x \cdot x\right)\right)\right), y\right), x\right) \]

       8. *-lowering-*.f64 50.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, x\right)\right)\right), y\right), x\right) \]

   11. Simplified 50.7%

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

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.66 \cdot 10^{+97}:\\ \;\;\;\;x \cdot \frac{y \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right)}{x}\\ \mathbf{elif}\;x \leq 4.3 \cdot 10^{+189}:\\ \;\;\;\;\frac{y \cdot \left(x \cdot \left(-0.16666666666666666 \cdot \left(x \cdot x\right)\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 56.9% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.85 \cdot 10^{+31}:\\ \;\;\;\;x \cdot \frac{1}{\frac{x}{y}}\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{+122}:\\ \;\;\;\;\frac{y \cdot \left(x \cdot \left(-0.16666666666666666 \cdot \left(x \cdot x\right)\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.85e+31)
   (* x (/ 1.0 (/ x y)))
   (if (<= y 2.2e+122)
     (/ (* y (* x (* -0.16666666666666666 (* x x)))) x)
     (* y (+ 1.0 (* 0.16666666666666666 (* y y)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.85e+31) {
		tmp = x * (1.0 / (x / y));
	} else if (y <= 2.2e+122) {
		tmp = (y * (x * (-0.16666666666666666 * (x * x)))) / x;
	} else {
		tmp = y * (1.0 + (0.16666666666666666 * (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 (y <= 1.85d+31) then
        tmp = x * (1.0d0 / (x / y))
    else if (y <= 2.2d+122) then
        tmp = (y * (x * ((-0.16666666666666666d0) * (x * x)))) / x
    else
        tmp = y * (1.0d0 + (0.16666666666666666d0 * (y * y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.85e+31) {
		tmp = x * (1.0 / (x / y));
	} else if (y <= 2.2e+122) {
		tmp = (y * (x * (-0.16666666666666666 * (x * x)))) / x;
	} else {
		tmp = y * (1.0 + (0.16666666666666666 * (y * y)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.85e+31:
		tmp = x * (1.0 / (x / y))
	elif y <= 2.2e+122:
		tmp = (y * (x * (-0.16666666666666666 * (x * x)))) / x
	else:
		tmp = y * (1.0 + (0.16666666666666666 * (y * y)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.85e+31)
		tmp = Float64(x * Float64(1.0 / Float64(x / y)));
	elseif (y <= 2.2e+122)
		tmp = Float64(Float64(y * Float64(x * Float64(-0.16666666666666666 * Float64(x * x)))) / x);
	else
		tmp = Float64(y * Float64(1.0 + Float64(0.16666666666666666 * Float64(y * y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.85e+31)
		tmp = x * (1.0 / (x / y));
	elseif (y <= 2.2e+122)
		tmp = (y * (x * (-0.16666666666666666 * (x * x)))) / x;
	else
		tmp = y * (1.0 + (0.16666666666666666 * (y * y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.85e+31], N[(x * N[(1.0 / N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.2e+122], N[(N[(y * N[(x * N[(-0.16666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(y * N[(1.0 + N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.8499999999999999e31

   1. Initial program 87.5%

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

      1. associate-/l* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin x, \color{blue}{\left(\frac{\sinh y}{x}\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\frac{\color{blue}{\sinh y}}{x}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\sinh y, \color{blue}{x}\right)\right) \]

      5. sinh-lowering-sinh.f64 99.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{x}, \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 75.9%

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

   8. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{y}{x}\right)}\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 57.3%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \color{blue}{x}\right)\right) \]

   10. Simplified 57.3%

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

       1. clear-num N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1}{\color{blue}{\frac{x}{y}}}\right)\right) \]

       2. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{x}{y}\right)}\right)\right) \]

       3. /-lowering-/.f64 58.0%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(x, \color{blue}{y}\right)\right)\right) \]

   12. Applied egg-rr 58.0%

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

   if 1.8499999999999999e31 < y < 2.1999999999999999e122

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \color{blue}{y}\right), x\right) \]
   4. Step-by-step derivation

   5. Simplified 3.1%

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

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)}, y\right), x\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right), y\right), x\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot {x}^{2}\right)\right)\right), y\right), x\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left({x}^{2} \cdot \frac{-1}{6}\right)\right)\right), y\right), x\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \frac{-1}{6}\right)\right)\right), y\right), x\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-1}{6}\right)\right)\right), y\right), x\right) \]

      6. *-lowering-*.f64 22.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{6}\right)\right)\right), y\right), x\right) \]

   8. Simplified 22.0%

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

   9. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(\frac{-1}{6} \cdot {x}^{3}\right)}, y\right), x\right) \]
   10. Step-by-step derivation

       1. unpow3 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{-1}{6} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), y\right), x\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{-1}{6} \cdot \left({x}^{2} \cdot x\right)\right), y\right), x\right) \]

       3. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\left(\frac{-1}{6} \cdot {x}^{2}\right) \cdot x\right), y\right), x\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(x \cdot \left(\frac{-1}{6} \cdot {x}^{2}\right)\right), y\right), x\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{-1}{6} \cdot {x}^{2}\right)\right), y\right), x\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{-1}{6}, \left({x}^{2}\right)\right)\right), y\right), x\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{-1}{6}, \left(x \cdot x\right)\right)\right), y\right), x\right) \]

       8. *-lowering-*.f64 21.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, x\right)\right)\right), y\right), x\right) \]

   11. Simplified 21.3%

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

   if 2.1999999999999999e122 < y

   1. Initial program 100.0%

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

      1. associate-/l* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin x, \color{blue}{\left(\frac{\sinh y}{x}\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\frac{\color{blue}{\sinh y}}{x}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\sinh y, \color{blue}{x}\right)\right) \]

      5. sinh-lowering-sinh.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{x}, \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 69.7%

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

   8. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)} \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(1 + \frac{1}{6} \cdot {y}^{2}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right)}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \color{blue}{\left({y}^{2}\right)}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \left(y \cdot \color{blue}{y}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 69.7%

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right)\right)\right) \]

   10. Simplified 69.7%

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

4. Final simplification 57.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.85 \cdot 10^{+31}:\\ \;\;\;\;x \cdot \frac{1}{\frac{x}{y}}\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{+122}:\\ \;\;\;\;\frac{y \cdot \left(x \cdot \left(-0.16666666666666666 \cdot \left(x \cdot x\right)\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 51.4% accurate, 12.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.66 \cdot 10^{+97}:\\ \;\;\;\;x \cdot \frac{1}{\frac{x}{y}}\\ \mathbf{elif}\;x \leq 4.05 \cdot 10^{+189}:\\ \;\;\;\;y \cdot \left(-0.16666666666666666 \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1.66e+97)
   (* x (/ 1.0 (/ x y)))
   (if (<= x 4.05e+189) (* y (* -0.16666666666666666 (* x x))) (* x (/ y x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.66e+97) {
		tmp = x * (1.0 / (x / y));
	} else if (x <= 4.05e+189) {
		tmp = y * (-0.16666666666666666 * (x * x));
	} else {
		tmp = x * (y / x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 1.66d+97) then
        tmp = x * (1.0d0 / (x / y))
    else if (x <= 4.05d+189) then
        tmp = y * ((-0.16666666666666666d0) * (x * x))
    else
        tmp = x * (y / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 1.66e+97) {
		tmp = x * (1.0 / (x / y));
	} else if (x <= 4.05e+189) {
		tmp = y * (-0.16666666666666666 * (x * x));
	} else {
		tmp = x * (y / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 1.66e+97:
		tmp = x * (1.0 / (x / y))
	elif x <= 4.05e+189:
		tmp = y * (-0.16666666666666666 * (x * x))
	else:
		tmp = x * (y / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.66e+97)
		tmp = Float64(x * Float64(1.0 / Float64(x / y)));
	elseif (x <= 4.05e+189)
		tmp = Float64(y * Float64(-0.16666666666666666 * Float64(x * x)));
	else
		tmp = Float64(x * Float64(y / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 1.66e+97)
		tmp = x * (1.0 / (x / y));
	elseif (x <= 4.05e+189)
		tmp = y * (-0.16666666666666666 * (x * x));
	else
		tmp = x * (y / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.66e+97], N[(x * N[(1.0 / N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.05e+189], N[(y * N[(-0.16666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(y / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.6599999999999999e97

   1. Initial program 88.0%

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

      1. associate-/l* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin x, \color{blue}{\left(\frac{\sinh y}{x}\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\frac{\color{blue}{\sinh y}}{x}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\sinh y, \color{blue}{x}\right)\right) \]

      5. sinh-lowering-sinh.f64 99.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{x}, \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 78.9%

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

   8. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{y}{x}\right)}\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 56.7%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \color{blue}{x}\right)\right) \]

   10. Simplified 56.7%

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

       1. clear-num N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1}{\color{blue}{\frac{x}{y}}}\right)\right) \]

       2. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{x}{y}\right)}\right)\right) \]

       3. /-lowering-/.f64 57.0%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(x, \color{blue}{y}\right)\right)\right) \]

   12. Applied egg-rr 57.0%

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

   if 1.6599999999999999e97 < x < 4.05000000000000016e189

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \color{blue}{y}\right), x\right) \]
   4. Step-by-step derivation

   5. Simplified 36.1%

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

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)}, y\right), x\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right), y\right), x\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot {x}^{2}\right)\right)\right), y\right), x\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left({x}^{2} \cdot \frac{-1}{6}\right)\right)\right), y\right), x\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \frac{-1}{6}\right)\right)\right), y\right), x\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-1}{6}\right)\right)\right), y\right), x\right) \]

      6. *-lowering-*.f64 50.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{6}\right)\right)\right), y\right), x\right) \]

   8. Simplified 50.7%

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

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{-1}{6} \cdot \left({x}^{2} \cdot y\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\frac{-1}{6} \cdot {x}^{2}\right) \cdot \color{blue}{y} \]

       2. *-commutative N/A

          \[\leadsto y \cdot \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2}\right)} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2}\right)}\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\frac{-1}{6}, \color{blue}{\left({x}^{2}\right)}\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\frac{-1}{6}, \left(x \cdot \color{blue}{x}\right)\right)\right) \]

       6. *-lowering-*.f64 50.9%

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right) \]

   11. Simplified 50.9%

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

   if 4.05000000000000016e189 < x

   1. Initial program 99.9%

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

      1. associate-/l* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin x, \color{blue}{\left(\frac{\sinh y}{x}\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\frac{\color{blue}{\sinh y}}{x}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\sinh y, \color{blue}{x}\right)\right) \]

      5. sinh-lowering-sinh.f64 99.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{x}, \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 77.4%

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

   8. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{y}{x}\right)}\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 43.5%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \color{blue}{x}\right)\right) \]

   10. Simplified 43.5%

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

Alternative 17: 57.4% accurate, 14.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.1 \cdot 10^{+100}:\\ \;\;\;\;x \cdot \frac{1}{\frac{x}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.1e+100)
   (* x (/ 1.0 (/ x y)))
   (* y (+ 1.0 (* 0.16666666666666666 (* y y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.1e+100) {
		tmp = x * (1.0 / (x / y));
	} else {
		tmp = y * (1.0 + (0.16666666666666666 * (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 (y <= 1.1d+100) then
        tmp = x * (1.0d0 / (x / y))
    else
        tmp = y * (1.0d0 + (0.16666666666666666d0 * (y * y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.1e+100) {
		tmp = x * (1.0 / (x / y));
	} else {
		tmp = y * (1.0 + (0.16666666666666666 * (y * y)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.1e+100:
		tmp = x * (1.0 / (x / y))
	else:
		tmp = y * (1.0 + (0.16666666666666666 * (y * y)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.1e+100)
		tmp = Float64(x * Float64(1.0 / Float64(x / y)));
	else
		tmp = Float64(y * Float64(1.0 + Float64(0.16666666666666666 * Float64(y * y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.1e+100)
		tmp = x * (1.0 / (x / y));
	else
		tmp = y * (1.0 + (0.16666666666666666 * (y * y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.1e+100], N[(x * N[(1.0 / N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(1.0 + N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.1e100

   1. Initial program 88.2%

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

      1. associate-/l* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin x, \color{blue}{\left(\frac{\sinh y}{x}\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\frac{\color{blue}{\sinh y}}{x}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\sinh y, \color{blue}{x}\right)\right) \]

      5. sinh-lowering-sinh.f64 99.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{x}, \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 75.4%

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

   8. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{y}{x}\right)}\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 55.2%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \color{blue}{x}\right)\right) \]

   10. Simplified 55.2%

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

       1. clear-num N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1}{\color{blue}{\frac{x}{y}}}\right)\right) \]

       2. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{x}{y}\right)}\right)\right) \]

       3. /-lowering-/.f64 55.8%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(x, \color{blue}{y}\right)\right)\right) \]

   12. Applied egg-rr 55.8%

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

   if 1.1e100 < y

   1. Initial program 100.0%

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

      1. associate-/l* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin x, \color{blue}{\left(\frac{\sinh y}{x}\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\frac{\color{blue}{\sinh y}}{x}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\sinh y, \color{blue}{x}\right)\right) \]

      5. sinh-lowering-sinh.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{x}, \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 66.7%

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

   8. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)} \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(1 + \frac{1}{6} \cdot {y}^{2}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right)}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \color{blue}{\left({y}^{2}\right)}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \left(y \cdot \color{blue}{y}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 66.7%

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right)\right)\right) \]

   10. Simplified 66.7%

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

Alternative 18: 51.2% accurate, 29.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.8%

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

   1. associate-/l* N/A

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

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\sin x, \color{blue}{\left(\frac{\sinh y}{x}\right)}\right) \]

   3. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\frac{\color{blue}{\sinh y}}{x}\right)\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\sinh y, \color{blue}{x}\right)\right) \]

   5. sinh-lowering-sinh.f64 99.9%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]

3. Simplified 99.9%

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

5. Taylor expanded in x around 0

   \[\leadsto \mathsf{*.f64}\left(\color{blue}{x}, \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]
6. Step-by-step derivation

7. Simplified 74.1%

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

8. Taylor expanded in y around 0

   \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{y}{x}\right)}\right) \]
9. Step-by-step derivation

   1. /-lowering-/.f64 51.7%

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \color{blue}{x}\right)\right) \]

10. Simplified 51.7%

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

    1. clear-num N/A

       \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1}{\color{blue}{\frac{x}{y}}}\right)\right) \]

    2. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{x}{y}\right)}\right)\right) \]

    3. /-lowering-/.f64 52.2%

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(x, \color{blue}{y}\right)\right)\right) \]

12. Applied egg-rr 52.2%

    \[\leadsto x \cdot \color{blue}{\frac{1}{\frac{x}{y}}} \]
13. Add Preprocessing

Alternative 19: 50.6% 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 89.8%

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

   1. associate-/l* N/A

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

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\sin x, \color{blue}{\left(\frac{\sinh y}{x}\right)}\right) \]

   3. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\frac{\color{blue}{\sinh y}}{x}\right)\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\sinh y, \color{blue}{x}\right)\right) \]

   5. sinh-lowering-sinh.f64 99.9%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]

3. Simplified 99.9%

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

5. Taylor expanded in x around 0

   \[\leadsto \mathsf{*.f64}\left(\color{blue}{x}, \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]
6. Step-by-step derivation

7. Simplified 74.1%

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

8. Taylor expanded in y around 0

   \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{y}{x}\right)}\right) \]
9. Step-by-step derivation

   1. /-lowering-/.f64 51.7%

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \color{blue}{x}\right)\right) \]

10. Simplified 51.7%

    \[\leadsto x \cdot \color{blue}{\frac{y}{x}} \]
11. Add Preprocessing

Alternative 20: 28.3% 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 89.8%

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

   1. associate-/l* N/A

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

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\sin x, \color{blue}{\left(\frac{\sinh y}{x}\right)}\right) \]

   3. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\frac{\color{blue}{\sinh y}}{x}\right)\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\sinh y, \color{blue}{x}\right)\right) \]

   5. sinh-lowering-sinh.f64 99.9%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]

3. Simplified 99.9%

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

5. Taylor expanded in x around 0

   \[\leadsto \mathsf{*.f64}\left(\color{blue}{x}, \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]
6. Step-by-step derivation

7. Simplified 74.1%

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

8. Taylor expanded in y around 0

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

10. Simplified 28.7%

    \[\leadsto \color{blue}{y} \]
11. Add Preprocessing

Developer Target 1: 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 2024163 
(FPCore (x y)
  :name "Linear.Quaternion:$ccosh from linear-1.19.1.3"
  :precision binary64

  :alt
  (! :herbie-platform default (* (sin x) (/ (sinh y) x)))

  (/ (* (sin x) (sinh y)) x))