Linear.Quaternion:$ccosh from linear-1.19.1.3

Percentage Accurate: 88.3% → 99.8%

Time: 15.3s

Alternatives: 24

Speedup: 1.0×

• 49.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 88.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% 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 89.5%

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

Alternative 2: 88.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\\ \mathbf{if}\;y \leq 0.0116:\\ \;\;\;\;y \cdot \left(t\_0 \cdot \frac{\sin x}{x}\right)\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+43}:\\ \;\;\;\;\frac{\frac{-1}{\frac{\frac{-1}{x}}{\sinh y}}}{x}\\ \mathbf{elif}\;y \leq 9.6 \cdot 10^{+102}:\\ \;\;\;\;\frac{\sinh y \cdot \left(x \cdot \left(1 + x \cdot \left(x \cdot -0.16666666666666666\right)\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(\sin x \cdot t\_0\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* 0.16666666666666666 (* y y)))))
   (if (<= y 0.0116)
     (* y (* t_0 (/ (sin x) x)))
     (if (<= y 5e+43)
       (/ (/ -1.0 (/ (/ -1.0 x) (sinh y))) x)
       (if (<= y 9.6e+102)
         (/ (* (sinh y) (* x (+ 1.0 (* x (* x -0.16666666666666666))))) x)
         (/ (* y (* (sin x) t_0)) x))))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (0.16666666666666666 * (y * y));
	double tmp;
	if (y <= 0.0116) {
		tmp = y * (t_0 * (sin(x) / x));
	} else if (y <= 5e+43) {
		tmp = (-1.0 / ((-1.0 / x) / sinh(y))) / x;
	} else if (y <= 9.6e+102) {
		tmp = (sinh(y) * (x * (1.0 + (x * (x * -0.16666666666666666))))) / x;
	} else {
		tmp = (y * (sin(x) * 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 = 1.0d0 + (0.16666666666666666d0 * (y * y))
    if (y <= 0.0116d0) then
        tmp = y * (t_0 * (sin(x) / x))
    else if (y <= 5d+43) then
        tmp = ((-1.0d0) / (((-1.0d0) / x) / sinh(y))) / x
    else if (y <= 9.6d+102) then
        tmp = (sinh(y) * (x * (1.0d0 + (x * (x * (-0.16666666666666666d0)))))) / x
    else
        tmp = (y * (sin(x) * t_0)) / 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 (y <= 0.0116) {
		tmp = y * (t_0 * (Math.sin(x) / x));
	} else if (y <= 5e+43) {
		tmp = (-1.0 / ((-1.0 / x) / Math.sinh(y))) / x;
	} else if (y <= 9.6e+102) {
		tmp = (Math.sinh(y) * (x * (1.0 + (x * (x * -0.16666666666666666))))) / x;
	} else {
		tmp = (y * (Math.sin(x) * t_0)) / x;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + (0.16666666666666666 * (y * y))
	tmp = 0
	if y <= 0.0116:
		tmp = y * (t_0 * (math.sin(x) / x))
	elif y <= 5e+43:
		tmp = (-1.0 / ((-1.0 / x) / math.sinh(y))) / x
	elif y <= 9.6e+102:
		tmp = (math.sinh(y) * (x * (1.0 + (x * (x * -0.16666666666666666))))) / x
	else:
		tmp = (y * (math.sin(x) * t_0)) / x
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(0.16666666666666666 * Float64(y * y)))
	tmp = 0.0
	if (y <= 0.0116)
		tmp = Float64(y * Float64(t_0 * Float64(sin(x) / x)));
	elseif (y <= 5e+43)
		tmp = Float64(Float64(-1.0 / Float64(Float64(-1.0 / x) / sinh(y))) / x);
	elseif (y <= 9.6e+102)
		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) * t_0)) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + (0.16666666666666666 * (y * y));
	tmp = 0.0;
	if (y <= 0.0116)
		tmp = y * (t_0 * (sin(x) / x));
	elseif (y <= 5e+43)
		tmp = (-1.0 / ((-1.0 / x) / sinh(y))) / x;
	elseif (y <= 9.6e+102)
		tmp = (sinh(y) * (x * (1.0 + (x * (x * -0.16666666666666666))))) / x;
	else
		tmp = (y * (sin(x) * t_0)) / 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[y, 0.0116], N[(y * N[(t$95$0 * N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5e+43], N[(N[(-1.0 / N[(N[(-1.0 / x), $MachinePrecision] / N[Sinh[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[y, 9.6e+102], 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] * t$95$0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if y < 0.0116

   1. Initial program 86.7%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
   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 90.6%

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

   if 0.0116 < y < 5.0000000000000004e43

   1. Initial program 99.8%

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

      1. sinh-def N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sin x \cdot \frac{e^{y} - e^{\mathsf{neg}\left(y\right)}}{2}\right), x\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\sin x \cdot \left(e^{y} - e^{\mathsf{neg}\left(y\right)}\right)}{2}\right), x\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{\frac{2}{\sin x \cdot \left(e^{y} - e^{\mathsf{neg}\left(y\right)}\right)}}\right), x\right) \]

      4. frac-2neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\frac{2}{\sin x \cdot \left(e^{y} - e^{\mathsf{neg}\left(y\right)}\right)}\right)}\right), x\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{\mathsf{neg}\left(\frac{2}{\sin x \cdot \left(e^{y} - e^{\mathsf{neg}\left(y\right)}\right)}\right)}\right), x\right) \]

      6. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{\mathsf{neg}\left(\frac{1}{\frac{\sin x \cdot \left(e^{y} - e^{\mathsf{neg}\left(y\right)}\right)}{2}}\right)}\right), x\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{\mathsf{neg}\left(\frac{1}{\sin x \cdot \frac{e^{y} - e^{\mathsf{neg}\left(y\right)}}{2}}\right)}\right), x\right) \]

      8. sinh-def N/A

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

      9. distribute-frac-neg2 N/A

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

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

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

      11. distribute-lft-neg-in N/A

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

      12. associate-/r* N/A

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

      13. metadata-eval N/A

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

      14. frac-2neg N/A

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

      15. /-lowering-/.f64 N/A

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

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

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

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

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

      18. sinh-lowering-sinh.f64 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

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

   7. Simplified 75.0%

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

   if 5.0000000000000004e43 < y < 9.59999999999999978e102

   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 81.8%

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

   5. Simplified 81.8%

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

   if 9.59999999999999978e102 < y

   1. Initial program 100.0%

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

      1. sinh-def N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sin x \cdot \frac{e^{y} - e^{\mathsf{neg}\left(y\right)}}{2}\right), x\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\sin x \cdot \left(e^{y} - e^{\mathsf{neg}\left(y\right)}\right)}{2}\right), x\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{\frac{2}{\sin x \cdot \left(e^{y} - e^{\mathsf{neg}\left(y\right)}\right)}}\right), x\right) \]

      4. frac-2neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\frac{2}{\sin x \cdot \left(e^{y} - e^{\mathsf{neg}\left(y\right)}\right)}\right)}\right), x\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{\mathsf{neg}\left(\frac{2}{\sin x \cdot \left(e^{y} - e^{\mathsf{neg}\left(y\right)}\right)}\right)}\right), x\right) \]

      6. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{\mathsf{neg}\left(\frac{1}{\frac{\sin x \cdot \left(e^{y} - e^{\mathsf{neg}\left(y\right)}\right)}{2}}\right)}\right), x\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{\mathsf{neg}\left(\frac{1}{\sin x \cdot \frac{e^{y} - e^{\mathsf{neg}\left(y\right)}}{2}}\right)}\right), x\right) \]

      8. sinh-def N/A

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

      9. distribute-frac-neg2 N/A

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

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

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

      11. distribute-lft-neg-in N/A

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

      12. associate-/r* N/A

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

      13. metadata-eval N/A

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

      14. frac-2neg N/A

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

      15. /-lowering-/.f64 N/A

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

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

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

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

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

      18. sinh-lowering-sinh.f64 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

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

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. distribute-rgt1-in N/A

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

      8. +-commutative N/A

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

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

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

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

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

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

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

      12. unpow2 N/A

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

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

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

      14. sin-lowering-sin.f64 97.4%

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

   7. Simplified 97.4%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.0116:\\ \;\;\;\;y \cdot \left(\left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \cdot \frac{\sin x}{x}\right)\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+43}:\\ \;\;\;\;\frac{\frac{-1}{\frac{\frac{-1}{x}}{\sinh y}}}{x}\\ \mathbf{elif}\;y \leq 9.6 \cdot 10^{+102}:\\ \;\;\;\;\frac{\sinh y \cdot \left(x \cdot \left(1 + x \cdot \left(x \cdot -0.16666666666666666\right)\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(\sin x \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right)\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\\ \mathbf{if}\;y \leq 0.035:\\ \;\;\;\;y \cdot \left(t\_0 \cdot \frac{\sin x}{x}\right)\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{+45}:\\ \;\;\;\;\frac{\frac{-1}{\frac{\frac{-1}{x}}{\sinh y}}}{x}\\ \mathbf{elif}\;y \leq 9.6 \cdot 10^{+102}:\\ \;\;\;\;\frac{\left(x \cdot \left(1 + x \cdot \left(x \cdot -0.16666666666666666\right)\right)\right) \cdot \left(y \cdot \left(1 + y \cdot \left(y \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(\sin x \cdot t\_0\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* 0.16666666666666666 (* y y)))))
   (if (<= y 0.035)
     (* y (* t_0 (/ (sin x) x)))
     (if (<= y 1.75e+45)
       (/ (/ -1.0 (/ (/ -1.0 x) (sinh y))) x)
       (if (<= y 9.6e+102)
         (/
          (*
           (* x (+ 1.0 (* x (* x -0.16666666666666666))))
           (*
            y
            (+
             1.0
             (*
              y
              (*
               y
               (+ 0.16666666666666666 (* (* y y) 0.008333333333333333)))))))
          x)
         (/ (* y (* (sin x) t_0)) x))))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (0.16666666666666666 * (y * y));
	double tmp;
	if (y <= 0.035) {
		tmp = y * (t_0 * (sin(x) / x));
	} else if (y <= 1.75e+45) {
		tmp = (-1.0 / ((-1.0 / x) / sinh(y))) / x;
	} else if (y <= 9.6e+102) {
		tmp = ((x * (1.0 + (x * (x * -0.16666666666666666)))) * (y * (1.0 + (y * (y * (0.16666666666666666 + ((y * y) * 0.008333333333333333))))))) / x;
	} else {
		tmp = (y * (sin(x) * 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 = 1.0d0 + (0.16666666666666666d0 * (y * y))
    if (y <= 0.035d0) then
        tmp = y * (t_0 * (sin(x) / x))
    else if (y <= 1.75d+45) then
        tmp = ((-1.0d0) / (((-1.0d0) / x) / sinh(y))) / x
    else if (y <= 9.6d+102) then
        tmp = ((x * (1.0d0 + (x * (x * (-0.16666666666666666d0))))) * (y * (1.0d0 + (y * (y * (0.16666666666666666d0 + ((y * y) * 0.008333333333333333d0))))))) / x
    else
        tmp = (y * (sin(x) * t_0)) / 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 (y <= 0.035) {
		tmp = y * (t_0 * (Math.sin(x) / x));
	} else if (y <= 1.75e+45) {
		tmp = (-1.0 / ((-1.0 / x) / Math.sinh(y))) / x;
	} else if (y <= 9.6e+102) {
		tmp = ((x * (1.0 + (x * (x * -0.16666666666666666)))) * (y * (1.0 + (y * (y * (0.16666666666666666 + ((y * y) * 0.008333333333333333))))))) / x;
	} else {
		tmp = (y * (Math.sin(x) * t_0)) / x;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + (0.16666666666666666 * (y * y))
	tmp = 0
	if y <= 0.035:
		tmp = y * (t_0 * (math.sin(x) / x))
	elif y <= 1.75e+45:
		tmp = (-1.0 / ((-1.0 / x) / math.sinh(y))) / x
	elif y <= 9.6e+102:
		tmp = ((x * (1.0 + (x * (x * -0.16666666666666666)))) * (y * (1.0 + (y * (y * (0.16666666666666666 + ((y * y) * 0.008333333333333333))))))) / x
	else:
		tmp = (y * (math.sin(x) * t_0)) / x
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(0.16666666666666666 * Float64(y * y)))
	tmp = 0.0
	if (y <= 0.035)
		tmp = Float64(y * Float64(t_0 * Float64(sin(x) / x)));
	elseif (y <= 1.75e+45)
		tmp = Float64(Float64(-1.0 / Float64(Float64(-1.0 / x) / sinh(y))) / x);
	elseif (y <= 9.6e+102)
		tmp = Float64(Float64(Float64(x * Float64(1.0 + Float64(x * Float64(x * -0.16666666666666666)))) * Float64(y * Float64(1.0 + Float64(y * Float64(y * Float64(0.16666666666666666 + Float64(Float64(y * y) * 0.008333333333333333))))))) / x);
	else
		tmp = Float64(Float64(y * Float64(sin(x) * t_0)) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + (0.16666666666666666 * (y * y));
	tmp = 0.0;
	if (y <= 0.035)
		tmp = y * (t_0 * (sin(x) / x));
	elseif (y <= 1.75e+45)
		tmp = (-1.0 / ((-1.0 / x) / sinh(y))) / x;
	elseif (y <= 9.6e+102)
		tmp = ((x * (1.0 + (x * (x * -0.16666666666666666)))) * (y * (1.0 + (y * (y * (0.16666666666666666 + ((y * y) * 0.008333333333333333))))))) / x;
	else
		tmp = (y * (sin(x) * t_0)) / 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[y, 0.035], N[(y * N[(t$95$0 * N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.75e+45], N[(N[(-1.0 / N[(N[(-1.0 / x), $MachinePrecision] / N[Sinh[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[y, 9.6e+102], N[(N[(N[(x * N[(1.0 + N[(x * N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y * 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], N[(N[(y * N[(N[Sin[x], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if y < 0.035000000000000003

   1. Initial program 86.7%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
   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 90.6%

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

   if 0.035000000000000003 < y < 1.75000000000000011e45

   1. Initial program 99.8%

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

      1. sinh-def N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sin x \cdot \frac{e^{y} - e^{\mathsf{neg}\left(y\right)}}{2}\right), x\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\sin x \cdot \left(e^{y} - e^{\mathsf{neg}\left(y\right)}\right)}{2}\right), x\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{\frac{2}{\sin x \cdot \left(e^{y} - e^{\mathsf{neg}\left(y\right)}\right)}}\right), x\right) \]

      4. frac-2neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\frac{2}{\sin x \cdot \left(e^{y} - e^{\mathsf{neg}\left(y\right)}\right)}\right)}\right), x\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{\mathsf{neg}\left(\frac{2}{\sin x \cdot \left(e^{y} - e^{\mathsf{neg}\left(y\right)}\right)}\right)}\right), x\right) \]

      6. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{\mathsf{neg}\left(\frac{1}{\frac{\sin x \cdot \left(e^{y} - e^{\mathsf{neg}\left(y\right)}\right)}{2}}\right)}\right), x\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{\mathsf{neg}\left(\frac{1}{\sin x \cdot \frac{e^{y} - e^{\mathsf{neg}\left(y\right)}}{2}}\right)}\right), x\right) \]

      8. sinh-def N/A

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

      9. distribute-frac-neg2 N/A

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

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

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

      11. distribute-lft-neg-in N/A

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

      12. associate-/r* N/A

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

      13. metadata-eval N/A

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

      14. frac-2neg N/A

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

      15. /-lowering-/.f64 N/A

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

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

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

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

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

      18. sinh-lowering-sinh.f64 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

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

   7. Simplified 75.0%

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

   if 1.75000000000000011e45 < y < 9.59999999999999978e102

   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 81.8%

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

   5. Simplified 81.8%

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

   6. Taylor expanded in y around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

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

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

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

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

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

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

      8. *-commutative N/A

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 73.4%

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

   8. Simplified 73.4%

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

   if 9.59999999999999978e102 < y

   1. Initial program 100.0%

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

      1. sinh-def N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sin x \cdot \frac{e^{y} - e^{\mathsf{neg}\left(y\right)}}{2}\right), x\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\sin x \cdot \left(e^{y} - e^{\mathsf{neg}\left(y\right)}\right)}{2}\right), x\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{\frac{2}{\sin x \cdot \left(e^{y} - e^{\mathsf{neg}\left(y\right)}\right)}}\right), x\right) \]

      4. frac-2neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\frac{2}{\sin x \cdot \left(e^{y} - e^{\mathsf{neg}\left(y\right)}\right)}\right)}\right), x\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{\mathsf{neg}\left(\frac{2}{\sin x \cdot \left(e^{y} - e^{\mathsf{neg}\left(y\right)}\right)}\right)}\right), x\right) \]

      6. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{\mathsf{neg}\left(\frac{1}{\frac{\sin x \cdot \left(e^{y} - e^{\mathsf{neg}\left(y\right)}\right)}{2}}\right)}\right), x\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{\mathsf{neg}\left(\frac{1}{\sin x \cdot \frac{e^{y} - e^{\mathsf{neg}\left(y\right)}}{2}}\right)}\right), x\right) \]

      8. sinh-def N/A

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

      9. distribute-frac-neg2 N/A

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

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

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

      11. distribute-lft-neg-in N/A

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

      12. associate-/r* N/A

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

      13. metadata-eval N/A

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

      14. frac-2neg N/A

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

      15. /-lowering-/.f64 N/A

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

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

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

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

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

      18. sinh-lowering-sinh.f64 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

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

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. distribute-rgt1-in N/A

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

      8. +-commutative N/A

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

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

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

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

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

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

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

      12. unpow2 N/A

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

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

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

      14. sin-lowering-sin.f64 97.4%

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

   7. Simplified 97.4%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.035:\\ \;\;\;\;y \cdot \left(\left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \cdot \frac{\sin x}{x}\right)\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{+45}:\\ \;\;\;\;\frac{\frac{-1}{\frac{\frac{-1}{x}}{\sinh y}}}{x}\\ \mathbf{elif}\;y \leq 9.6 \cdot 10^{+102}:\\ \;\;\;\;\frac{\left(x \cdot \left(1 + x \cdot \left(x \cdot -0.16666666666666666\right)\right)\right) \cdot \left(y \cdot \left(1 + y \cdot \left(y \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(\sin x \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right)\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(\left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \cdot \frac{\sin x}{x}\right)\\ \mathbf{if}\;y \leq 0.15:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 3.4 \cdot 10^{+128}:\\ \;\;\;\;\frac{\frac{-1}{\frac{\frac{-1}{x}}{\sinh y}}}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* (+ 1.0 (* 0.16666666666666666 (* y y))) (/ (sin x) x)))))
   (if (<= y 0.15)
     t_0
     (if (<= y 3.4e+128) (/ (/ -1.0 (/ (/ -1.0 x) (sinh y))) x) t_0))))

C

double code(double x, double y) {
	double t_0 = y * ((1.0 + (0.16666666666666666 * (y * y))) * (sin(x) / x));
	double tmp;
	if (y <= 0.15) {
		tmp = t_0;
	} else if (y <= 3.4e+128) {
		tmp = (-1.0 / ((-1.0 / x) / sinh(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 = y * ((1.0d0 + (0.16666666666666666d0 * (y * y))) * (sin(x) / x))
    if (y <= 0.15d0) then
        tmp = t_0
    else if (y <= 3.4d+128) then
        tmp = ((-1.0d0) / (((-1.0d0) / x) / sinh(y))) / x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = y * ((1.0 + (0.16666666666666666 * (y * y))) * (Math.sin(x) / x));
	double tmp;
	if (y <= 0.15) {
		tmp = t_0;
	} else if (y <= 3.4e+128) {
		tmp = (-1.0 / ((-1.0 / x) / Math.sinh(y))) / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * ((1.0 + (0.16666666666666666 * (y * y))) * (math.sin(x) / x))
	tmp = 0
	if y <= 0.15:
		tmp = t_0
	elif y <= 3.4e+128:
		tmp = (-1.0 / ((-1.0 / x) / math.sinh(y))) / x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * Float64(Float64(1.0 + Float64(0.16666666666666666 * Float64(y * y))) * Float64(sin(x) / x)))
	tmp = 0.0
	if (y <= 0.15)
		tmp = t_0;
	elseif (y <= 3.4e+128)
		tmp = Float64(Float64(-1.0 / Float64(Float64(-1.0 / x) / sinh(y))) / x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y * ((1.0 + (0.16666666666666666 * (y * y))) * (sin(x) / x));
	tmp = 0.0;
	if (y <= 0.15)
		tmp = t_0;
	elseif (y <= 3.4e+128)
		tmp = (-1.0 / ((-1.0 / x) / sinh(y))) / x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = 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, 0.15], t$95$0, If[LessEqual[y, 3.4e+128], N[(N[(-1.0 / N[(N[(-1.0 / x), $MachinePrecision] / N[Sinh[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < 0.149999999999999994 or 3.3999999999999999e128 < y

   1. Initial program 88.4%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 90.5%

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

   if 0.149999999999999994 < y < 3.3999999999999999e128

   1. Initial program 99.9%

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

      1. sinh-def N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sin x \cdot \frac{e^{y} - e^{\mathsf{neg}\left(y\right)}}{2}\right), x\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\sin x \cdot \left(e^{y} - e^{\mathsf{neg}\left(y\right)}\right)}{2}\right), x\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{\frac{2}{\sin x \cdot \left(e^{y} - e^{\mathsf{neg}\left(y\right)}\right)}}\right), x\right) \]

      4. frac-2neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\frac{2}{\sin x \cdot \left(e^{y} - e^{\mathsf{neg}\left(y\right)}\right)}\right)}\right), x\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{\mathsf{neg}\left(\frac{2}{\sin x \cdot \left(e^{y} - e^{\mathsf{neg}\left(y\right)}\right)}\right)}\right), x\right) \]

      6. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{\mathsf{neg}\left(\frac{1}{\frac{\sin x \cdot \left(e^{y} - e^{\mathsf{neg}\left(y\right)}\right)}{2}}\right)}\right), x\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{\mathsf{neg}\left(\frac{1}{\sin x \cdot \frac{e^{y} - e^{\mathsf{neg}\left(y\right)}}{2}}\right)}\right), x\right) \]

      8. sinh-def N/A

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

      9. distribute-frac-neg2 N/A

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

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

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

      11. distribute-lft-neg-in N/A

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

      12. associate-/r* N/A

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

      13. metadata-eval N/A

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

      14. frac-2neg N/A

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

      15. /-lowering-/.f64 N/A

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

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

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

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

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

      18. sinh-lowering-sinh.f64 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

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

   7. Simplified 75.0%

      \[\leadsto \frac{\frac{-1}{\frac{\frac{-1}{\color{blue}{x}}}{\sinh y}}}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 69.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.0027:\\ \;\;\;\;\frac{y}{\frac{x}{\sin x}}\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{+45}:\\ \;\;\;\;\frac{\frac{-1}{\frac{\frac{-1}{x}}{\sinh y}}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x \cdot \left(1 + x \cdot \left(x \cdot -0.16666666666666666\right)\right)\right) \cdot \left(y \cdot \left(1 + y \cdot \left(y \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right)\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 0.0027)
   (/ y (/ x (sin x)))
   (if (<= y 1.7e+45)
     (/ (/ -1.0 (/ (/ -1.0 x) (sinh y))) x)
     (/
      (*
       (* x (+ 1.0 (* x (* x -0.16666666666666666))))
       (*
        y
        (+
         1.0
         (*
          y
          (* y (+ 0.16666666666666666 (* (* y y) 0.008333333333333333)))))))
      x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 0.0027) {
		tmp = y / (x / sin(x));
	} else if (y <= 1.7e+45) {
		tmp = (-1.0 / ((-1.0 / x) / sinh(y))) / x;
	} else {
		tmp = ((x * (1.0 + (x * (x * -0.16666666666666666)))) * (y * (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 <= 0.0027d0) then
        tmp = y / (x / sin(x))
    else if (y <= 1.7d+45) then
        tmp = ((-1.0d0) / (((-1.0d0) / x) / sinh(y))) / x
    else
        tmp = ((x * (1.0d0 + (x * (x * (-0.16666666666666666d0))))) * (y * (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 <= 0.0027) {
		tmp = y / (x / Math.sin(x));
	} else if (y <= 1.7e+45) {
		tmp = (-1.0 / ((-1.0 / x) / Math.sinh(y))) / x;
	} else {
		tmp = ((x * (1.0 + (x * (x * -0.16666666666666666)))) * (y * (1.0 + (y * (y * (0.16666666666666666 + ((y * y) * 0.008333333333333333))))))) / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 0.0027:
		tmp = y / (x / math.sin(x))
	elif y <= 1.7e+45:
		tmp = (-1.0 / ((-1.0 / x) / math.sinh(y))) / x
	else:
		tmp = ((x * (1.0 + (x * (x * -0.16666666666666666)))) * (y * (1.0 + (y * (y * (0.16666666666666666 + ((y * y) * 0.008333333333333333))))))) / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 0.0027)
		tmp = Float64(y / Float64(x / sin(x)));
	elseif (y <= 1.7e+45)
		tmp = Float64(Float64(-1.0 / Float64(Float64(-1.0 / x) / sinh(y))) / x);
	else
		tmp = Float64(Float64(Float64(x * Float64(1.0 + Float64(x * Float64(x * -0.16666666666666666)))) * Float64(y * 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 <= 0.0027)
		tmp = y / (x / sin(x));
	elseif (y <= 1.7e+45)
		tmp = (-1.0 / ((-1.0 / x) / sinh(y))) / x;
	else
		tmp = ((x * (1.0 + (x * (x * -0.16666666666666666)))) * (y * (1.0 + (y * (y * (0.16666666666666666 + ((y * y) * 0.008333333333333333))))))) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 0.0027], N[(y / N[(x / N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.7e+45], N[(N[(-1.0 / N[(N[(-1.0 / x), $MachinePrecision] / N[Sinh[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(x * N[(1.0 + N[(x * N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y * 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 < 0.0027000000000000001

   1. Initial program 86.7%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 57.4%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

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

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

      6. div-inv N/A

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

      7. clear-num N/A

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

      8. div-inv N/A

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

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

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

      10. *-lft-identity N/A

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

      11. metadata-eval N/A

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

      12. associate-/r/ N/A

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

      13. un-div-inv N/A

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

      14. clear-num N/A

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

      15. div-inv N/A

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

      16. frac-2neg N/A

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

      17. metadata-eval N/A

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

      18. associate-/r/ N/A

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

      19. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(y, \mathsf{/.f64}\left(x, \left(-1 \cdot \left(\mathsf{neg}\left(\color{blue}{\sin x}\right)\right)\right)\right)\right) \]

      20. neg-mul-1 N/A

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

      21. remove-double-neg N/A

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

      22. sin-lowering-sin.f64 70.6%

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

   7. Applied egg-rr 70.6%

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

   if 0.0027000000000000001 < y < 1.7e45

   1. Initial program 99.8%

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

      1. sinh-def N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sin x \cdot \frac{e^{y} - e^{\mathsf{neg}\left(y\right)}}{2}\right), x\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\sin x \cdot \left(e^{y} - e^{\mathsf{neg}\left(y\right)}\right)}{2}\right), x\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{\frac{2}{\sin x \cdot \left(e^{y} - e^{\mathsf{neg}\left(y\right)}\right)}}\right), x\right) \]

      4. frac-2neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\frac{2}{\sin x \cdot \left(e^{y} - e^{\mathsf{neg}\left(y\right)}\right)}\right)}\right), x\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{\mathsf{neg}\left(\frac{2}{\sin x \cdot \left(e^{y} - e^{\mathsf{neg}\left(y\right)}\right)}\right)}\right), x\right) \]

      6. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{\mathsf{neg}\left(\frac{1}{\frac{\sin x \cdot \left(e^{y} - e^{\mathsf{neg}\left(y\right)}\right)}{2}}\right)}\right), x\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{\mathsf{neg}\left(\frac{1}{\sin x \cdot \frac{e^{y} - e^{\mathsf{neg}\left(y\right)}}{2}}\right)}\right), x\right) \]

      8. sinh-def N/A

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

      9. distribute-frac-neg2 N/A

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

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

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

      11. distribute-lft-neg-in N/A

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

      12. associate-/r* N/A

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

      13. metadata-eval N/A

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

      14. frac-2neg N/A

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

      15. /-lowering-/.f64 N/A

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

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

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

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

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

      18. sinh-lowering-sinh.f64 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

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

   7. Simplified 75.0%

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

   if 1.7e45 < y

   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 80.0%

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

   5. Simplified 80.0%

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

   6. Taylor expanded in y around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

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

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

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

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

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

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

      8. *-commutative N/A

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 77.9%

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

   8. Simplified 77.9%

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

Alternative 6: 69.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.0065:\\ \;\;\;\;\frac{y}{\frac{x}{\sin x}}\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{+45}:\\ \;\;\;\;\sinh y\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x \cdot \left(1 + x \cdot \left(x \cdot -0.16666666666666666\right)\right)\right) \cdot \left(y \cdot \left(1 + y \cdot \left(y \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right)\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 0.0065)
   (/ y (/ x (sin x)))
   (if (<= y 1.7e+45)
     (sinh y)
     (/
      (*
       (* x (+ 1.0 (* x (* x -0.16666666666666666))))
       (*
        y
        (+
         1.0
         (*
          y
          (* y (+ 0.16666666666666666 (* (* y y) 0.008333333333333333)))))))
      x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 0.0065) {
		tmp = y / (x / sin(x));
	} else if (y <= 1.7e+45) {
		tmp = sinh(y);
	} else {
		tmp = ((x * (1.0 + (x * (x * -0.16666666666666666)))) * (y * (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 <= 0.0065d0) then
        tmp = y / (x / sin(x))
    else if (y <= 1.7d+45) then
        tmp = sinh(y)
    else
        tmp = ((x * (1.0d0 + (x * (x * (-0.16666666666666666d0))))) * (y * (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 <= 0.0065) {
		tmp = y / (x / Math.sin(x));
	} else if (y <= 1.7e+45) {
		tmp = Math.sinh(y);
	} else {
		tmp = ((x * (1.0 + (x * (x * -0.16666666666666666)))) * (y * (1.0 + (y * (y * (0.16666666666666666 + ((y * y) * 0.008333333333333333))))))) / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 0.0065:
		tmp = y / (x / math.sin(x))
	elif y <= 1.7e+45:
		tmp = math.sinh(y)
	else:
		tmp = ((x * (1.0 + (x * (x * -0.16666666666666666)))) * (y * (1.0 + (y * (y * (0.16666666666666666 + ((y * y) * 0.008333333333333333))))))) / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 0.0065)
		tmp = Float64(y / Float64(x / sin(x)));
	elseif (y <= 1.7e+45)
		tmp = sinh(y);
	else
		tmp = Float64(Float64(Float64(x * Float64(1.0 + Float64(x * Float64(x * -0.16666666666666666)))) * Float64(y * 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 <= 0.0065)
		tmp = y / (x / sin(x));
	elseif (y <= 1.7e+45)
		tmp = sinh(y);
	else
		tmp = ((x * (1.0 + (x * (x * -0.16666666666666666)))) * (y * (1.0 + (y * (y * (0.16666666666666666 + ((y * y) * 0.008333333333333333))))))) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 0.0065], N[(y / N[(x / N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.7e+45], N[Sinh[y], $MachinePrecision], N[(N[(N[(x * N[(1.0 + N[(x * N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y * 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 < 0.0064999999999999997

   1. Initial program 86.7%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 57.4%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

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

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

      6. div-inv N/A

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

      7. clear-num N/A

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

      8. div-inv N/A

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

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

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

      10. *-lft-identity N/A

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

      11. metadata-eval N/A

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

      12. associate-/r/ N/A

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

      13. un-div-inv N/A

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

      14. clear-num N/A

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

      15. div-inv N/A

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

      16. frac-2neg N/A

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

      17. metadata-eval N/A

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

      18. associate-/r/ N/A

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

      19. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(y, \mathsf{/.f64}\left(x, \left(-1 \cdot \left(\mathsf{neg}\left(\color{blue}{\sin x}\right)\right)\right)\right)\right) \]

      20. neg-mul-1 N/A

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

      21. remove-double-neg N/A

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

      22. sin-lowering-sin.f64 70.6%

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

   7. Applied egg-rr 70.6%

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

   if 0.0064999999999999997 < y < 1.7e45

   1. Initial program 99.8%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
   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 74.8%

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. associate-/l* N/A

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

      4. *-inverses N/A

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

      5. *-rgt-identity N/A

         \[\leadsto \sinh y \]

      6. sinh-lowering-sinh.f64 74.8%

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

   9. Applied egg-rr 74.8%

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

   if 1.7e45 < y

   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 80.0%

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

   5. Simplified 80.0%

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

   6. Taylor expanded in y around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

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

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

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

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

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

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

      8. *-commutative N/A

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 77.9%

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

   8. Simplified 77.9%

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

Alternative 7: 74.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.0027:\\ \;\;\;\;\sin x \cdot \frac{y}{x}\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{+45}:\\ \;\;\;\;\sinh y\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x \cdot \left(1 + x \cdot \left(x \cdot -0.16666666666666666\right)\right)\right) \cdot \left(y \cdot \left(1 + y \cdot \left(y \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right)\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 0.0027)
   (* (sin x) (/ y x))
   (if (<= y 1.75e+45)
     (sinh y)
     (/
      (*
       (* x (+ 1.0 (* x (* x -0.16666666666666666))))
       (*
        y
        (+
         1.0
         (*
          y
          (* y (+ 0.16666666666666666 (* (* y y) 0.008333333333333333)))))))
      x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 0.0027) {
		tmp = sin(x) * (y / x);
	} else if (y <= 1.75e+45) {
		tmp = sinh(y);
	} else {
		tmp = ((x * (1.0 + (x * (x * -0.16666666666666666)))) * (y * (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 <= 0.0027d0) then
        tmp = sin(x) * (y / x)
    else if (y <= 1.75d+45) then
        tmp = sinh(y)
    else
        tmp = ((x * (1.0d0 + (x * (x * (-0.16666666666666666d0))))) * (y * (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 <= 0.0027) {
		tmp = Math.sin(x) * (y / x);
	} else if (y <= 1.75e+45) {
		tmp = Math.sinh(y);
	} else {
		tmp = ((x * (1.0 + (x * (x * -0.16666666666666666)))) * (y * (1.0 + (y * (y * (0.16666666666666666 + ((y * y) * 0.008333333333333333))))))) / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 0.0027:
		tmp = math.sin(x) * (y / x)
	elif y <= 1.75e+45:
		tmp = math.sinh(y)
	else:
		tmp = ((x * (1.0 + (x * (x * -0.16666666666666666)))) * (y * (1.0 + (y * (y * (0.16666666666666666 + ((y * y) * 0.008333333333333333))))))) / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 0.0027)
		tmp = Float64(sin(x) * Float64(y / x));
	elseif (y <= 1.75e+45)
		tmp = sinh(y);
	else
		tmp = Float64(Float64(Float64(x * Float64(1.0 + Float64(x * Float64(x * -0.16666666666666666)))) * Float64(y * 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 <= 0.0027)
		tmp = sin(x) * (y / x);
	elseif (y <= 1.75e+45)
		tmp = sinh(y);
	else
		tmp = ((x * (1.0 + (x * (x * -0.16666666666666666)))) * (y * (1.0 + (y * (y * (0.16666666666666666 + ((y * y) * 0.008333333333333333))))))) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 0.0027], N[(N[Sin[x], $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.75e+45], N[Sinh[y], $MachinePrecision], N[(N[(N[(x * N[(1.0 + N[(x * N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y * 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 < 0.0027000000000000001

   1. Initial program 86.7%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
   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 76.4%

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

   7. Simplified 76.4%

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

   if 0.0027000000000000001 < y < 1.75000000000000011e45

   1. Initial program 99.8%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
   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 74.8%

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. associate-/l* N/A

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

      4. *-inverses N/A

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

      5. *-rgt-identity N/A

         \[\leadsto \sinh y \]

      6. sinh-lowering-sinh.f64 74.8%

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

   9. Applied egg-rr 74.8%

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

   if 1.75000000000000011e45 < y

   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 80.0%

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

   5. Simplified 80.0%

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

   6. Taylor expanded in y around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

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

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

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

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

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

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

      8. *-commutative N/A

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 77.9%

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

   8. Simplified 77.9%

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

Alternative 8: 62.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 9.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{x}{\left(1 - -0.16666666666666666 \cdot \left(x \cdot x\right)\right) \cdot \frac{x}{y}}\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{+45}:\\ \;\;\;\;\sinh y\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x \cdot \left(1 + x \cdot \left(x \cdot -0.16666666666666666\right)\right)\right) \cdot \left(y \cdot \left(1 + y \cdot \left(y \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right)\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 9.5e-5)
   (/ x (* (- 1.0 (* -0.16666666666666666 (* x x))) (/ x y)))
   (if (<= y 1.1e+45)
     (sinh y)
     (/
      (*
       (* x (+ 1.0 (* x (* x -0.16666666666666666))))
       (*
        y
        (+
         1.0
         (*
          y
          (* y (+ 0.16666666666666666 (* (* y y) 0.008333333333333333)))))))
      x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 9.5e-5) {
		tmp = x / ((1.0 - (-0.16666666666666666 * (x * x))) * (x / y));
	} else if (y <= 1.1e+45) {
		tmp = sinh(y);
	} else {
		tmp = ((x * (1.0 + (x * (x * -0.16666666666666666)))) * (y * (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 <= 9.5d-5) then
        tmp = x / ((1.0d0 - ((-0.16666666666666666d0) * (x * x))) * (x / y))
    else if (y <= 1.1d+45) then
        tmp = sinh(y)
    else
        tmp = ((x * (1.0d0 + (x * (x * (-0.16666666666666666d0))))) * (y * (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 <= 9.5e-5) {
		tmp = x / ((1.0 - (-0.16666666666666666 * (x * x))) * (x / y));
	} else if (y <= 1.1e+45) {
		tmp = Math.sinh(y);
	} else {
		tmp = ((x * (1.0 + (x * (x * -0.16666666666666666)))) * (y * (1.0 + (y * (y * (0.16666666666666666 + ((y * y) * 0.008333333333333333))))))) / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 9.5e-5:
		tmp = x / ((1.0 - (-0.16666666666666666 * (x * x))) * (x / y))
	elif y <= 1.1e+45:
		tmp = math.sinh(y)
	else:
		tmp = ((x * (1.0 + (x * (x * -0.16666666666666666)))) * (y * (1.0 + (y * (y * (0.16666666666666666 + ((y * y) * 0.008333333333333333))))))) / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 9.5e-5)
		tmp = Float64(x / Float64(Float64(1.0 - Float64(-0.16666666666666666 * Float64(x * x))) * Float64(x / y)));
	elseif (y <= 1.1e+45)
		tmp = sinh(y);
	else
		tmp = Float64(Float64(Float64(x * Float64(1.0 + Float64(x * Float64(x * -0.16666666666666666)))) * Float64(y * 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 <= 9.5e-5)
		tmp = x / ((1.0 - (-0.16666666666666666 * (x * x))) * (x / y));
	elseif (y <= 1.1e+45)
		tmp = sinh(y);
	else
		tmp = ((x * (1.0 + (x * (x * -0.16666666666666666)))) * (y * (1.0 + (y * (y * (0.16666666666666666 + ((y * y) * 0.008333333333333333))))))) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 9.5e-5], N[(x / N[(N[(1.0 - N[(-0.16666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.1e+45], N[Sinh[y], $MachinePrecision], N[(N[(N[(x * N[(1.0 + N[(x * N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y * 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 < 9.5000000000000005e-5

   1. Initial program 86.7%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 46.6%

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

   5. Simplified 46.6%

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

   6. Taylor expanded in y around 0

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

   8. Simplified 31.7%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. flip-+ N/A

         \[\leadsto \left(\frac{1 \cdot 1 - \left(x \cdot \left(x \cdot \frac{-1}{6}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{-1}{6}\right)\right)}{1 - x \cdot \left(x \cdot \frac{-1}{6}\right)} \cdot x\right) \cdot \frac{y}{x} \]

      4. associate-*l/ N/A

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

      5. clear-num N/A

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

      6. frac-times N/A

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

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

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

   10. Applied egg-rr 44.5%

       \[\leadsto \color{blue}{\frac{\left(\left(1 + \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot -0.027777777777777776\right) \cdot x\right) \cdot 1}{\left(1 - -0.16666666666666666 \cdot \left(x \cdot x\right)\right) \cdot \frac{x}{y}}} \]

   11. Taylor expanded in x around 0

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

   13. Simplified 59.6%

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

   if 9.5000000000000005e-5 < y < 1.1e45

   1. Initial program 99.8%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
   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 74.8%

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. associate-/l* N/A

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

      4. *-inverses N/A

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

      5. *-rgt-identity N/A

         \[\leadsto \sinh y \]

      6. sinh-lowering-sinh.f64 74.8%

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

   9. Applied egg-rr 74.8%

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

   if 1.1e45 < y

   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 80.0%

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

   5. Simplified 80.0%

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

   6. Taylor expanded in y around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

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

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

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

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

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

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

      8. *-commutative N/A

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 77.9%

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

   8. Simplified 77.9%

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

Alternative 9: 74.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.1 \cdot 10^{+45}:\\ \;\;\;\;x \cdot \frac{\sinh y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x \cdot \left(1 + x \cdot \left(x \cdot -0.16666666666666666\right)\right)\right) \cdot \left(y \cdot \left(1 + y \cdot \left(y \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right)\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.1e+45)
   (* x (/ (sinh y) x))
   (/
    (*
     (* x (+ 1.0 (* x (* x -0.16666666666666666))))
     (*
      y
      (+
       1.0
       (* y (* y (+ 0.16666666666666666 (* (* y y) 0.008333333333333333)))))))
    x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.1e+45) {
		tmp = x * (sinh(y) / x);
	} else {
		tmp = ((x * (1.0 + (x * (x * -0.16666666666666666)))) * (y * (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 <= 1.1d+45) then
        tmp = x * (sinh(y) / x)
    else
        tmp = ((x * (1.0d0 + (x * (x * (-0.16666666666666666d0))))) * (y * (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 <= 1.1e+45) {
		tmp = x * (Math.sinh(y) / x);
	} else {
		tmp = ((x * (1.0 + (x * (x * -0.16666666666666666)))) * (y * (1.0 + (y * (y * (0.16666666666666666 + ((y * y) * 0.008333333333333333))))))) / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.1e+45:
		tmp = x * (math.sinh(y) / x)
	else:
		tmp = ((x * (1.0 + (x * (x * -0.16666666666666666)))) * (y * (1.0 + (y * (y * (0.16666666666666666 + ((y * y) * 0.008333333333333333))))))) / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.1e+45)
		tmp = Float64(x * Float64(sinh(y) / x));
	else
		tmp = Float64(Float64(Float64(x * Float64(1.0 + Float64(x * Float64(x * -0.16666666666666666)))) * Float64(y * 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 <= 1.1e+45)
		tmp = x * (sinh(y) / x);
	else
		tmp = ((x * (1.0 + (x * (x * -0.16666666666666666)))) * (y * (1.0 + (y * (y * (0.16666666666666666 + ((y * y) * 0.008333333333333333))))))) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.1e+45], N[(x * N[(N[Sinh[y], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * N[(1.0 + N[(x * N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y * 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 2 regimes

2. if y < 1.1e45

   1. Initial program 87.2%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
   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 74.3%

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

   if 1.1e45 < y

   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 80.0%

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

   5. Simplified 80.0%

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

   6. Taylor expanded in y around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

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

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

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

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

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

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

      8. *-commutative N/A

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 77.9%

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

   8. Simplified 77.9%

      \[\leadsto \frac{\left(x \cdot \left(1 + x \cdot \left(x \cdot -0.16666666666666666\right)\right)\right) \cdot \color{blue}{\left(y \cdot \left(1 + y \cdot \left(y \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right)\right)}}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 60.2% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.9 \cdot 10^{-23}:\\ \;\;\;\;\frac{x}{\left(1 - -0.16666666666666666 \cdot \left(x \cdot x\right)\right) \cdot \frac{x}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x \cdot \left(1 + x \cdot \left(x \cdot -0.16666666666666666\right)\right)\right) \cdot \left(y \cdot \left(1 + y \cdot \left(y \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right)\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 3.9e-23)
   (/ x (* (- 1.0 (* -0.16666666666666666 (* x x))) (/ x y)))
   (/
    (*
     (* x (+ 1.0 (* x (* x -0.16666666666666666))))
     (*
      y
      (+
       1.0
       (* y (* y (+ 0.16666666666666666 (* (* y y) 0.008333333333333333)))))))
    x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 3.9e-23) {
		tmp = x / ((1.0 - (-0.16666666666666666 * (x * x))) * (x / y));
	} else {
		tmp = ((x * (1.0 + (x * (x * -0.16666666666666666)))) * (y * (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 <= 3.9d-23) then
        tmp = x / ((1.0d0 - ((-0.16666666666666666d0) * (x * x))) * (x / y))
    else
        tmp = ((x * (1.0d0 + (x * (x * (-0.16666666666666666d0))))) * (y * (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 <= 3.9e-23) {
		tmp = x / ((1.0 - (-0.16666666666666666 * (x * x))) * (x / y));
	} else {
		tmp = ((x * (1.0 + (x * (x * -0.16666666666666666)))) * (y * (1.0 + (y * (y * (0.16666666666666666 + ((y * y) * 0.008333333333333333))))))) / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 3.9e-23:
		tmp = x / ((1.0 - (-0.16666666666666666 * (x * x))) * (x / y))
	else:
		tmp = ((x * (1.0 + (x * (x * -0.16666666666666666)))) * (y * (1.0 + (y * (y * (0.16666666666666666 + ((y * y) * 0.008333333333333333))))))) / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 3.9e-23)
		tmp = Float64(x / Float64(Float64(1.0 - Float64(-0.16666666666666666 * Float64(x * x))) * Float64(x / y)));
	else
		tmp = Float64(Float64(Float64(x * Float64(1.0 + Float64(x * Float64(x * -0.16666666666666666)))) * Float64(y * 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 <= 3.9e-23)
		tmp = x / ((1.0 - (-0.16666666666666666 * (x * x))) * (x / y));
	else
		tmp = ((x * (1.0 + (x * (x * -0.16666666666666666)))) * (y * (1.0 + (y * (y * (0.16666666666666666 + ((y * y) * 0.008333333333333333))))))) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 3.9e-23], N[(x / N[(N[(1.0 - N[(-0.16666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * N[(1.0 + N[(x * N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y * 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 2 regimes

2. if y < 3.9e-23

   1. Initial program 86.6%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 46.0%

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

   5. Simplified 46.0%

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

   6. Taylor expanded in y around 0

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

   8. Simplified 31.0%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. flip-+ N/A

         \[\leadsto \left(\frac{1 \cdot 1 - \left(x \cdot \left(x \cdot \frac{-1}{6}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{-1}{6}\right)\right)}{1 - x \cdot \left(x \cdot \frac{-1}{6}\right)} \cdot x\right) \cdot \frac{y}{x} \]

      4. associate-*l/ N/A

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

      5. clear-num N/A

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

      6. frac-times N/A

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

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

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

   10. Applied egg-rr 44.0%

       \[\leadsto \color{blue}{\frac{\left(\left(1 + \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot -0.027777777777777776\right) \cdot x\right) \cdot 1}{\left(1 - -0.16666666666666666 \cdot \left(x \cdot x\right)\right) \cdot \frac{x}{y}}} \]

   11. Taylor expanded in x around 0

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

   13. Simplified 59.2%

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

   if 3.9e-23 < y

   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 78.2%

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

   5. Simplified 78.2%

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

   6. Taylor expanded in y around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

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

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

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

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

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

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

      8. *-commutative N/A

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 71.8%

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

   8. Simplified 71.8%

      \[\leadsto \frac{\left(x \cdot \left(1 + x \cdot \left(x \cdot -0.16666666666666666\right)\right)\right) \cdot \color{blue}{\left(y \cdot \left(1 + y \cdot \left(y \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right)\right)}}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 59.7% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 9.2 \cdot 10^{-24}:\\ \;\;\;\;\frac{x}{\left(1 - -0.16666666666666666 \cdot \left(x \cdot x\right)\right) \cdot \frac{x}{y}}\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{+62}:\\ \;\;\;\;y \cdot \left(\left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \cdot \left(1 + x \cdot \left(x \cdot -0.16666666666666666\right)\right)\right)\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+124}:\\ \;\;\;\;y \cdot \left(1 + y \cdot \left(y \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(y \cdot y\right) \cdot \left(0.16666666666666666 + \left(x \cdot x\right) \cdot -0.027777777777777776\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 9.2e-24)
   (/ x (* (- 1.0 (* -0.16666666666666666 (* x x))) (/ x y)))
   (if (<= y 1.15e+62)
     (*
      y
      (*
       (+ 1.0 (* 0.16666666666666666 (* y y)))
       (+ 1.0 (* x (* x -0.16666666666666666)))))
     (if (<= y 5e+124)
       (*
        y
        (+
         1.0
         (* y (* y (+ 0.16666666666666666 (* (* y y) 0.008333333333333333))))))
       (*
        y
        (*
         (* y y)
         (+ 0.16666666666666666 (* (* x x) -0.027777777777777776))))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 9.2e-24) {
		tmp = x / ((1.0 - (-0.16666666666666666 * (x * x))) * (x / y));
	} else if (y <= 1.15e+62) {
		tmp = y * ((1.0 + (0.16666666666666666 * (y * y))) * (1.0 + (x * (x * -0.16666666666666666))));
	} else if (y <= 5e+124) {
		tmp = y * (1.0 + (y * (y * (0.16666666666666666 + ((y * y) * 0.008333333333333333)))));
	} else {
		tmp = y * ((y * y) * (0.16666666666666666 + ((x * x) * -0.027777777777777776)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 9.2d-24) then
        tmp = x / ((1.0d0 - ((-0.16666666666666666d0) * (x * x))) * (x / y))
    else if (y <= 1.15d+62) then
        tmp = y * ((1.0d0 + (0.16666666666666666d0 * (y * y))) * (1.0d0 + (x * (x * (-0.16666666666666666d0)))))
    else if (y <= 5d+124) then
        tmp = y * (1.0d0 + (y * (y * (0.16666666666666666d0 + ((y * y) * 0.008333333333333333d0)))))
    else
        tmp = y * ((y * y) * (0.16666666666666666d0 + ((x * x) * (-0.027777777777777776d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 9.2e-24) {
		tmp = x / ((1.0 - (-0.16666666666666666 * (x * x))) * (x / y));
	} else if (y <= 1.15e+62) {
		tmp = y * ((1.0 + (0.16666666666666666 * (y * y))) * (1.0 + (x * (x * -0.16666666666666666))));
	} else if (y <= 5e+124) {
		tmp = y * (1.0 + (y * (y * (0.16666666666666666 + ((y * y) * 0.008333333333333333)))));
	} else {
		tmp = y * ((y * y) * (0.16666666666666666 + ((x * x) * -0.027777777777777776)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 9.2e-24:
		tmp = x / ((1.0 - (-0.16666666666666666 * (x * x))) * (x / y))
	elif y <= 1.15e+62:
		tmp = y * ((1.0 + (0.16666666666666666 * (y * y))) * (1.0 + (x * (x * -0.16666666666666666))))
	elif y <= 5e+124:
		tmp = y * (1.0 + (y * (y * (0.16666666666666666 + ((y * y) * 0.008333333333333333)))))
	else:
		tmp = y * ((y * y) * (0.16666666666666666 + ((x * x) * -0.027777777777777776)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 9.2e-24)
		tmp = Float64(x / Float64(Float64(1.0 - Float64(-0.16666666666666666 * Float64(x * x))) * Float64(x / y)));
	elseif (y <= 1.15e+62)
		tmp = Float64(y * Float64(Float64(1.0 + Float64(0.16666666666666666 * Float64(y * y))) * Float64(1.0 + Float64(x * Float64(x * -0.16666666666666666)))));
	elseif (y <= 5e+124)
		tmp = Float64(y * Float64(1.0 + Float64(y * Float64(y * Float64(0.16666666666666666 + Float64(Float64(y * y) * 0.008333333333333333))))));
	else
		tmp = Float64(y * Float64(Float64(y * y) * Float64(0.16666666666666666 + Float64(Float64(x * x) * -0.027777777777777776))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 9.2e-24)
		tmp = x / ((1.0 - (-0.16666666666666666 * (x * x))) * (x / y));
	elseif (y <= 1.15e+62)
		tmp = y * ((1.0 + (0.16666666666666666 * (y * y))) * (1.0 + (x * (x * -0.16666666666666666))));
	elseif (y <= 5e+124)
		tmp = y * (1.0 + (y * (y * (0.16666666666666666 + ((y * y) * 0.008333333333333333)))));
	else
		tmp = y * ((y * y) * (0.16666666666666666 + ((x * x) * -0.027777777777777776)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 9.2e-24], N[(x / N[(N[(1.0 - N[(-0.16666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.15e+62], N[(y * N[(N[(1.0 + N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(x * N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5e+124], N[(y * N[(1.0 + N[(y * N[(y * N[(0.16666666666666666 + N[(N[(y * y), $MachinePrecision] * 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[(y * y), $MachinePrecision] * N[(0.16666666666666666 + N[(N[(x * x), $MachinePrecision] * -0.027777777777777776), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < 9.2000000000000004e-24

   1. Initial program 86.6%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 46.0%

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

   5. Simplified 46.0%

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

   6. Taylor expanded in y around 0

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

   8. Simplified 31.0%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. flip-+ N/A

         \[\leadsto \left(\frac{1 \cdot 1 - \left(x \cdot \left(x \cdot \frac{-1}{6}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{-1}{6}\right)\right)}{1 - x \cdot \left(x \cdot \frac{-1}{6}\right)} \cdot x\right) \cdot \frac{y}{x} \]

      4. associate-*l/ N/A

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

      5. clear-num N/A

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

      6. frac-times N/A

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

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

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

   10. Applied egg-rr 44.0%

       \[\leadsto \color{blue}{\frac{\left(\left(1 + \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot -0.027777777777777776\right) \cdot x\right) \cdot 1}{\left(1 - -0.16666666666666666 \cdot \left(x \cdot x\right)\right) \cdot \frac{x}{y}}} \]

   11. Taylor expanded in x around 0

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

   13. Simplified 59.2%

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

   if 9.2000000000000004e-24 < y < 1.14999999999999992e62

   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 74.9%

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

   5. Simplified 74.9%

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

   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-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-in 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 29.3%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

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

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

      4. *-lowering-*.f64 29.3%

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

   10. Applied egg-rr 29.3%

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

   if 1.14999999999999992e62 < y < 4.9999999999999996e124

   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 78.6%

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

   8. Taylor expanded in y around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

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

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

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

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

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

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

      8. *-commutative N/A

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 78.6%

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

   10. Simplified 78.6%

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

   if 4.9999999999999996e124 < y

   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 86.2%

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

   5. Simplified 86.2%

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

   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-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-in 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 86.2%

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

   9. Taylor expanded in y around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*l* N/A

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

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

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

       5. unpow2 N/A

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

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

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

       7. distribute-lft-in N/A

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

       8. metadata-eval N/A

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

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

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

       10. *-commutative N/A

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

       11. *-commutative N/A

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

       12. associate-*l* N/A

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

       13. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \left({x}^{2} \cdot \frac{-1}{36}\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{-1}{36}}\right)\right)\right)\right) \]

       15. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-1}{36}\right)\right)\right)\right) \]

       16. *-lowering-*.f64 86.2%

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{36}\right)\right)\right)\right) \]

   11. Simplified 86.2%

       \[\leadsto y \cdot \color{blue}{\left(\left(y \cdot y\right) \cdot \left(0.16666666666666666 + \left(x \cdot x\right) \cdot -0.027777777777777776\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 9.2 \cdot 10^{-24}:\\ \;\;\;\;\frac{x}{\left(1 - -0.16666666666666666 \cdot \left(x \cdot x\right)\right) \cdot \frac{x}{y}}\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{+62}:\\ \;\;\;\;y \cdot \left(\left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \cdot \left(1 + x \cdot \left(x \cdot -0.16666666666666666\right)\right)\right)\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+124}:\\ \;\;\;\;y \cdot \left(1 + y \cdot \left(y \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(y \cdot y\right) \cdot \left(0.16666666666666666 + \left(x \cdot x\right) \cdot -0.027777777777777776\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 59.5% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(\left(y \cdot y\right) \cdot \left(0.16666666666666666 + \left(x \cdot x\right) \cdot -0.027777777777777776\right)\right)\\ \mathbf{if}\;y \leq 6.4 \cdot 10^{+16}:\\ \;\;\;\;\frac{x}{\left(1 - -0.16666666666666666 \cdot \left(x \cdot x\right)\right) \cdot \frac{x}{y}}\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{+62}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+124}:\\ \;\;\;\;y \cdot \left(1 + y \cdot \left(y \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (*
          y
          (*
           (* y y)
           (+ 0.16666666666666666 (* (* x x) -0.027777777777777776))))))
   (if (<= y 6.4e+16)
     (/ x (* (- 1.0 (* -0.16666666666666666 (* x x))) (/ x y)))
     (if (<= y 1.15e+62)
       t_0
       (if (<= y 5e+124)
         (*
          y
          (+
           1.0
           (*
            y
            (* y (+ 0.16666666666666666 (* (* y y) 0.008333333333333333))))))
         t_0)))))

C

double code(double x, double y) {
	double t_0 = y * ((y * y) * (0.16666666666666666 + ((x * x) * -0.027777777777777776)));
	double tmp;
	if (y <= 6.4e+16) {
		tmp = x / ((1.0 - (-0.16666666666666666 * (x * x))) * (x / y));
	} else if (y <= 1.15e+62) {
		tmp = t_0;
	} else if (y <= 5e+124) {
		tmp = y * (1.0 + (y * (y * (0.16666666666666666 + ((y * y) * 0.008333333333333333)))));
	} 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 = y * ((y * y) * (0.16666666666666666d0 + ((x * x) * (-0.027777777777777776d0))))
    if (y <= 6.4d+16) then
        tmp = x / ((1.0d0 - ((-0.16666666666666666d0) * (x * x))) * (x / y))
    else if (y <= 1.15d+62) then
        tmp = t_0
    else if (y <= 5d+124) then
        tmp = y * (1.0d0 + (y * (y * (0.16666666666666666d0 + ((y * y) * 0.008333333333333333d0)))))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = y * ((y * y) * (0.16666666666666666 + ((x * x) * -0.027777777777777776)));
	double tmp;
	if (y <= 6.4e+16) {
		tmp = x / ((1.0 - (-0.16666666666666666 * (x * x))) * (x / y));
	} else if (y <= 1.15e+62) {
		tmp = t_0;
	} else if (y <= 5e+124) {
		tmp = y * (1.0 + (y * (y * (0.16666666666666666 + ((y * y) * 0.008333333333333333)))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * ((y * y) * (0.16666666666666666 + ((x * x) * -0.027777777777777776)))
	tmp = 0
	if y <= 6.4e+16:
		tmp = x / ((1.0 - (-0.16666666666666666 * (x * x))) * (x / y))
	elif y <= 1.15e+62:
		tmp = t_0
	elif y <= 5e+124:
		tmp = y * (1.0 + (y * (y * (0.16666666666666666 + ((y * y) * 0.008333333333333333)))))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * Float64(Float64(y * y) * Float64(0.16666666666666666 + Float64(Float64(x * x) * -0.027777777777777776))))
	tmp = 0.0
	if (y <= 6.4e+16)
		tmp = Float64(x / Float64(Float64(1.0 - Float64(-0.16666666666666666 * Float64(x * x))) * Float64(x / y)));
	elseif (y <= 1.15e+62)
		tmp = t_0;
	elseif (y <= 5e+124)
		tmp = Float64(y * Float64(1.0 + Float64(y * Float64(y * Float64(0.16666666666666666 + Float64(Float64(y * y) * 0.008333333333333333))))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y * ((y * y) * (0.16666666666666666 + ((x * x) * -0.027777777777777776)));
	tmp = 0.0;
	if (y <= 6.4e+16)
		tmp = x / ((1.0 - (-0.16666666666666666 * (x * x))) * (x / y));
	elseif (y <= 1.15e+62)
		tmp = t_0;
	elseif (y <= 5e+124)
		tmp = y * (1.0 + (y * (y * (0.16666666666666666 + ((y * y) * 0.008333333333333333)))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(N[(y * y), $MachinePrecision] * N[(0.16666666666666666 + N[(N[(x * x), $MachinePrecision] * -0.027777777777777776), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 6.4e+16], N[(x / N[(N[(1.0 - N[(-0.16666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.15e+62], t$95$0, If[LessEqual[y, 5e+124], N[(y * N[(1.0 + N[(y * N[(y * N[(0.16666666666666666 + N[(N[(y * y), $MachinePrecision] * 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y < 6.4e16

   1. Initial program 87.0%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 47.1%

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

   5. Simplified 47.1%

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

   6. Taylor expanded in y around 0

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

   8. Simplified 31.2%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. flip-+ N/A

         \[\leadsto \left(\frac{1 \cdot 1 - \left(x \cdot \left(x \cdot \frac{-1}{6}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{-1}{6}\right)\right)}{1 - x \cdot \left(x \cdot \frac{-1}{6}\right)} \cdot x\right) \cdot \frac{y}{x} \]

      4. associate-*l/ N/A

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

      5. clear-num N/A

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

      6. frac-times N/A

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

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

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

   10. Applied egg-rr 43.8%

       \[\leadsto \color{blue}{\frac{\left(\left(1 + \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot -0.027777777777777776\right) \cdot x\right) \cdot 1}{\left(1 - -0.16666666666666666 \cdot \left(x \cdot x\right)\right) \cdot \frac{x}{y}}} \]

   11. Taylor expanded in x around 0

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

   13. Simplified 58.6%

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

   if 6.4e16 < y < 1.14999999999999992e62 or 4.9999999999999996e124 < y

   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 82.9%

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

   5. Simplified 82.9%

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

   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-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-in 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 74.8%

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

   9. Taylor expanded in y around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*l* N/A

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

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

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

       5. unpow2 N/A

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

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

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

       7. distribute-lft-in N/A

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

       8. metadata-eval N/A

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

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

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

       10. *-commutative N/A

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

       11. *-commutative N/A

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

       12. associate-*l* N/A

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

       13. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \left({x}^{2} \cdot \frac{-1}{36}\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{-1}{36}}\right)\right)\right)\right) \]

       15. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-1}{36}\right)\right)\right)\right) \]

       16. *-lowering-*.f64 74.8%

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{36}\right)\right)\right)\right) \]

   11. Simplified 74.8%

       \[\leadsto y \cdot \color{blue}{\left(\left(y \cdot y\right) \cdot \left(0.16666666666666666 + \left(x \cdot x\right) \cdot -0.027777777777777776\right)\right)} \]

   if 1.14999999999999992e62 < y < 4.9999999999999996e124

   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 78.6%

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

   8. Taylor expanded in y around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

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

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

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

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

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

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

      8. *-commutative N/A

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 78.6%

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

   10. Simplified 78.6%

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

Alternative 13: 71.6% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{+127}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{1 + y \cdot \left(y \cdot \left(0.16666666666666666 + y \cdot \left(y \cdot \left(0.008333333333333333 + y \cdot \left(y \cdot 0.0001984126984126984\right)\right)\right)\right)\right)}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(y \cdot y\right) \cdot \left(0.16666666666666666 + \left(x \cdot x\right) \cdot -0.027777777777777776\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 5e+127)
   (*
    x
    (*
     y
     (/
      (+
       1.0
       (*
        y
        (*
         y
         (+
          0.16666666666666666
          (*
           y
           (*
            y
            (+ 0.008333333333333333 (* y (* y 0.0001984126984126984)))))))))
      x)))
   (*
    y
    (* (* y y) (+ 0.16666666666666666 (* (* x x) -0.027777777777777776))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 5e+127) {
		tmp = x * (y * ((1.0 + (y * (y * (0.16666666666666666 + (y * (y * (0.008333333333333333 + (y * (y * 0.0001984126984126984))))))))) / x));
	} else {
		tmp = y * ((y * y) * (0.16666666666666666 + ((x * x) * -0.027777777777777776)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 5d+127) then
        tmp = x * (y * ((1.0d0 + (y * (y * (0.16666666666666666d0 + (y * (y * (0.008333333333333333d0 + (y * (y * 0.0001984126984126984d0))))))))) / x))
    else
        tmp = y * ((y * y) * (0.16666666666666666d0 + ((x * x) * (-0.027777777777777776d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 5e+127) {
		tmp = x * (y * ((1.0 + (y * (y * (0.16666666666666666 + (y * (y * (0.008333333333333333 + (y * (y * 0.0001984126984126984))))))))) / x));
	} else {
		tmp = y * ((y * y) * (0.16666666666666666 + ((x * x) * -0.027777777777777776)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 5e+127:
		tmp = x * (y * ((1.0 + (y * (y * (0.16666666666666666 + (y * (y * (0.008333333333333333 + (y * (y * 0.0001984126984126984))))))))) / x))
	else:
		tmp = y * ((y * y) * (0.16666666666666666 + ((x * x) * -0.027777777777777776)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 5e+127)
		tmp = Float64(x * Float64(y * Float64(Float64(1.0 + Float64(y * Float64(y * Float64(0.16666666666666666 + Float64(y * Float64(y * Float64(0.008333333333333333 + Float64(y * Float64(y * 0.0001984126984126984))))))))) / x)));
	else
		tmp = Float64(y * Float64(Float64(y * y) * Float64(0.16666666666666666 + Float64(Float64(x * x) * -0.027777777777777776))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 5e+127)
		tmp = x * (y * ((1.0 + (y * (y * (0.16666666666666666 + (y * (y * (0.008333333333333333 + (y * (y * 0.0001984126984126984))))))))) / x));
	else
		tmp = y * ((y * y) * (0.16666666666666666 + ((x * x) * -0.027777777777777776)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 5e+127], N[(x * N[(y * N[(N[(1.0 + N[(y * N[(y * N[(0.16666666666666666 + N[(y * N[(y * N[(0.008333333333333333 + N[(y * N[(y * 0.0001984126984126984), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[(y * y), $MachinePrecision] * N[(0.16666666666666666 + N[(N[(x * x), $MachinePrecision] * -0.027777777777777776), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.0000000000000004e127

   1. Initial program 88.1%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 74.4%

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

   8. Taylor expanded in y around 0

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

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

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

      2. +-commutative N/A

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

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

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

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

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

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

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

      6. unpow2 N/A

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

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

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

      8. +-commutative N/A

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

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

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. /-lowering-/.f64 N/A

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

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

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

   10. Simplified 70.3%

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

   11. Taylor expanded in x around 0

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

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

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

   13. Simplified 70.3%

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

   if 5.0000000000000004e127 < y

   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 86.2%

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

   5. Simplified 86.2%

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

   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-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-in 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 86.2%

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

   9. Taylor expanded in y around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*l* N/A

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

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

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

       5. unpow2 N/A

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

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

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

       7. distribute-lft-in N/A

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

       8. metadata-eval N/A

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

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

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

       10. *-commutative N/A

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

       11. *-commutative N/A

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

       12. associate-*l* N/A

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

       13. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \left({x}^{2} \cdot \frac{-1}{36}\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{-1}{36}}\right)\right)\right)\right) \]

       15. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-1}{36}\right)\right)\right)\right) \]

       16. *-lowering-*.f64 86.2%

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{36}\right)\right)\right)\right) \]

   11. Simplified 86.2%

       \[\leadsto y \cdot \color{blue}{\left(\left(y \cdot y\right) \cdot \left(0.16666666666666666 + \left(x \cdot x\right) \cdot -0.027777777777777776\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 70.2% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{+127}:\\ \;\;\;\;x \cdot \frac{y \cdot \left(1 + y \cdot \left(y \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(y \cdot y\right) \cdot \left(0.16666666666666666 + \left(x \cdot x\right) \cdot -0.027777777777777776\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 5e+127)
   (*
    x
    (/
     (*
      y
      (+
       1.0
       (* y (* y (+ 0.16666666666666666 (* (* y y) 0.008333333333333333))))))
     x))
   (*
    y
    (* (* y y) (+ 0.16666666666666666 (* (* x x) -0.027777777777777776))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 5e+127) {
		tmp = x * ((y * (1.0 + (y * (y * (0.16666666666666666 + ((y * y) * 0.008333333333333333)))))) / x);
	} else {
		tmp = y * ((y * y) * (0.16666666666666666 + ((x * x) * -0.027777777777777776)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 5d+127) then
        tmp = x * ((y * (1.0d0 + (y * (y * (0.16666666666666666d0 + ((y * y) * 0.008333333333333333d0)))))) / x)
    else
        tmp = y * ((y * y) * (0.16666666666666666d0 + ((x * x) * (-0.027777777777777776d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 5e+127) {
		tmp = x * ((y * (1.0 + (y * (y * (0.16666666666666666 + ((y * y) * 0.008333333333333333)))))) / x);
	} else {
		tmp = y * ((y * y) * (0.16666666666666666 + ((x * x) * -0.027777777777777776)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 5e+127:
		tmp = x * ((y * (1.0 + (y * (y * (0.16666666666666666 + ((y * y) * 0.008333333333333333)))))) / x)
	else:
		tmp = y * ((y * y) * (0.16666666666666666 + ((x * x) * -0.027777777777777776)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 5e+127)
		tmp = Float64(x * Float64(Float64(y * Float64(1.0 + Float64(y * Float64(y * Float64(0.16666666666666666 + Float64(Float64(y * y) * 0.008333333333333333)))))) / x));
	else
		tmp = Float64(y * Float64(Float64(y * y) * Float64(0.16666666666666666 + Float64(Float64(x * x) * -0.027777777777777776))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 5e+127)
		tmp = x * ((y * (1.0 + (y * (y * (0.16666666666666666 + ((y * y) * 0.008333333333333333)))))) / x);
	else
		tmp = y * ((y * y) * (0.16666666666666666 + ((x * x) * -0.027777777777777776)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 5e+127], N[(x * N[(N[(y * N[(1.0 + N[(y * N[(y * N[(0.16666666666666666 + N[(N[(y * y), $MachinePrecision] * 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[(y * y), $MachinePrecision] * N[(0.16666666666666666 + N[(N[(x * x), $MachinePrecision] * -0.027777777777777776), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.0000000000000004e127

   1. Initial program 88.1%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 74.4%

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

   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 70.3%

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

   10. Simplified 70.3%

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

   if 5.0000000000000004e127 < y

   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 86.2%

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

   5. Simplified 86.2%

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

   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-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-in 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 86.2%

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

   9. Taylor expanded in y around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*l* N/A

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

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

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

       5. unpow2 N/A

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

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

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

       7. distribute-lft-in N/A

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

       8. metadata-eval N/A

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

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

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

       10. *-commutative N/A

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

       11. *-commutative N/A

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

       12. associate-*l* N/A

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

       13. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \left({x}^{2} \cdot \frac{-1}{36}\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{-1}{36}}\right)\right)\right)\right) \]

       15. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-1}{36}\right)\right)\right)\right) \]

       16. *-lowering-*.f64 86.2%

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{36}\right)\right)\right)\right) \]

   11. Simplified 86.2%

       \[\leadsto y \cdot \color{blue}{\left(\left(y \cdot y\right) \cdot \left(0.16666666666666666 + \left(x \cdot x\right) \cdot -0.027777777777777776\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 57.0% accurate, 10.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 6.7 \cdot 10^{+16}:\\ \;\;\;\;\frac{x}{\frac{x}{y}}\\ \mathbf{elif}\;y \leq 4.4 \cdot 10^{+97}:\\ \;\;\;\;y \cdot \left(1 + -0.16666666666666666 \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot \left(y \cdot \left(y \cdot y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 6.7e+16)
   (/ x (/ x y))
   (if (<= y 4.4e+97)
     (* y (+ 1.0 (* -0.16666666666666666 (* x x))))
     (* 0.16666666666666666 (* y (* y y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 6.7e+16) {
		tmp = x / (x / y);
	} else if (y <= 4.4e+97) {
		tmp = y * (1.0 + (-0.16666666666666666 * (x * x)));
	} else {
		tmp = 0.16666666666666666 * (y * (y * y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 6.7d+16) then
        tmp = x / (x / y)
    else if (y <= 4.4d+97) then
        tmp = y * (1.0d0 + ((-0.16666666666666666d0) * (x * x)))
    else
        tmp = 0.16666666666666666d0 * (y * (y * y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 6.7e+16) {
		tmp = x / (x / y);
	} else if (y <= 4.4e+97) {
		tmp = y * (1.0 + (-0.16666666666666666 * (x * x)));
	} else {
		tmp = 0.16666666666666666 * (y * (y * y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 6.7e+16:
		tmp = x / (x / y)
	elif y <= 4.4e+97:
		tmp = y * (1.0 + (-0.16666666666666666 * (x * x)))
	else:
		tmp = 0.16666666666666666 * (y * (y * y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 6.7e+16)
		tmp = Float64(x / Float64(x / y));
	elseif (y <= 4.4e+97)
		tmp = Float64(y * Float64(1.0 + Float64(-0.16666666666666666 * Float64(x * x))));
	else
		tmp = Float64(0.16666666666666666 * Float64(y * Float64(y * y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 6.7e+16)
		tmp = x / (x / y);
	elseif (y <= 4.4e+97)
		tmp = y * (1.0 + (-0.16666666666666666 * (x * x)));
	else
		tmp = 0.16666666666666666 * (y * (y * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 6.7e+16], N[(x / N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.4e+97], N[(y * N[(1.0 + N[(-0.16666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.16666666666666666 * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 6.7e16

   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 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
   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 74.8%

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

   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 58.3%

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

   10. Simplified 58.3%

       \[\leadsto x \cdot \color{blue}{\frac{y}{x}} \]
   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 58.4%

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

   12. Applied egg-rr 58.4%

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

   if 6.7e16 < y < 4.4000000000000002e97

   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 3.0%

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

   6. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-in N/A

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

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

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

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

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

      6. *-commutative N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 23.5%

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

   8. Simplified 23.5%

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

   if 4.4000000000000002e97 < 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 68.6%

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

   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 66.4%

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

   10. Simplified 66.4%

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

   11. Taylor expanded in y around inf

       \[\leadsto \color{blue}{\frac{1}{6} \cdot {y}^{3}} \]
   12. Step-by-step derivation

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \color{blue}{\left({y}^{3}\right)}\right) \]

       2. cube-mult N/A

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

       3. unpow2 N/A

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

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

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

       5. unpow2 N/A

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

       6. *-lowering-*.f64 66.0%

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

   13. Simplified 66.0%

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

4. Final simplification 57.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6.7 \cdot 10^{+16}:\\ \;\;\;\;\frac{x}{\frac{x}{y}}\\ \mathbf{elif}\;y \leq 4.4 \cdot 10^{+97}:\\ \;\;\;\;y \cdot \left(1 + -0.16666666666666666 \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot \left(y \cdot \left(y \cdot y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 58.1% accurate, 11.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 6.4 \cdot 10^{+16}:\\ \;\;\;\;\frac{x}{\left(1 - -0.16666666666666666 \cdot \left(x \cdot x\right)\right) \cdot \frac{x}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(y \cdot y\right) \cdot \left(0.16666666666666666 + \left(x \cdot x\right) \cdot -0.027777777777777776\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 6.4e+16)
   (/ x (* (- 1.0 (* -0.16666666666666666 (* x x))) (/ x y)))
   (*
    y
    (* (* y y) (+ 0.16666666666666666 (* (* x x) -0.027777777777777776))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 6.4e+16) {
		tmp = x / ((1.0 - (-0.16666666666666666 * (x * x))) * (x / y));
	} else {
		tmp = y * ((y * y) * (0.16666666666666666 + ((x * x) * -0.027777777777777776)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 6.4d+16) then
        tmp = x / ((1.0d0 - ((-0.16666666666666666d0) * (x * x))) * (x / y))
    else
        tmp = y * ((y * y) * (0.16666666666666666d0 + ((x * x) * (-0.027777777777777776d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 6.4e+16) {
		tmp = x / ((1.0 - (-0.16666666666666666 * (x * x))) * (x / y));
	} else {
		tmp = y * ((y * y) * (0.16666666666666666 + ((x * x) * -0.027777777777777776)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 6.4e+16:
		tmp = x / ((1.0 - (-0.16666666666666666 * (x * x))) * (x / y))
	else:
		tmp = y * ((y * y) * (0.16666666666666666 + ((x * x) * -0.027777777777777776)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 6.4e+16)
		tmp = Float64(x / Float64(Float64(1.0 - Float64(-0.16666666666666666 * Float64(x * x))) * Float64(x / y)));
	else
		tmp = Float64(y * Float64(Float64(y * y) * Float64(0.16666666666666666 + Float64(Float64(x * x) * -0.027777777777777776))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 6.4e+16)
		tmp = x / ((1.0 - (-0.16666666666666666 * (x * x))) * (x / y));
	else
		tmp = y * ((y * y) * (0.16666666666666666 + ((x * x) * -0.027777777777777776)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 6.4e+16], N[(x / N[(N[(1.0 - N[(-0.16666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[(y * y), $MachinePrecision] * N[(0.16666666666666666 + N[(N[(x * x), $MachinePrecision] * -0.027777777777777776), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6.4e16

   1. Initial program 87.0%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 47.1%

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

   5. Simplified 47.1%

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

   6. Taylor expanded in y around 0

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

   8. Simplified 31.2%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. flip-+ N/A

         \[\leadsto \left(\frac{1 \cdot 1 - \left(x \cdot \left(x \cdot \frac{-1}{6}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{-1}{6}\right)\right)}{1 - x \cdot \left(x \cdot \frac{-1}{6}\right)} \cdot x\right) \cdot \frac{y}{x} \]

      4. associate-*l/ N/A

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

      5. clear-num N/A

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

      6. frac-times N/A

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

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

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

   10. Applied egg-rr 43.8%

       \[\leadsto \color{blue}{\frac{\left(\left(1 + \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot -0.027777777777777776\right) \cdot x\right) \cdot 1}{\left(1 - -0.16666666666666666 \cdot \left(x \cdot x\right)\right) \cdot \frac{x}{y}}} \]

   11. Taylor expanded in x around 0

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

   13. Simplified 58.6%

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

   if 6.4e16 < y

   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 77.6%

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

   5. Simplified 77.6%

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

   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-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-in 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 62.4%

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

   9. Taylor expanded in y around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*l* N/A

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

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

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

       5. unpow2 N/A

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

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

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

       7. distribute-lft-in N/A

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

       8. metadata-eval N/A

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

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

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

       10. *-commutative N/A

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

       11. *-commutative N/A

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

       12. associate-*l* N/A

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

       13. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \left({x}^{2} \cdot \frac{-1}{36}\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{-1}{36}}\right)\right)\right)\right) \]

       15. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-1}{36}\right)\right)\right)\right) \]

       16. *-lowering-*.f64 62.4%

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{36}\right)\right)\right)\right) \]

   11. Simplified 62.4%

       \[\leadsto y \cdot \color{blue}{\left(\left(y \cdot y\right) \cdot \left(0.16666666666666666 + \left(x \cdot x\right) \cdot -0.027777777777777776\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 57.8% accurate, 11.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 6.4 \cdot 10^{+16}:\\ \;\;\;\;\frac{x}{\frac{x}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(y \cdot y\right) \cdot \left(0.16666666666666666 + \left(x \cdot x\right) \cdot -0.027777777777777776\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 6.4e+16)
   (/ x (/ x y))
   (*
    y
    (* (* y y) (+ 0.16666666666666666 (* (* x x) -0.027777777777777776))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 6.4e+16) {
		tmp = x / (x / y);
	} else {
		tmp = y * ((y * y) * (0.16666666666666666 + ((x * x) * -0.027777777777777776)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 6.4d+16) then
        tmp = x / (x / y)
    else
        tmp = y * ((y * y) * (0.16666666666666666d0 + ((x * x) * (-0.027777777777777776d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 6.4e+16) {
		tmp = x / (x / y);
	} else {
		tmp = y * ((y * y) * (0.16666666666666666 + ((x * x) * -0.027777777777777776)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 6.4e+16:
		tmp = x / (x / y)
	else:
		tmp = y * ((y * y) * (0.16666666666666666 + ((x * x) * -0.027777777777777776)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 6.4e+16)
		tmp = Float64(x / Float64(x / y));
	else
		tmp = Float64(y * Float64(Float64(y * y) * Float64(0.16666666666666666 + Float64(Float64(x * x) * -0.027777777777777776))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 6.4e+16)
		tmp = x / (x / y);
	else
		tmp = y * ((y * y) * (0.16666666666666666 + ((x * x) * -0.027777777777777776)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 6.4e+16], N[(x / N[(x / y), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[(y * y), $MachinePrecision] * N[(0.16666666666666666 + N[(N[(x * x), $MachinePrecision] * -0.027777777777777776), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6.4e16

   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 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
   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 74.8%

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

   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 58.3%

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

   10. Simplified 58.3%

       \[\leadsto x \cdot \color{blue}{\frac{y}{x}} \]
   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 58.4%

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

   12. Applied egg-rr 58.4%

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

   if 6.4e16 < y

   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 77.6%

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

   5. Simplified 77.6%

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

   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-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-in 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 62.4%

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

   9. Taylor expanded in y around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*l* N/A

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

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

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

       5. unpow2 N/A

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

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

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

       7. distribute-lft-in N/A

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

       8. metadata-eval N/A

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

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

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

       10. *-commutative N/A

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

       11. *-commutative N/A

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

       12. associate-*l* N/A

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

       13. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \left({x}^{2} \cdot \frac{-1}{36}\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{-1}{36}}\right)\right)\right)\right) \]

       15. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-1}{36}\right)\right)\right)\right) \]

       16. *-lowering-*.f64 62.4%

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{36}\right)\right)\right)\right) \]

   11. Simplified 62.4%

       \[\leadsto y \cdot \color{blue}{\left(\left(y \cdot y\right) \cdot \left(0.16666666666666666 + \left(x \cdot x\right) \cdot -0.027777777777777776\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 56.9% accurate, 12.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 6.4 \cdot 10^{+16}:\\ \;\;\;\;\frac{x}{\frac{x}{y}}\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{+97}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(y \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot \left(y \cdot \left(y \cdot y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 6.4e+16)
   (/ x (/ x y))
   (if (<= y 3.8e+97)
     (* -0.16666666666666666 (* y (* x x)))
     (* 0.16666666666666666 (* y (* y y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 6.4e+16) {
		tmp = x / (x / y);
	} else if (y <= 3.8e+97) {
		tmp = -0.16666666666666666 * (y * (x * x));
	} else {
		tmp = 0.16666666666666666 * (y * (y * y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 6.4d+16) then
        tmp = x / (x / y)
    else if (y <= 3.8d+97) then
        tmp = (-0.16666666666666666d0) * (y * (x * x))
    else
        tmp = 0.16666666666666666d0 * (y * (y * y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 6.4e+16) {
		tmp = x / (x / y);
	} else if (y <= 3.8e+97) {
		tmp = -0.16666666666666666 * (y * (x * x));
	} else {
		tmp = 0.16666666666666666 * (y * (y * y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 6.4e+16:
		tmp = x / (x / y)
	elif y <= 3.8e+97:
		tmp = -0.16666666666666666 * (y * (x * x))
	else:
		tmp = 0.16666666666666666 * (y * (y * y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 6.4e+16)
		tmp = Float64(x / Float64(x / y));
	elseif (y <= 3.8e+97)
		tmp = Float64(-0.16666666666666666 * Float64(y * Float64(x * x)));
	else
		tmp = Float64(0.16666666666666666 * Float64(y * Float64(y * y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 6.4e+16)
		tmp = x / (x / y);
	elseif (y <= 3.8e+97)
		tmp = -0.16666666666666666 * (y * (x * x));
	else
		tmp = 0.16666666666666666 * (y * (y * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 6.4e+16], N[(x / N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.8e+97], N[(-0.16666666666666666 * N[(y * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.16666666666666666 * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 6.4e16

   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 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
   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 74.8%

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

   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 58.3%

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

   10. Simplified 58.3%

       \[\leadsto x \cdot \color{blue}{\frac{y}{x}} \]
   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 58.4%

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

   12. Applied egg-rr 58.4%

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

   if 6.4e16 < y < 3.80000000000000036e97

   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 71.4%

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

   5. Simplified 71.4%

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

   6. Taylor expanded in y around 0

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

   8. Simplified 23.5%

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

   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. *-lowering-*.f64 N/A

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

       2. *-commutative N/A

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

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

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

       4. unpow2 N/A

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

       5. *-lowering-*.f64 22.6%

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

   11. Simplified 22.6%

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

   if 3.80000000000000036e97 < 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 68.6%

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

   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 66.4%

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

   10. Simplified 66.4%

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

   11. Taylor expanded in y around inf

       \[\leadsto \color{blue}{\frac{1}{6} \cdot {y}^{3}} \]
   12. Step-by-step derivation

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \color{blue}{\left({y}^{3}\right)}\right) \]

       2. cube-mult N/A

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

       3. unpow2 N/A

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

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

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

       5. unpow2 N/A

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

       6. *-lowering-*.f64 66.0%

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

   13. Simplified 66.0%

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

Alternative 19: 50.3% accurate, 12.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 6.4 \cdot 10^{+16}:\\ \;\;\;\;\frac{x}{\frac{x}{y}}\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{+184}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(y \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 6.4e+16)
   (/ x (/ x y))
   (if (<= y 1.8e+184) (* -0.16666666666666666 (* y (* x x))) (* x (/ y x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 6.4e+16) {
		tmp = x / (x / y);
	} else if (y <= 1.8e+184) {
		tmp = -0.16666666666666666 * (y * (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 (y <= 6.4d+16) then
        tmp = x / (x / y)
    else if (y <= 1.8d+184) then
        tmp = (-0.16666666666666666d0) * (y * (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 (y <= 6.4e+16) {
		tmp = x / (x / y);
	} else if (y <= 1.8e+184) {
		tmp = -0.16666666666666666 * (y * (x * x));
	} else {
		tmp = x * (y / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 6.4e+16:
		tmp = x / (x / y)
	elif y <= 1.8e+184:
		tmp = -0.16666666666666666 * (y * (x * x))
	else:
		tmp = x * (y / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 6.4e+16)
		tmp = Float64(x / Float64(x / y));
	elseif (y <= 1.8e+184)
		tmp = Float64(-0.16666666666666666 * Float64(y * 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 (y <= 6.4e+16)
		tmp = x / (x / y);
	elseif (y <= 1.8e+184)
		tmp = -0.16666666666666666 * (y * (x * x));
	else
		tmp = x * (y / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 6.4e+16], N[(x / N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.8e+184], N[(-0.16666666666666666 * N[(y * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(y / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 6.4e16

   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 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
   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 74.8%

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

   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 58.3%

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

   10. Simplified 58.3%

       \[\leadsto x \cdot \color{blue}{\frac{y}{x}} \]
   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 58.4%

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

   12. Applied egg-rr 58.4%

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

   if 6.4e16 < y < 1.80000000000000007e184

   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 73.3%

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

   5. Simplified 73.3%

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

   6. Taylor expanded in y around 0

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

   8. Simplified 25.5%

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

   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. *-lowering-*.f64 N/A

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

       2. *-commutative N/A

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

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

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

       4. unpow2 N/A

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

       5. *-lowering-*.f64 24.1%

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

   11. Simplified 24.1%

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

   if 1.80000000000000007e184 < 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 63.2%

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

   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 38.6%

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

   10. Simplified 38.6%

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

Alternative 20: 58.3% accurate, 12.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.4:\\ \;\;\;\;\frac{x}{\frac{x}{y}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y \cdot \left(0.16666666666666666 \cdot \left(y \cdot y\right)\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2.4)
   (/ x (/ x y))
   (* x (/ (* y (* 0.16666666666666666 (* y y))) x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.4) {
		tmp = x / (x / y);
	} else {
		tmp = x * ((y * (0.16666666666666666 * (y * 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 <= 2.4d0) then
        tmp = x / (x / y)
    else
        tmp = x * ((y * (0.16666666666666666d0 * (y * y))) / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2.4) {
		tmp = x / (x / y);
	} else {
		tmp = x * ((y * (0.16666666666666666 * (y * y))) / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.4:
		tmp = x / (x / y)
	else:
		tmp = x * ((y * (0.16666666666666666 * (y * y))) / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.4)
		tmp = Float64(x / Float64(x / y));
	else
		tmp = Float64(x * Float64(Float64(y * Float64(0.16666666666666666 * Float64(y * y))) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.4)
		tmp = x / (x / y);
	else
		tmp = x * ((y * (0.16666666666666666 * (y * y))) / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.4], N[(x / N[(x / y), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(y * N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.39999999999999991

   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 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
   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 74.4%

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

   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 59.1%

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

   10. Simplified 59.1%

       \[\leadsto x \cdot \color{blue}{\frac{y}{x}} \]
   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 59.2%

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

   12. Applied egg-rr 59.2%

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

   if 2.39999999999999991 < 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 67.3%

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

   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 47.6%

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

   10. Simplified 47.6%

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

   11. Taylor expanded in y around inf

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

       1. *-commutative N/A

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

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

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

       3. unpow2 N/A

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

       4. *-lowering-*.f64 47.6%

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

   13. Simplified 47.6%

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

4. Final simplification 56.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.4:\\ \;\;\;\;\frac{x}{\frac{x}{y}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y \cdot \left(0.16666666666666666 \cdot \left(y \cdot y\right)\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 66.9% accurate, 15.8× speedup

Math

\[\begin{array}{l} \\ x \cdot \frac{y \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right)}{x} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (* x (/ (* y (+ 1.0 (* 0.16666666666666666 (* y y)))) x)))

C

double code(double x, double y) {
	return x * ((y * (1.0 + (0.16666666666666666 * (y * y)))) / x);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x * ((y * (1.0d0 + (0.16666666666666666d0 * (y * y)))) / x)
end function

Java

public static double code(double x, double y) {
	return x * ((y * (1.0 + (0.16666666666666666 * (y * y)))) / x);
}

Python

def code(x, y):
	return x * ((y * (1.0 + (0.16666666666666666 * (y * y)))) / x)

Julia

function code(x, y)
	return Float64(x * Float64(Float64(y * Float64(1.0 + Float64(0.16666666666666666 * Float64(y * y)))) / x))
end

MATLAB

function tmp = code(x, y)
	tmp = x * ((y * (1.0 + (0.16666666666666666 * (y * y)))) / x);
end

Wolfram

code[x_, y_] := N[(x * N[(N[(y * N[(1.0 + N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 89.5%

   \[\frac{\sin x \cdot \sinh y}{x} \]
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 73.0%

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

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 66.1%

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

10. Simplified 66.1%

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

Alternative 22: 49.9% accurate, 41.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.5%

   \[\frac{\sin x \cdot \sinh y}{x} \]
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 73.0%

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

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 51.1%

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

10. Simplified 51.1%

    \[\leadsto x \cdot \color{blue}{\frac{y}{x}} \]
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 51.2%

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

12. Applied egg-rr 51.2%

    \[\leadsto \color{blue}{\frac{x}{\frac{x}{y}}} \]
13. Add Preprocessing

Alternative 23: 50.3% 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 89.5%

   \[\frac{\sin x \cdot \sinh y}{x} \]
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 73.0%

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

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 51.1%

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

10. Simplified 51.1%

    \[\leadsto x \cdot \color{blue}{\frac{y}{x}} \]
11. Add Preprocessing

Alternative 24: 28.7% 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 89.5%

   \[\frac{\sin x \cdot \sinh y}{x} \]
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 73.0%

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

8. Taylor expanded in y around 0

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

10. Simplified 31.1%

    \[\leadsto \color{blue}{y} \]
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 2024161 
(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))