Linear.Quaternion:$csinh from linear-1.19.1.3

Percentage Accurate: 99.9% → 99.9%

Time: 11.1s

Alternatives: 21

Speedup: 1.0×

• 49.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. associate-*r/ N/A

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

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

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

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

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

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

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

   5. sin-lowering-sin.f64 99.9%

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

4. Applied egg-rr 99.9%

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

Alternative 2: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

Alternative 3: 92.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot x\right)\\ t_1 := \frac{\sin y}{y} \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot 0.041666666666666664\right)\right)\right)\\ t_2 := \left(x \cdot x\right) \cdot 0.001388888888888889\\ \mathbf{if}\;x \leq 0.125:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{+41}:\\ \;\;\;\;\frac{\cosh x \cdot y}{y}\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{+77}:\\ \;\;\;\;\left(1 + \left(x \cdot x\right) \cdot \left(0.5 + \frac{\left(x \cdot x\right) \cdot \left(7.233796296296296 \cdot 10^{-5} + t\_0 \cdot \left(t\_0 \cdot 2.6791838134430728 \cdot 10^{-9}\right)\right)}{0.001736111111111111 + t\_2 \cdot \left(t\_2 - 0.041666666666666664\right)}\right)\right) \cdot \left(1 + \left(y \cdot y\right) \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* x (* x x)))
        (t_1
         (*
          (/ (sin y) y)
          (+ 1.0 (* (* x x) (+ 0.5 (* x (* x 0.041666666666666664)))))))
        (t_2 (* (* x x) 0.001388888888888889)))
   (if (<= x 0.125)
     t_1
     (if (<= x 5.5e+41)
       (/ (* (cosh x) y) y)
       (if (<= x 2.6e+77)
         (*
          (+
           1.0
           (*
            (* x x)
            (+
             0.5
             (/
              (*
               (* x x)
               (+ 7.233796296296296e-5 (* t_0 (* t_0 2.6791838134430728e-9))))
              (+ 0.001736111111111111 (* t_2 (- t_2 0.041666666666666664)))))))
          (+ 1.0 (* (* y y) -0.16666666666666666)))
         t_1)))))

C

double code(double x, double y) {
	double t_0 = x * (x * x);
	double t_1 = (sin(y) / y) * (1.0 + ((x * x) * (0.5 + (x * (x * 0.041666666666666664)))));
	double t_2 = (x * x) * 0.001388888888888889;
	double tmp;
	if (x <= 0.125) {
		tmp = t_1;
	} else if (x <= 5.5e+41) {
		tmp = (cosh(x) * y) / y;
	} else if (x <= 2.6e+77) {
		tmp = (1.0 + ((x * x) * (0.5 + (((x * x) * (7.233796296296296e-5 + (t_0 * (t_0 * 2.6791838134430728e-9)))) / (0.001736111111111111 + (t_2 * (t_2 - 0.041666666666666664))))))) * (1.0 + ((y * y) * -0.16666666666666666));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = x * (x * x)
    t_1 = (sin(y) / y) * (1.0d0 + ((x * x) * (0.5d0 + (x * (x * 0.041666666666666664d0)))))
    t_2 = (x * x) * 0.001388888888888889d0
    if (x <= 0.125d0) then
        tmp = t_1
    else if (x <= 5.5d+41) then
        tmp = (cosh(x) * y) / y
    else if (x <= 2.6d+77) then
        tmp = (1.0d0 + ((x * x) * (0.5d0 + (((x * x) * (7.233796296296296d-5 + (t_0 * (t_0 * 2.6791838134430728d-9)))) / (0.001736111111111111d0 + (t_2 * (t_2 - 0.041666666666666664d0))))))) * (1.0d0 + ((y * y) * (-0.16666666666666666d0)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x * (x * x);
	double t_1 = (Math.sin(y) / y) * (1.0 + ((x * x) * (0.5 + (x * (x * 0.041666666666666664)))));
	double t_2 = (x * x) * 0.001388888888888889;
	double tmp;
	if (x <= 0.125) {
		tmp = t_1;
	} else if (x <= 5.5e+41) {
		tmp = (Math.cosh(x) * y) / y;
	} else if (x <= 2.6e+77) {
		tmp = (1.0 + ((x * x) * (0.5 + (((x * x) * (7.233796296296296e-5 + (t_0 * (t_0 * 2.6791838134430728e-9)))) / (0.001736111111111111 + (t_2 * (t_2 - 0.041666666666666664))))))) * (1.0 + ((y * y) * -0.16666666666666666));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x * (x * x)
	t_1 = (math.sin(y) / y) * (1.0 + ((x * x) * (0.5 + (x * (x * 0.041666666666666664)))))
	t_2 = (x * x) * 0.001388888888888889
	tmp = 0
	if x <= 0.125:
		tmp = t_1
	elif x <= 5.5e+41:
		tmp = (math.cosh(x) * y) / y
	elif x <= 2.6e+77:
		tmp = (1.0 + ((x * x) * (0.5 + (((x * x) * (7.233796296296296e-5 + (t_0 * (t_0 * 2.6791838134430728e-9)))) / (0.001736111111111111 + (t_2 * (t_2 - 0.041666666666666664))))))) * (1.0 + ((y * y) * -0.16666666666666666))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x * Float64(x * x))
	t_1 = Float64(Float64(sin(y) / y) * Float64(1.0 + Float64(Float64(x * x) * Float64(0.5 + Float64(x * Float64(x * 0.041666666666666664))))))
	t_2 = Float64(Float64(x * x) * 0.001388888888888889)
	tmp = 0.0
	if (x <= 0.125)
		tmp = t_1;
	elseif (x <= 5.5e+41)
		tmp = Float64(Float64(cosh(x) * y) / y);
	elseif (x <= 2.6e+77)
		tmp = Float64(Float64(1.0 + Float64(Float64(x * x) * Float64(0.5 + Float64(Float64(Float64(x * x) * Float64(7.233796296296296e-5 + Float64(t_0 * Float64(t_0 * 2.6791838134430728e-9)))) / Float64(0.001736111111111111 + Float64(t_2 * Float64(t_2 - 0.041666666666666664))))))) * Float64(1.0 + Float64(Float64(y * y) * -0.16666666666666666)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x * (x * x);
	t_1 = (sin(y) / y) * (1.0 + ((x * x) * (0.5 + (x * (x * 0.041666666666666664)))));
	t_2 = (x * x) * 0.001388888888888889;
	tmp = 0.0;
	if (x <= 0.125)
		tmp = t_1;
	elseif (x <= 5.5e+41)
		tmp = (cosh(x) * y) / y;
	elseif (x <= 2.6e+77)
		tmp = (1.0 + ((x * x) * (0.5 + (((x * x) * (7.233796296296296e-5 + (t_0 * (t_0 * 2.6791838134430728e-9)))) / (0.001736111111111111 + (t_2 * (t_2 - 0.041666666666666664))))))) * (1.0 + ((y * y) * -0.16666666666666666));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision] * N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(0.5 + N[(x * N[(x * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * x), $MachinePrecision] * 0.001388888888888889), $MachinePrecision]}, If[LessEqual[x, 0.125], t$95$1, If[LessEqual[x, 5.5e+41], N[(N[(N[Cosh[x], $MachinePrecision] * y), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[x, 2.6e+77], N[(N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(0.5 + N[(N[(N[(x * x), $MachinePrecision] * N[(7.233796296296296e-5 + N[(t$95$0 * N[(t$95$0 * 2.6791838134430728e-9), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.001736111111111111 + N[(t$95$2 * N[(t$95$2 - 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(y * y), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 0.125 or 2.6000000000000002e77 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)}, \mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), y\right)\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left({x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)\right), \mathsf{/.f64}\left(\color{blue}{\mathsf{sin.f64}\left(y\right)}, y\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), y\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), y\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), y\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{24} \cdot {x}^{2}\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), y\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left({x}^{2} \cdot \frac{1}{24}\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), y\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(x \cdot x\right) \cdot \frac{1}{24}\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), y\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(x \cdot \left(x \cdot \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), y\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \left(x \cdot \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), y\right)\right) \]

      10. *-lowering-*.f64 91.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), y\right)\right) \]

   5. Simplified 91.8%

      \[\leadsto \color{blue}{\left(1 + \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot 0.041666666666666664\right)\right)\right)} \cdot \frac{\sin y}{y} \]

   if 0.125 < x < 5.5000000000000003e41

   1. Initial program 99.8%

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

      1. associate-*r/ N/A

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

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

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

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

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

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

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

      5. sin-lowering-sin.f64 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in y around 0

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

   7. Simplified 100.0%

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

   if 5.5000000000000003e41 < x < 2.6000000000000002e77

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

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

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

      2. *-commutative N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 75.0%

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

   5. Simplified 75.0%

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

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left({x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(x \cdot x\right) \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(x \cdot \left(x \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \left(\frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \left({x}^{2} \cdot \frac{1}{720}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({x}^{2}\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

      14. *-lowering-*.f64 29.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

   8. Simplified 29.0%

      \[\leadsto \color{blue}{\left(1 + \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot \left(0.041666666666666664 + \left(x \cdot x\right) \cdot 0.001388888888888889\right)\right)\right)\right)} \cdot \left(1 + \left(y \cdot y\right) \cdot -0.16666666666666666\right) \]
   9. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(x \cdot x\right) \cdot \left(\frac{1}{24} + \left(x \cdot x\right) \cdot \frac{1}{720}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

      2. flip3-+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(x \cdot x\right) \cdot \frac{{\frac{1}{24}}^{3} + {\left(\left(x \cdot x\right) \cdot \frac{1}{720}\right)}^{3}}{\frac{1}{24} \cdot \frac{1}{24} + \left(\left(\left(x \cdot x\right) \cdot \frac{1}{720}\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{720}\right) - \frac{1}{24} \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{720}\right)\right)}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\left(x \cdot x\right) \cdot \left({\frac{1}{24}}^{3} + {\left(\left(x \cdot x\right) \cdot \frac{1}{720}\right)}^{3}\right)}{\frac{1}{24} \cdot \frac{1}{24} + \left(\left(\left(x \cdot x\right) \cdot \frac{1}{720}\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{720}\right) - \frac{1}{24} \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{720}\right)\right)}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\left(\left(x \cdot x\right) \cdot \left({\frac{1}{24}}^{3} + {\left(\left(x \cdot x\right) \cdot \frac{1}{720}\right)}^{3}\right)\right), \left(\frac{1}{24} \cdot \frac{1}{24} + \left(\left(\left(x \cdot x\right) \cdot \frac{1}{720}\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{720}\right) - \frac{1}{24} \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{720}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

   10. Applied egg-rr 75.0%

       \[\leadsto \left(1 + \left(x \cdot x\right) \cdot \left(0.5 + \color{blue}{\frac{\left(x \cdot x\right) \cdot \left(7.233796296296296 \cdot 10^{-5} + \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot 2.6791838134430728 \cdot 10^{-9}\right)\right)}{0.001736111111111111 + \left(\left(x \cdot x\right) \cdot 0.001388888888888889\right) \cdot \left(\left(x \cdot x\right) \cdot 0.001388888888888889 - 0.041666666666666664\right)}}\right)\right) \cdot \left(1 + \left(y \cdot y\right) \cdot -0.16666666666666666\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.125:\\ \;\;\;\;\frac{\sin y}{y} \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot 0.041666666666666664\right)\right)\right)\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{+41}:\\ \;\;\;\;\frac{\cosh x \cdot y}{y}\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{+77}:\\ \;\;\;\;\left(1 + \left(x \cdot x\right) \cdot \left(0.5 + \frac{\left(x \cdot x\right) \cdot \left(7.233796296296296 \cdot 10^{-5} + \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot 2.6791838134430728 \cdot 10^{-9}\right)\right)}{0.001736111111111111 + \left(\left(x \cdot x\right) \cdot 0.001388888888888889\right) \cdot \left(\left(x \cdot x\right) \cdot 0.001388888888888889 - 0.041666666666666664\right)}\right)\right) \cdot \left(1 + \left(y \cdot y\right) \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin y}{y} \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot 0.041666666666666664\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y} \cdot \left(1 + \left(x \cdot x\right) \cdot 0.5\right)\\ \mathbf{if}\;x \leq 0.04:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{+41}:\\ \;\;\;\;\frac{\cosh x \cdot y}{y}\\ \mathbf{elif}\;x \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;\cosh x \cdot \left(1 + \left(y \cdot y\right) \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (/ (sin y) y) (+ 1.0 (* (* x x) 0.5)))))
   (if (<= x 0.04)
     t_0
     (if (<= x 3.3e+41)
       (/ (* (cosh x) y) y)
       (if (<= x 1.32e+154)
         (* (cosh x) (+ 1.0 (* (* y y) -0.16666666666666666)))
         t_0)))))

C

double code(double x, double y) {
	double t_0 = (sin(y) / y) * (1.0 + ((x * x) * 0.5));
	double tmp;
	if (x <= 0.04) {
		tmp = t_0;
	} else if (x <= 3.3e+41) {
		tmp = (cosh(x) * y) / y;
	} else if (x <= 1.32e+154) {
		tmp = cosh(x) * (1.0 + ((y * y) * -0.16666666666666666));
	} 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 = (sin(y) / y) * (1.0d0 + ((x * x) * 0.5d0))
    if (x <= 0.04d0) then
        tmp = t_0
    else if (x <= 3.3d+41) then
        tmp = (cosh(x) * y) / y
    else if (x <= 1.32d+154) then
        tmp = cosh(x) * (1.0d0 + ((y * y) * (-0.16666666666666666d0)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (Math.sin(y) / y) * (1.0 + ((x * x) * 0.5));
	double tmp;
	if (x <= 0.04) {
		tmp = t_0;
	} else if (x <= 3.3e+41) {
		tmp = (Math.cosh(x) * y) / y;
	} else if (x <= 1.32e+154) {
		tmp = Math.cosh(x) * (1.0 + ((y * y) * -0.16666666666666666));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (math.sin(y) / y) * (1.0 + ((x * x) * 0.5))
	tmp = 0
	if x <= 0.04:
		tmp = t_0
	elif x <= 3.3e+41:
		tmp = (math.cosh(x) * y) / y
	elif x <= 1.32e+154:
		tmp = math.cosh(x) * (1.0 + ((y * y) * -0.16666666666666666))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(sin(y) / y) * Float64(1.0 + Float64(Float64(x * x) * 0.5)))
	tmp = 0.0
	if (x <= 0.04)
		tmp = t_0;
	elseif (x <= 3.3e+41)
		tmp = Float64(Float64(cosh(x) * y) / y);
	elseif (x <= 1.32e+154)
		tmp = Float64(cosh(x) * Float64(1.0 + Float64(Float64(y * y) * -0.16666666666666666)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (sin(y) / y) * (1.0 + ((x * x) * 0.5));
	tmp = 0.0;
	if (x <= 0.04)
		tmp = t_0;
	elseif (x <= 3.3e+41)
		tmp = (cosh(x) * y) / y;
	elseif (x <= 1.32e+154)
		tmp = cosh(x) * (1.0 + ((y * y) * -0.16666666666666666));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision] * N[(1.0 + N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 0.04], t$95$0, If[LessEqual[x, 3.3e+41], N[(N[(N[Cosh[x], $MachinePrecision] * y), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[x, 1.32e+154], N[(N[Cosh[x], $MachinePrecision] * N[(1.0 + N[(N[(y * y), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < 0.0400000000000000008 or 1.31999999999999998e154 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{{x}^{2} \cdot \sin y}{y} + \frac{\sin y}{y}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{{x}^{2} \cdot \sin y}{y} \cdot \frac{1}{2} + \frac{\color{blue}{\sin y}}{y} \]

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. distribute-lft1-in N/A

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

      8. +-commutative N/A

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

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

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

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

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

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

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

      12. unpow2 N/A

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

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

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

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

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

      15. sin-lowering-sin.f64 83.8%

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

   5. Simplified 83.8%

      \[\leadsto \color{blue}{\left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \frac{\sin y}{y}} \]

   if 0.0400000000000000008 < x < 3.3e41

   1. Initial program 99.8%

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

      1. associate-*r/ N/A

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

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

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

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

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

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

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

      5. sin-lowering-sin.f64 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in y around 0

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

   7. Simplified 100.0%

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

   if 3.3e41 < x < 1.31999999999999998e154

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

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

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

      2. *-commutative N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 91.7%

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

   5. Simplified 91.7%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.04:\\ \;\;\;\;\frac{\sin y}{y} \cdot \left(1 + \left(x \cdot x\right) \cdot 0.5\right)\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{+41}:\\ \;\;\;\;\frac{\cosh x \cdot y}{y}\\ \mathbf{elif}\;x \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;\cosh x \cdot \left(1 + \left(y \cdot y\right) \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin y}{y} \cdot \left(1 + \left(x \cdot x\right) \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 69.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.015:\\ \;\;\;\;\frac{\sin y}{y}\\ \mathbf{elif}\;x \leq 4 \cdot 10^{+41}:\\ \;\;\;\;\frac{\cosh x \cdot y}{y}\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot \left(1 + \left(y \cdot y\right) \cdot -0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.015)
   (/ (sin y) y)
   (if (<= x 4e+41)
     (/ (* (cosh x) y) y)
     (* (cosh x) (+ 1.0 (* (* y y) -0.16666666666666666))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.015) {
		tmp = sin(y) / y;
	} else if (x <= 4e+41) {
		tmp = (cosh(x) * y) / y;
	} else {
		tmp = cosh(x) * (1.0 + ((y * y) * -0.16666666666666666));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 0.015d0) then
        tmp = sin(y) / y
    else if (x <= 4d+41) then
        tmp = (cosh(x) * y) / y
    else
        tmp = cosh(x) * (1.0d0 + ((y * y) * (-0.16666666666666666d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 0.015) {
		tmp = Math.sin(y) / y;
	} else if (x <= 4e+41) {
		tmp = (Math.cosh(x) * y) / y;
	} else {
		tmp = Math.cosh(x) * (1.0 + ((y * y) * -0.16666666666666666));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 0.015:
		tmp = math.sin(y) / y
	elif x <= 4e+41:
		tmp = (math.cosh(x) * y) / y
	else:
		tmp = math.cosh(x) * (1.0 + ((y * y) * -0.16666666666666666))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.015)
		tmp = Float64(sin(y) / y);
	elseif (x <= 4e+41)
		tmp = Float64(Float64(cosh(x) * y) / y);
	else
		tmp = Float64(cosh(x) * Float64(1.0 + Float64(Float64(y * y) * -0.16666666666666666)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 0.015)
		tmp = sin(y) / y;
	elseif (x <= 4e+41)
		tmp = (cosh(x) * y) / y;
	else
		tmp = cosh(x) * (1.0 + ((y * y) * -0.16666666666666666));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.015], N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], If[LessEqual[x, 4e+41], N[(N[(N[Cosh[x], $MachinePrecision] * y), $MachinePrecision] / y), $MachinePrecision], N[(N[Cosh[x], $MachinePrecision] * N[(1.0 + N[(N[(y * y), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 0.014999999999999999

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

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

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

      2. sin-lowering-sin.f64 59.2%

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

   5. Simplified 59.2%

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

   if 0.014999999999999999 < x < 4.00000000000000002e41

   1. Initial program 99.8%

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

      1. associate-*r/ N/A

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

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

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

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

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

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

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

      5. sin-lowering-sin.f64 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in y around 0

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

   7. Simplified 100.0%

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

   if 4.00000000000000002e41 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

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

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

      2. *-commutative N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 86.8%

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

   5. Simplified 86.8%

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

Alternative 6: 69.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0095:\\ \;\;\;\;\frac{\sin y}{y}\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{+51}:\\ \;\;\;\;\frac{\cosh x \cdot y}{y}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(y \cdot y\right) \cdot -0.16666666666666666\right) \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot \left(0.041666666666666664 + \left(x \cdot x\right) \cdot 0.001388888888888889\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.0095)
   (/ (sin y) y)
   (if (<= x 7.2e+51)
     (/ (* (cosh x) y) y)
     (*
      (+ 1.0 (* (* y y) -0.16666666666666666))
      (+
       1.0
       (*
        (* x x)
        (+
         0.5
         (*
          x
          (*
           x
           (+ 0.041666666666666664 (* (* x x) 0.001388888888888889)))))))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.0095) {
		tmp = sin(y) / y;
	} else if (x <= 7.2e+51) {
		tmp = (cosh(x) * y) / y;
	} else {
		tmp = (1.0 + ((y * y) * -0.16666666666666666)) * (1.0 + ((x * x) * (0.5 + (x * (x * (0.041666666666666664 + ((x * x) * 0.001388888888888889)))))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 0.0095d0) then
        tmp = sin(y) / y
    else if (x <= 7.2d+51) then
        tmp = (cosh(x) * y) / y
    else
        tmp = (1.0d0 + ((y * y) * (-0.16666666666666666d0))) * (1.0d0 + ((x * x) * (0.5d0 + (x * (x * (0.041666666666666664d0 + ((x * x) * 0.001388888888888889d0)))))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 0.0095) {
		tmp = Math.sin(y) / y;
	} else if (x <= 7.2e+51) {
		tmp = (Math.cosh(x) * y) / y;
	} else {
		tmp = (1.0 + ((y * y) * -0.16666666666666666)) * (1.0 + ((x * x) * (0.5 + (x * (x * (0.041666666666666664 + ((x * x) * 0.001388888888888889)))))));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 0.0095:
		tmp = math.sin(y) / y
	elif x <= 7.2e+51:
		tmp = (math.cosh(x) * y) / y
	else:
		tmp = (1.0 + ((y * y) * -0.16666666666666666)) * (1.0 + ((x * x) * (0.5 + (x * (x * (0.041666666666666664 + ((x * x) * 0.001388888888888889)))))))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.0095)
		tmp = Float64(sin(y) / y);
	elseif (x <= 7.2e+51)
		tmp = Float64(Float64(cosh(x) * y) / y);
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(y * y) * -0.16666666666666666)) * Float64(1.0 + Float64(Float64(x * x) * Float64(0.5 + Float64(x * Float64(x * Float64(0.041666666666666664 + Float64(Float64(x * x) * 0.001388888888888889))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 0.0095)
		tmp = sin(y) / y;
	elseif (x <= 7.2e+51)
		tmp = (cosh(x) * y) / y;
	else
		tmp = (1.0 + ((y * y) * -0.16666666666666666)) * (1.0 + ((x * x) * (0.5 + (x * (x * (0.041666666666666664 + ((x * x) * 0.001388888888888889)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.0095], N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], If[LessEqual[x, 7.2e+51], N[(N[(N[Cosh[x], $MachinePrecision] * y), $MachinePrecision] / y), $MachinePrecision], N[(N[(1.0 + N[(N[(y * y), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(0.5 + N[(x * N[(x * N[(0.041666666666666664 + N[(N[(x * x), $MachinePrecision] * 0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 0.00949999999999999976

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

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

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

      2. sin-lowering-sin.f64 59.2%

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

   5. Simplified 59.2%

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

   if 0.00949999999999999976 < x < 7.20000000000000022e51

   1. Initial program 99.9%

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

      1. associate-*r/ N/A

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

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

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

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

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

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

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

      5. sin-lowering-sin.f64 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in y around 0

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

   7. Simplified 90.9%

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

   if 7.20000000000000022e51 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

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

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

      2. *-commutative N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 85.7%

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

   5. Simplified 85.7%

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

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left({x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(x \cdot x\right) \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(x \cdot \left(x \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \left(\frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \left({x}^{2} \cdot \frac{1}{720}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({x}^{2}\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

      14. *-lowering-*.f64 85.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

   8. Simplified 85.7%

      \[\leadsto \color{blue}{\left(1 + \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot \left(0.041666666666666664 + \left(x \cdot x\right) \cdot 0.001388888888888889\right)\right)\right)\right)} \cdot \left(1 + \left(y \cdot y\right) \cdot -0.16666666666666666\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0095:\\ \;\;\;\;\frac{\sin y}{y}\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{+51}:\\ \;\;\;\;\frac{\cosh x \cdot y}{y}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(y \cdot y\right) \cdot -0.16666666666666666\right) \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot \left(0.041666666666666664 + \left(x \cdot x\right) \cdot 0.001388888888888889\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 69.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.005:\\ \;\;\;\;\frac{\sin y}{y}\\ \mathbf{elif}\;x \leq 8 \cdot 10^{+51}:\\ \;\;\;\;\cosh x\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(y \cdot y\right) \cdot -0.16666666666666666\right) \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot \left(0.041666666666666664 + \left(x \cdot x\right) \cdot 0.001388888888888889\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.005)
   (/ (sin y) y)
   (if (<= x 8e+51)
     (cosh x)
     (*
      (+ 1.0 (* (* y y) -0.16666666666666666))
      (+
       1.0
       (*
        (* x x)
        (+
         0.5
         (*
          x
          (*
           x
           (+ 0.041666666666666664 (* (* x x) 0.001388888888888889)))))))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.005) {
		tmp = sin(y) / y;
	} else if (x <= 8e+51) {
		tmp = cosh(x);
	} else {
		tmp = (1.0 + ((y * y) * -0.16666666666666666)) * (1.0 + ((x * x) * (0.5 + (x * (x * (0.041666666666666664 + ((x * x) * 0.001388888888888889)))))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 0.005d0) then
        tmp = sin(y) / y
    else if (x <= 8d+51) then
        tmp = cosh(x)
    else
        tmp = (1.0d0 + ((y * y) * (-0.16666666666666666d0))) * (1.0d0 + ((x * x) * (0.5d0 + (x * (x * (0.041666666666666664d0 + ((x * x) * 0.001388888888888889d0)))))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 0.005) {
		tmp = Math.sin(y) / y;
	} else if (x <= 8e+51) {
		tmp = Math.cosh(x);
	} else {
		tmp = (1.0 + ((y * y) * -0.16666666666666666)) * (1.0 + ((x * x) * (0.5 + (x * (x * (0.041666666666666664 + ((x * x) * 0.001388888888888889)))))));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 0.005:
		tmp = math.sin(y) / y
	elif x <= 8e+51:
		tmp = math.cosh(x)
	else:
		tmp = (1.0 + ((y * y) * -0.16666666666666666)) * (1.0 + ((x * x) * (0.5 + (x * (x * (0.041666666666666664 + ((x * x) * 0.001388888888888889)))))))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.005)
		tmp = Float64(sin(y) / y);
	elseif (x <= 8e+51)
		tmp = cosh(x);
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(y * y) * -0.16666666666666666)) * Float64(1.0 + Float64(Float64(x * x) * Float64(0.5 + Float64(x * Float64(x * Float64(0.041666666666666664 + Float64(Float64(x * x) * 0.001388888888888889))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 0.005)
		tmp = sin(y) / y;
	elseif (x <= 8e+51)
		tmp = cosh(x);
	else
		tmp = (1.0 + ((y * y) * -0.16666666666666666)) * (1.0 + ((x * x) * (0.5 + (x * (x * (0.041666666666666664 + ((x * x) * 0.001388888888888889)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.005], N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], If[LessEqual[x, 8e+51], N[Cosh[x], $MachinePrecision], N[(N[(1.0 + N[(N[(y * y), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(0.5 + N[(x * N[(x * N[(0.041666666666666664 + N[(N[(x * x), $MachinePrecision] * 0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 0.0050000000000000001

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

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

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

      2. sin-lowering-sin.f64 59.2%

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

   5. Simplified 59.2%

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

   if 0.0050000000000000001 < x < 8e51

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(x\right), \color{blue}{1}\right) \]
   4. Step-by-step derivation

   5. Simplified 90.8%

      \[\leadsto \cosh x \cdot \color{blue}{1} \]
   6. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto \cosh x \]

      2. cosh-lowering-cosh.f64 90.8%

         \[\leadsto \mathsf{cosh.f64}\left(x\right) \]

   7. Applied egg-rr 90.8%

      \[\leadsto \color{blue}{\cosh x} \]

   if 8e51 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

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

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

      2. *-commutative N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 85.7%

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

   5. Simplified 85.7%

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

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left({x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(x \cdot x\right) \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(x \cdot \left(x \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \left(\frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \left({x}^{2} \cdot \frac{1}{720}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({x}^{2}\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

      14. *-lowering-*.f64 85.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

   8. Simplified 85.7%

      \[\leadsto \color{blue}{\left(1 + \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot \left(0.041666666666666664 + \left(x \cdot x\right) \cdot 0.001388888888888889\right)\right)\right)\right)} \cdot \left(1 + \left(y \cdot y\right) \cdot -0.16666666666666666\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.005:\\ \;\;\;\;\frac{\sin y}{y}\\ \mathbf{elif}\;x \leq 8 \cdot 10^{+51}:\\ \;\;\;\;\cosh x\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(y \cdot y\right) \cdot -0.16666666666666666\right) \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot \left(0.041666666666666664 + \left(x \cdot x\right) \cdot 0.001388888888888889\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 62.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 7.2 \cdot 10^{+51}:\\ \;\;\;\;\cosh x\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(y \cdot y\right) \cdot -0.16666666666666666\right) \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot \left(0.041666666666666664 + \left(x \cdot x\right) \cdot 0.001388888888888889\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 7.2e+51)
   (cosh x)
   (*
    (+ 1.0 (* (* y y) -0.16666666666666666))
    (+
     1.0
     (*
      (* x x)
      (+
       0.5
       (*
        x
        (* x (+ 0.041666666666666664 (* (* x x) 0.001388888888888889))))))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 7.2e+51) {
		tmp = cosh(x);
	} else {
		tmp = (1.0 + ((y * y) * -0.16666666666666666)) * (1.0 + ((x * x) * (0.5 + (x * (x * (0.041666666666666664 + ((x * x) * 0.001388888888888889)))))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 7.2d+51) then
        tmp = cosh(x)
    else
        tmp = (1.0d0 + ((y * y) * (-0.16666666666666666d0))) * (1.0d0 + ((x * x) * (0.5d0 + (x * (x * (0.041666666666666664d0 + ((x * x) * 0.001388888888888889d0)))))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 7.2e+51) {
		tmp = Math.cosh(x);
	} else {
		tmp = (1.0 + ((y * y) * -0.16666666666666666)) * (1.0 + ((x * x) * (0.5 + (x * (x * (0.041666666666666664 + ((x * x) * 0.001388888888888889)))))));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 7.2e+51:
		tmp = math.cosh(x)
	else:
		tmp = (1.0 + ((y * y) * -0.16666666666666666)) * (1.0 + ((x * x) * (0.5 + (x * (x * (0.041666666666666664 + ((x * x) * 0.001388888888888889)))))))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 7.2e+51)
		tmp = cosh(x);
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(y * y) * -0.16666666666666666)) * Float64(1.0 + Float64(Float64(x * x) * Float64(0.5 + Float64(x * Float64(x * Float64(0.041666666666666664 + Float64(Float64(x * x) * 0.001388888888888889))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 7.2e+51)
		tmp = cosh(x);
	else
		tmp = (1.0 + ((y * y) * -0.16666666666666666)) * (1.0 + ((x * x) * (0.5 + (x * (x * (0.041666666666666664 + ((x * x) * 0.001388888888888889)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 7.2e+51], N[Cosh[x], $MachinePrecision], N[(N[(1.0 + N[(N[(y * y), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(0.5 + N[(x * N[(x * N[(0.041666666666666664 + N[(N[(x * x), $MachinePrecision] * 0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 7.20000000000000022e51

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(x\right), \color{blue}{1}\right) \]
   4. Step-by-step derivation

   5. Simplified 59.1%

      \[\leadsto \cosh x \cdot \color{blue}{1} \]
   6. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto \cosh x \]

      2. cosh-lowering-cosh.f64 59.1%

         \[\leadsto \mathsf{cosh.f64}\left(x\right) \]

   7. Applied egg-rr 59.1%

      \[\leadsto \color{blue}{\cosh x} \]

   if 7.20000000000000022e51 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

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

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

      2. *-commutative N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 85.7%

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

   5. Simplified 85.7%

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

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left({x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(x \cdot x\right) \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(x \cdot \left(x \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \left(\frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \left({x}^{2} \cdot \frac{1}{720}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({x}^{2}\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

      14. *-lowering-*.f64 85.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

   8. Simplified 85.7%

      \[\leadsto \color{blue}{\left(1 + \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot \left(0.041666666666666664 + \left(x \cdot x\right) \cdot 0.001388888888888889\right)\right)\right)\right)} \cdot \left(1 + \left(y \cdot y\right) \cdot -0.16666666666666666\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 7.2 \cdot 10^{+51}:\\ \;\;\;\;\cosh x\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(y \cdot y\right) \cdot -0.16666666666666666\right) \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot \left(0.041666666666666664 + \left(x \cdot x\right) \cdot 0.001388888888888889\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 46.2% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot x\right)\\ t_1 := \left(x \cdot x\right) \cdot 0.001388888888888889\\ t_2 := 1 + \left(y \cdot y\right) \cdot -0.16666666666666666\\ \mathbf{if}\;x \leq 1.12 \cdot 10^{+77}:\\ \;\;\;\;\left(1 + \left(x \cdot x\right) \cdot \left(0.5 + \frac{\left(x \cdot x\right) \cdot \left(7.233796296296296 \cdot 10^{-5} + t\_0 \cdot \left(t\_0 \cdot 2.6791838134430728 \cdot 10^{-9}\right)\right)}{0.001736111111111111 + t\_1 \cdot \left(t\_1 - 0.041666666666666664\right)}\right)\right) \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot 0.041666666666666664\right)\right)\right) \cdot t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* x (* x x)))
        (t_1 (* (* x x) 0.001388888888888889))
        (t_2 (+ 1.0 (* (* y y) -0.16666666666666666))))
   (if (<= x 1.12e+77)
     (*
      (+
       1.0
       (*
        (* x x)
        (+
         0.5
         (/
          (*
           (* x x)
           (+ 7.233796296296296e-5 (* t_0 (* t_0 2.6791838134430728e-9))))
          (+ 0.001736111111111111 (* t_1 (- t_1 0.041666666666666664)))))))
      t_2)
     (* (+ 1.0 (* (* x x) (+ 0.5 (* x (* x 0.041666666666666664))))) t_2))))

C

double code(double x, double y) {
	double t_0 = x * (x * x);
	double t_1 = (x * x) * 0.001388888888888889;
	double t_2 = 1.0 + ((y * y) * -0.16666666666666666);
	double tmp;
	if (x <= 1.12e+77) {
		tmp = (1.0 + ((x * x) * (0.5 + (((x * x) * (7.233796296296296e-5 + (t_0 * (t_0 * 2.6791838134430728e-9)))) / (0.001736111111111111 + (t_1 * (t_1 - 0.041666666666666664))))))) * t_2;
	} else {
		tmp = (1.0 + ((x * x) * (0.5 + (x * (x * 0.041666666666666664))))) * t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = x * (x * x)
    t_1 = (x * x) * 0.001388888888888889d0
    t_2 = 1.0d0 + ((y * y) * (-0.16666666666666666d0))
    if (x <= 1.12d+77) then
        tmp = (1.0d0 + ((x * x) * (0.5d0 + (((x * x) * (7.233796296296296d-5 + (t_0 * (t_0 * 2.6791838134430728d-9)))) / (0.001736111111111111d0 + (t_1 * (t_1 - 0.041666666666666664d0))))))) * t_2
    else
        tmp = (1.0d0 + ((x * x) * (0.5d0 + (x * (x * 0.041666666666666664d0))))) * t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x * (x * x);
	double t_1 = (x * x) * 0.001388888888888889;
	double t_2 = 1.0 + ((y * y) * -0.16666666666666666);
	double tmp;
	if (x <= 1.12e+77) {
		tmp = (1.0 + ((x * x) * (0.5 + (((x * x) * (7.233796296296296e-5 + (t_0 * (t_0 * 2.6791838134430728e-9)))) / (0.001736111111111111 + (t_1 * (t_1 - 0.041666666666666664))))))) * t_2;
	} else {
		tmp = (1.0 + ((x * x) * (0.5 + (x * (x * 0.041666666666666664))))) * t_2;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x * (x * x)
	t_1 = (x * x) * 0.001388888888888889
	t_2 = 1.0 + ((y * y) * -0.16666666666666666)
	tmp = 0
	if x <= 1.12e+77:
		tmp = (1.0 + ((x * x) * (0.5 + (((x * x) * (7.233796296296296e-5 + (t_0 * (t_0 * 2.6791838134430728e-9)))) / (0.001736111111111111 + (t_1 * (t_1 - 0.041666666666666664))))))) * t_2
	else:
		tmp = (1.0 + ((x * x) * (0.5 + (x * (x * 0.041666666666666664))))) * t_2
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x * Float64(x * x))
	t_1 = Float64(Float64(x * x) * 0.001388888888888889)
	t_2 = Float64(1.0 + Float64(Float64(y * y) * -0.16666666666666666))
	tmp = 0.0
	if (x <= 1.12e+77)
		tmp = Float64(Float64(1.0 + Float64(Float64(x * x) * Float64(0.5 + Float64(Float64(Float64(x * x) * Float64(7.233796296296296e-5 + Float64(t_0 * Float64(t_0 * 2.6791838134430728e-9)))) / Float64(0.001736111111111111 + Float64(t_1 * Float64(t_1 - 0.041666666666666664))))))) * t_2);
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(x * x) * Float64(0.5 + Float64(x * Float64(x * 0.041666666666666664))))) * t_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x * (x * x);
	t_1 = (x * x) * 0.001388888888888889;
	t_2 = 1.0 + ((y * y) * -0.16666666666666666);
	tmp = 0.0;
	if (x <= 1.12e+77)
		tmp = (1.0 + ((x * x) * (0.5 + (((x * x) * (7.233796296296296e-5 + (t_0 * (t_0 * 2.6791838134430728e-9)))) / (0.001736111111111111 + (t_1 * (t_1 - 0.041666666666666664))))))) * t_2;
	else
		tmp = (1.0 + ((x * x) * (0.5 + (x * (x * 0.041666666666666664))))) * t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * x), $MachinePrecision] * 0.001388888888888889), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 + N[(N[(y * y), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 1.12e+77], N[(N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(0.5 + N[(N[(N[(x * x), $MachinePrecision] * N[(7.233796296296296e-5 + N[(t$95$0 * N[(t$95$0 * 2.6791838134430728e-9), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.001736111111111111 + N[(t$95$1 * N[(t$95$1 - 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision], N[(N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(0.5 + N[(x * N[(x * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.1199999999999999e77

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

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

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

      2. *-commutative N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 58.2%

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

   5. Simplified 58.2%

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

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left({x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(x \cdot x\right) \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(x \cdot \left(x \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \left(\frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \left({x}^{2} \cdot \frac{1}{720}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({x}^{2}\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

      14. *-lowering-*.f64 52.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

   8. Simplified 52.5%

      \[\leadsto \color{blue}{\left(1 + \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot \left(0.041666666666666664 + \left(x \cdot x\right) \cdot 0.001388888888888889\right)\right)\right)\right)} \cdot \left(1 + \left(y \cdot y\right) \cdot -0.16666666666666666\right) \]
   9. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(x \cdot x\right) \cdot \left(\frac{1}{24} + \left(x \cdot x\right) \cdot \frac{1}{720}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

      2. flip3-+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(x \cdot x\right) \cdot \frac{{\frac{1}{24}}^{3} + {\left(\left(x \cdot x\right) \cdot \frac{1}{720}\right)}^{3}}{\frac{1}{24} \cdot \frac{1}{24} + \left(\left(\left(x \cdot x\right) \cdot \frac{1}{720}\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{720}\right) - \frac{1}{24} \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{720}\right)\right)}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\left(x \cdot x\right) \cdot \left({\frac{1}{24}}^{3} + {\left(\left(x \cdot x\right) \cdot \frac{1}{720}\right)}^{3}\right)}{\frac{1}{24} \cdot \frac{1}{24} + \left(\left(\left(x \cdot x\right) \cdot \frac{1}{720}\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{720}\right) - \frac{1}{24} \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{720}\right)\right)}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\left(\left(x \cdot x\right) \cdot \left({\frac{1}{24}}^{3} + {\left(\left(x \cdot x\right) \cdot \frac{1}{720}\right)}^{3}\right)\right), \left(\frac{1}{24} \cdot \frac{1}{24} + \left(\left(\left(x \cdot x\right) \cdot \frac{1}{720}\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{720}\right) - \frac{1}{24} \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{720}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

   10. Applied egg-rr 34.2%

       \[\leadsto \left(1 + \left(x \cdot x\right) \cdot \left(0.5 + \color{blue}{\frac{\left(x \cdot x\right) \cdot \left(7.233796296296296 \cdot 10^{-5} + \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot 2.6791838134430728 \cdot 10^{-9}\right)\right)}{0.001736111111111111 + \left(\left(x \cdot x\right) \cdot 0.001388888888888889\right) \cdot \left(\left(x \cdot x\right) \cdot 0.001388888888888889 - 0.041666666666666664\right)}}\right)\right) \cdot \left(1 + \left(y \cdot y\right) \cdot -0.16666666666666666\right) \]

   if 1.1199999999999999e77 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)}, \mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), y\right)\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left({x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)\right), \mathsf{/.f64}\left(\color{blue}{\mathsf{sin.f64}\left(y\right)}, y\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), y\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), y\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), y\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{24} \cdot {x}^{2}\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), y\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left({x}^{2} \cdot \frac{1}{24}\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), y\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(x \cdot x\right) \cdot \frac{1}{24}\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), y\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(x \cdot \left(x \cdot \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), y\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \left(x \cdot \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), y\right)\right) \]

      10. *-lowering-*.f64 97.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), y\right)\right) \]

   5. Simplified 97.9%

      \[\leadsto \color{blue}{\left(1 + \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot 0.041666666666666664\right)\right)\right)} \cdot \frac{\sin y}{y} \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1 + \left(\frac{-1}{6} \cdot \left({y}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)\right) + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto 1 + \left({x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) + \color{blue}{\frac{-1}{6} \cdot \left({y}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)\right)}\right) \]

      2. associate-+r+ N/A

         \[\leadsto \left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right) + \color{blue}{\frac{-1}{6} \cdot \left({y}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)\right)} \]

      3. associate-*r* N/A

         \[\leadsto \left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right) + \left(\frac{-1}{6} \cdot {y}^{2}\right) \cdot \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)} \]

      4. distribute-rgt1-in N/A

         \[\leadsto \left(\frac{-1}{6} \cdot {y}^{2} + 1\right) \cdot \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)} \]

      5. +-commutative N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) \cdot \left(\color{blue}{1} + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(1 + \frac{-1}{6} \cdot {y}^{2}\right), \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot {y}^{2}\right)\right), \left(\color{blue}{1} + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left({y}^{2} \cdot \frac{-1}{6}\right)\right), \left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({y}^{2}\right), \frac{-1}{6}\right)\right), \left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(y \cdot y\right), \frac{-1}{6}\right)\right), \left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right), \left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)}\right)\right)\right) \]

   8. Simplified 89.1%

      \[\leadsto \color{blue}{\left(1 + \left(y \cdot y\right) \cdot -0.16666666666666666\right) \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot 0.041666666666666664\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.12 \cdot 10^{+77}:\\ \;\;\;\;\left(1 + \left(x \cdot x\right) \cdot \left(0.5 + \frac{\left(x \cdot x\right) \cdot \left(7.233796296296296 \cdot 10^{-5} + \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot 2.6791838134430728 \cdot 10^{-9}\right)\right)}{0.001736111111111111 + \left(\left(x \cdot x\right) \cdot 0.001388888888888889\right) \cdot \left(\left(x \cdot x\right) \cdot 0.001388888888888889 - 0.041666666666666664\right)}\right)\right) \cdot \left(1 + \left(y \cdot y\right) \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot 0.041666666666666664\right)\right)\right) \cdot \left(1 + \left(y \cdot y\right) \cdot -0.16666666666666666\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 59.2% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \left(1 + \left(y \cdot y\right) \cdot -0.16666666666666666\right) \cdot \left(\left(1 + x \cdot \left(x \cdot 0.5\right)\right) + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(0.041666666666666664 + \left(x \cdot x\right) \cdot 0.001388888888888889\right)\right)\right) \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (*
  (+ 1.0 (* (* y y) -0.16666666666666666))
  (+
   (+ 1.0 (* x (* x 0.5)))
   (*
    (* x x)
    (* (* x x) (+ 0.041666666666666664 (* (* x x) 0.001388888888888889)))))))

C

double code(double x, double y) {
	return (1.0 + ((y * y) * -0.16666666666666666)) * ((1.0 + (x * (x * 0.5))) + ((x * x) * ((x * x) * (0.041666666666666664 + ((x * x) * 0.001388888888888889)))));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (1.0d0 + ((y * y) * (-0.16666666666666666d0))) * ((1.0d0 + (x * (x * 0.5d0))) + ((x * x) * ((x * x) * (0.041666666666666664d0 + ((x * x) * 0.001388888888888889d0)))))
end function

Java

public static double code(double x, double y) {
	return (1.0 + ((y * y) * -0.16666666666666666)) * ((1.0 + (x * (x * 0.5))) + ((x * x) * ((x * x) * (0.041666666666666664 + ((x * x) * 0.001388888888888889)))));
}

Python

def code(x, y):
	return (1.0 + ((y * y) * -0.16666666666666666)) * ((1.0 + (x * (x * 0.5))) + ((x * x) * ((x * x) * (0.041666666666666664 + ((x * x) * 0.001388888888888889)))))

Julia

function code(x, y)
	return Float64(Float64(1.0 + Float64(Float64(y * y) * -0.16666666666666666)) * Float64(Float64(1.0 + Float64(x * Float64(x * 0.5))) + Float64(Float64(x * x) * Float64(Float64(x * x) * Float64(0.041666666666666664 + Float64(Float64(x * x) * 0.001388888888888889))))))
end

MATLAB

function tmp = code(x, y)
	tmp = (1.0 + ((y * y) * -0.16666666666666666)) * ((1.0 + (x * (x * 0.5))) + ((x * x) * ((x * x) * (0.041666666666666664 + ((x * x) * 0.001388888888888889)))));
end

Wolfram

code[x_, y_] := N[(N[(1.0 + N[(N[(y * y), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[(x * N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(0.041666666666666664 + N[(N[(x * x), $MachinePrecision] * 0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in y around 0

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

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

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

   2. *-commutative N/A

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

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

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

   4. unpow2 N/A

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

   5. *-lowering-*.f64 63.8%

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

5. Simplified 63.8%

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

6. Taylor expanded in x around 0

   \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]
7. Step-by-step derivation

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left({x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

   3. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(x \cdot x\right) \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

   7. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(x \cdot \left(x \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \left(\frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \left({x}^{2} \cdot \frac{1}{720}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({x}^{2}\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

   13. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

   14. *-lowering-*.f64 59.1%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

8. Simplified 59.1%

   \[\leadsto \color{blue}{\left(1 + \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot \left(0.041666666666666664 + \left(x \cdot x\right) \cdot 0.001388888888888889\right)\right)\right)\right)} \cdot \left(1 + \left(y \cdot y\right) \cdot -0.16666666666666666\right) \]
9. Step-by-step derivation

   1. distribute-rgt-in N/A

      \[\leadsto \mathsf{*.f64}\left(\left(1 + \left(\frac{1}{2} \cdot \left(x \cdot x\right) + \left(x \cdot \left(x \cdot \left(\frac{1}{24} + \left(x \cdot x\right) \cdot \frac{1}{720}\right)\right)\right) \cdot \left(x \cdot x\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

   2. associate-+r+ N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\left(1 + \frac{1}{2} \cdot \left(x \cdot x\right)\right) + \left(x \cdot \left(x \cdot \left(\frac{1}{24} + \left(x \cdot x\right) \cdot \frac{1}{720}\right)\right)\right) \cdot \left(x \cdot x\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(1 + \frac{1}{2} \cdot \left(x \cdot x\right)\right), \left(\left(x \cdot \left(x \cdot \left(\frac{1}{24} + \left(x \cdot x\right) \cdot \frac{1}{720}\right)\right)\right) \cdot \left(x \cdot x\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot \left(x \cdot x\right)\right)\right), \left(\left(x \cdot \left(x \cdot \left(\frac{1}{24} + \left(x \cdot x\right) \cdot \frac{1}{720}\right)\right)\right) \cdot \left(x \cdot x\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

   5. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\left(\frac{1}{2} \cdot x\right) \cdot x\right)\right), \left(\left(x \cdot \left(x \cdot \left(\frac{1}{24} + \left(x \cdot x\right) \cdot \frac{1}{720}\right)\right)\right) \cdot \left(x \cdot x\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(x \cdot \left(\frac{1}{2} \cdot x\right)\right)\right), \left(\left(x \cdot \left(x \cdot \left(\frac{1}{24} + \left(x \cdot x\right) \cdot \frac{1}{720}\right)\right)\right) \cdot \left(x \cdot x\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{1}{2} \cdot x\right)\right)\right), \left(\left(x \cdot \left(x \cdot \left(\frac{1}{24} + \left(x \cdot x\right) \cdot \frac{1}{720}\right)\right)\right) \cdot \left(x \cdot x\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \frac{1}{2}\right)\right)\right), \left(\left(x \cdot \left(x \cdot \left(\frac{1}{24} + \left(x \cdot x\right) \cdot \frac{1}{720}\right)\right)\right) \cdot \left(x \cdot x\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), \left(\left(x \cdot \left(x \cdot \left(\frac{1}{24} + \left(x \cdot x\right) \cdot \frac{1}{720}\right)\right)\right) \cdot \left(x \cdot x\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(\frac{1}{24} + \left(x \cdot x\right) \cdot \frac{1}{720}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), \mathsf{*.f64}\left(\left(x \cdot x\right), \left(x \cdot \left(x \cdot \left(\frac{1}{24} + \left(x \cdot x\right) \cdot \frac{1}{720}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(x \cdot \left(x \cdot \left(\frac{1}{24} + \left(x \cdot x\right) \cdot \frac{1}{720}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

   13. associate-*r* N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\left(x \cdot x\right) \cdot \left(\frac{1}{24} + \left(x \cdot x\right) \cdot \frac{1}{720}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\frac{1}{24} + \left(x \cdot x\right) \cdot \frac{1}{720}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{24} + \left(x \cdot x\right) \cdot \frac{1}{720}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \left(\left(x \cdot x\right) \cdot \frac{1}{720}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{720}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

   18. *-lowering-*.f64 59.1%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{720}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

10. Applied egg-rr 59.1%

    \[\leadsto \color{blue}{\left(\left(1 + x \cdot \left(x \cdot 0.5\right)\right) + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(0.041666666666666664 + \left(x \cdot x\right) \cdot 0.001388888888888889\right)\right)\right)} \cdot \left(1 + \left(y \cdot y\right) \cdot -0.16666666666666666\right) \]

11. Final simplification 59.1%

    \[\leadsto \left(1 + \left(y \cdot y\right) \cdot -0.16666666666666666\right) \cdot \left(\left(1 + x \cdot \left(x \cdot 0.5\right)\right) + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(0.041666666666666664 + \left(x \cdot x\right) \cdot 0.001388888888888889\right)\right)\right) \]
12. Add Preprocessing

Alternative 11: 59.2% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \left(1 + \left(y \cdot y\right) \cdot -0.16666666666666666\right) \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot \left(0.041666666666666664 + \left(x \cdot x\right) \cdot 0.001388888888888889\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (*
  (+ 1.0 (* (* y y) -0.16666666666666666))
  (+
   1.0
   (*
    (* x x)
    (+
     0.5
     (* x (* x (+ 0.041666666666666664 (* (* x x) 0.001388888888888889)))))))))

C

double code(double x, double y) {
	return (1.0 + ((y * y) * -0.16666666666666666)) * (1.0 + ((x * x) * (0.5 + (x * (x * (0.041666666666666664 + ((x * x) * 0.001388888888888889)))))));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (1.0d0 + ((y * y) * (-0.16666666666666666d0))) * (1.0d0 + ((x * x) * (0.5d0 + (x * (x * (0.041666666666666664d0 + ((x * x) * 0.001388888888888889d0)))))))
end function

Java

public static double code(double x, double y) {
	return (1.0 + ((y * y) * -0.16666666666666666)) * (1.0 + ((x * x) * (0.5 + (x * (x * (0.041666666666666664 + ((x * x) * 0.001388888888888889)))))));
}

Python

def code(x, y):
	return (1.0 + ((y * y) * -0.16666666666666666)) * (1.0 + ((x * x) * (0.5 + (x * (x * (0.041666666666666664 + ((x * x) * 0.001388888888888889)))))))

Julia

function code(x, y)
	return Float64(Float64(1.0 + Float64(Float64(y * y) * -0.16666666666666666)) * Float64(1.0 + Float64(Float64(x * x) * Float64(0.5 + Float64(x * Float64(x * Float64(0.041666666666666664 + Float64(Float64(x * x) * 0.001388888888888889))))))))
end

MATLAB

function tmp = code(x, y)
	tmp = (1.0 + ((y * y) * -0.16666666666666666)) * (1.0 + ((x * x) * (0.5 + (x * (x * (0.041666666666666664 + ((x * x) * 0.001388888888888889)))))));
end

Wolfram

code[x_, y_] := N[(N[(1.0 + N[(N[(y * y), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(0.5 + N[(x * N[(x * N[(0.041666666666666664 + N[(N[(x * x), $MachinePrecision] * 0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in y around 0

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

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

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

   2. *-commutative N/A

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

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

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

   4. unpow2 N/A

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

   5. *-lowering-*.f64 63.8%

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

5. Simplified 63.8%

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

6. Taylor expanded in x around 0

   \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]
7. Step-by-step derivation

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left({x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

   3. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(x \cdot x\right) \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

   7. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(x \cdot \left(x \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \left(\frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \left({x}^{2} \cdot \frac{1}{720}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({x}^{2}\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

   13. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

   14. *-lowering-*.f64 59.1%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

8. Simplified 59.1%

   \[\leadsto \color{blue}{\left(1 + \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot \left(0.041666666666666664 + \left(x \cdot x\right) \cdot 0.001388888888888889\right)\right)\right)\right)} \cdot \left(1 + \left(y \cdot y\right) \cdot -0.16666666666666666\right) \]

9. Final simplification 59.1%

   \[\leadsto \left(1 + \left(y \cdot y\right) \cdot -0.16666666666666666\right) \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot \left(0.041666666666666664 + \left(x \cdot x\right) \cdot 0.001388888888888889\right)\right)\right)\right) \]
10. Add Preprocessing

Alternative 12: 57.7% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 380:\\ \;\;\;\;1 + x \cdot \left(x \cdot \left(0.5 + \left(x \cdot x\right) \cdot \left(0.041666666666666664 + x \cdot \left(x \cdot 0.001388888888888889\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(0.5 + \left(y \cdot y\right) \cdot -0.08333333333333333\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 380.0)
   (+
    1.0
    (*
     x
     (*
      x
      (+
       0.5
       (*
        (* x x)
        (+ 0.041666666666666664 (* x (* x 0.001388888888888889))))))))
   (* x (* x (+ 0.5 (* (* y y) -0.08333333333333333))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 380.0) {
		tmp = 1.0 + (x * (x * (0.5 + ((x * x) * (0.041666666666666664 + (x * (x * 0.001388888888888889)))))));
	} else {
		tmp = x * (x * (0.5 + ((y * y) * -0.08333333333333333)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 380.0d0) then
        tmp = 1.0d0 + (x * (x * (0.5d0 + ((x * x) * (0.041666666666666664d0 + (x * (x * 0.001388888888888889d0)))))))
    else
        tmp = x * (x * (0.5d0 + ((y * y) * (-0.08333333333333333d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 380.0) {
		tmp = 1.0 + (x * (x * (0.5 + ((x * x) * (0.041666666666666664 + (x * (x * 0.001388888888888889)))))));
	} else {
		tmp = x * (x * (0.5 + ((y * y) * -0.08333333333333333)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 380.0:
		tmp = 1.0 + (x * (x * (0.5 + ((x * x) * (0.041666666666666664 + (x * (x * 0.001388888888888889)))))))
	else:
		tmp = x * (x * (0.5 + ((y * y) * -0.08333333333333333)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 380.0)
		tmp = Float64(1.0 + Float64(x * Float64(x * Float64(0.5 + Float64(Float64(x * x) * Float64(0.041666666666666664 + Float64(x * Float64(x * 0.001388888888888889))))))));
	else
		tmp = Float64(x * Float64(x * Float64(0.5 + Float64(Float64(y * y) * -0.08333333333333333))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 380.0)
		tmp = 1.0 + (x * (x * (0.5 + ((x * x) * (0.041666666666666664 + (x * (x * 0.001388888888888889)))))));
	else
		tmp = x * (x * (0.5 + ((y * y) * -0.08333333333333333)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 380.0], N[(1.0 + N[(x * N[(x * N[(0.5 + N[(N[(x * x), $MachinePrecision] * N[(0.041666666666666664 + N[(x * N[(x * 0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(x * N[(0.5 + N[(N[(y * y), $MachinePrecision] * -0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 380

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

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

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

      2. *-commutative N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 74.1%

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

   5. Simplified 74.1%

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

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left({x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(x \cdot x\right) \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(x \cdot \left(x \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \left(\frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \left({x}^{2} \cdot \frac{1}{720}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({x}^{2}\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

      14. *-lowering-*.f64 67.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

   8. Simplified 67.8%

      \[\leadsto \color{blue}{\left(1 + \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot \left(0.041666666666666664 + \left(x \cdot x\right) \cdot 0.001388888888888889\right)\right)\right)\right)} \cdot \left(1 + \left(y \cdot y\right) \cdot -0.16666666666666666\right) \]

   9. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)} \]
   10. Step-by-step derivation

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

          \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)}\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\frac{1}{2}} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)}\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)}\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{1}{24}} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{1}{24}} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \color{blue}{\left(\frac{1}{720} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

       11. *-commutative N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \left({x}^{2} \cdot \color{blue}{\frac{1}{720}}\right)\right)\right)\right)\right)\right)\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \left(\left(x \cdot x\right) \cdot \frac{1}{720}\right)\right)\right)\right)\right)\right)\right) \]

       13. associate-*l* N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \left(x \cdot \color{blue}{\left(x \cdot \frac{1}{720}\right)}\right)\right)\right)\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \frac{1}{720}\right)}\right)\right)\right)\right)\right)\right)\right) \]

       15. *-lowering-*.f64 65.5%

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\frac{1}{720}}\right)\right)\right)\right)\right)\right)\right)\right) \]

   11. Simplified 65.5%

       \[\leadsto \color{blue}{1 + x \cdot \left(x \cdot \left(0.5 + \left(x \cdot x\right) \cdot \left(0.041666666666666664 + x \cdot \left(x \cdot 0.001388888888888889\right)\right)\right)\right)} \]

   if 380 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

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

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

      2. *-commutative N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 33.3%

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

   5. Simplified 33.3%

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

   6. Taylor expanded in x around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 30.5%

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

   8. Simplified 30.5%

      \[\leadsto \color{blue}{\left(1 + 0.5 \cdot \left(x \cdot x\right)\right)} \cdot \left(1 + \left(y \cdot y\right) \cdot -0.16666666666666666\right) \]

   9. Taylor expanded in x around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. unpow2 N/A

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

       5. associate-*l* N/A

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

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

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

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

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

       8. distribute-lft-in N/A

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

       9. metadata-eval N/A

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

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

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

       11. associate-*r* N/A

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

       12. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{-1}{12} \cdot {\color{blue}{y}}^{2}\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{-1}{12}, \color{blue}{\left({y}^{2}\right)}\right)\right)\right)\right) \]

       14. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{-1}{12}, \left(y \cdot \color{blue}{y}\right)\right)\right)\right)\right) \]

       15. *-lowering-*.f64 30.8%

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{-1}{12}, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right)\right)\right)\right) \]

   11. Simplified 30.8%

       \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(0.5 + -0.08333333333333333 \cdot \left(y \cdot y\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 380:\\ \;\;\;\;1 + x \cdot \left(x \cdot \left(0.5 + \left(x \cdot x\right) \cdot \left(0.041666666666666664 + x \cdot \left(x \cdot 0.001388888888888889\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(0.5 + \left(y \cdot y\right) \cdot -0.08333333333333333\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 44.5% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.75 \cdot 10^{+77}:\\ \;\;\;\;1 + \left(y \cdot y\right) \cdot -0.16666666666666666\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{+149}:\\ \;\;\;\;x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(0.5 + \left(y \cdot y\right) \cdot -0.08333333333333333\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1.75e+77)
   (+ 1.0 (* (* y y) -0.16666666666666666))
   (if (<= x 4.8e+149)
     (* x (* x (* (* x x) 0.041666666666666664)))
     (* x (* x (+ 0.5 (* (* y y) -0.08333333333333333)))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.75e+77) {
		tmp = 1.0 + ((y * y) * -0.16666666666666666);
	} else if (x <= 4.8e+149) {
		tmp = x * (x * ((x * x) * 0.041666666666666664));
	} else {
		tmp = x * (x * (0.5 + ((y * y) * -0.08333333333333333)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 1.75d+77) then
        tmp = 1.0d0 + ((y * y) * (-0.16666666666666666d0))
    else if (x <= 4.8d+149) then
        tmp = x * (x * ((x * x) * 0.041666666666666664d0))
    else
        tmp = x * (x * (0.5d0 + ((y * y) * (-0.08333333333333333d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 1.75e+77) {
		tmp = 1.0 + ((y * y) * -0.16666666666666666);
	} else if (x <= 4.8e+149) {
		tmp = x * (x * ((x * x) * 0.041666666666666664));
	} else {
		tmp = x * (x * (0.5 + ((y * y) * -0.08333333333333333)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 1.75e+77:
		tmp = 1.0 + ((y * y) * -0.16666666666666666)
	elif x <= 4.8e+149:
		tmp = x * (x * ((x * x) * 0.041666666666666664))
	else:
		tmp = x * (x * (0.5 + ((y * y) * -0.08333333333333333)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.75e+77)
		tmp = Float64(1.0 + Float64(Float64(y * y) * -0.16666666666666666));
	elseif (x <= 4.8e+149)
		tmp = Float64(x * Float64(x * Float64(Float64(x * x) * 0.041666666666666664)));
	else
		tmp = Float64(x * Float64(x * Float64(0.5 + Float64(Float64(y * y) * -0.08333333333333333))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 1.75e+77)
		tmp = 1.0 + ((y * y) * -0.16666666666666666);
	elseif (x <= 4.8e+149)
		tmp = x * (x * ((x * x) * 0.041666666666666664));
	else
		tmp = x * (x * (0.5 + ((y * y) * -0.08333333333333333)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.75e+77], N[(1.0 + N[(N[(y * y), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.8e+149], N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(x * N[(0.5 + N[(N[(y * y), $MachinePrecision] * -0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.7500000000000001e77

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

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

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

      2. *-commutative N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 58.4%

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

   5. Simplified 58.4%

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

   6. Taylor expanded in x around 0

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

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

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

      2. *-commutative N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 32.4%

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

   8. Simplified 32.4%

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

   if 1.7500000000000001e77 < x < 4.80000000000000024e149

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(x\right), \color{blue}{1}\right) \]
   4. Step-by-step derivation

   5. Simplified 64.3%

      \[\leadsto \cosh x \cdot \color{blue}{1} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)} \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{1}{2}} + \frac{1}{24} \cdot {x}^{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{1}{2}} + \frac{1}{24} \cdot {x}^{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{24} \cdot {x}^{2}\right)}\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left({x}^{2} \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(x \cdot x\right) \cdot \frac{1}{24}\right)\right)\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(x \cdot \color{blue}{\left(x \cdot \frac{1}{24}\right)}\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(x \cdot \left(\frac{1}{24} \cdot \color{blue}{x}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{24} \cdot x\right)}\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 64.3%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right) \]

   8. Simplified 64.3%

      \[\leadsto \color{blue}{1 + \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot 0.041666666666666664\right)\right)} \]

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{24} \cdot {x}^{4}} \]
   10. Step-by-step derivation

       1. metadata-eval N/A

          \[\leadsto \frac{1}{24} \cdot {x}^{\left(2 \cdot \color{blue}{2}\right)} \]

       2. pow-sqr N/A

          \[\leadsto \frac{1}{24} \cdot \left({x}^{2} \cdot \color{blue}{{x}^{2}}\right) \]

       3. associate-*l* N/A

          \[\leadsto \left(\frac{1}{24} \cdot {x}^{2}\right) \cdot \color{blue}{{x}^{2}} \]

       4. *-commutative N/A

          \[\leadsto {x}^{2} \cdot \color{blue}{\left(\frac{1}{24} \cdot {x}^{2}\right)} \]

       5. unpow2 N/A

          \[\leadsto \left(x \cdot x\right) \cdot \left(\color{blue}{\frac{1}{24}} \cdot {x}^{2}\right) \]

       6. associate-*l* N/A

          \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{24} \cdot {x}^{2}\right)\right)} \]

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

          \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{1}{24} \cdot {x}^{2}\right)\right)}\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{24} \cdot {x}^{2}\right)}\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \color{blue}{\frac{1}{24}}\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{24}}\right)\right)\right) \]

       11. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{24}\right)\right)\right) \]

       12. *-lowering-*.f64 64.3%

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right) \]

   11. Simplified 64.3%

       \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)} \]

   if 4.80000000000000024e149 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

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

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

      2. *-commutative N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 83.9%

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

   5. Simplified 83.9%

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

   6. Taylor expanded in x around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 83.9%

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

   8. Simplified 83.9%

      \[\leadsto \color{blue}{\left(1 + 0.5 \cdot \left(x \cdot x\right)\right)} \cdot \left(1 + \left(y \cdot y\right) \cdot -0.16666666666666666\right) \]

   9. Taylor expanded in x around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. unpow2 N/A

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

       5. associate-*l* N/A

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

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

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

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

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

       8. distribute-lft-in N/A

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

       9. metadata-eval N/A

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

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

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

       11. associate-*r* N/A

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

       12. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{-1}{12} \cdot {\color{blue}{y}}^{2}\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{-1}{12}, \color{blue}{\left({y}^{2}\right)}\right)\right)\right)\right) \]

       14. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{-1}{12}, \left(y \cdot \color{blue}{y}\right)\right)\right)\right)\right) \]

       15. *-lowering-*.f64 83.9%

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{-1}{12}, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right)\right)\right)\right) \]

   11. Simplified 83.9%

       \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(0.5 + -0.08333333333333333 \cdot \left(y \cdot y\right)\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 40.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.75 \cdot 10^{+77}:\\ \;\;\;\;1 + \left(y \cdot y\right) \cdot -0.16666666666666666\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{+149}:\\ \;\;\;\;x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(0.5 + \left(y \cdot y\right) \cdot -0.08333333333333333\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 56.9% accurate, 9.8× speedup

Math

\[\begin{array}{l} \\ \left(1 + \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot 0.041666666666666664\right)\right)\right) \cdot \left(1 + \left(y \cdot y\right) \cdot -0.16666666666666666\right) \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (*
  (+ 1.0 (* (* x x) (+ 0.5 (* x (* x 0.041666666666666664)))))
  (+ 1.0 (* (* y y) -0.16666666666666666))))

C

double code(double x, double y) {
	return (1.0 + ((x * x) * (0.5 + (x * (x * 0.041666666666666664))))) * (1.0 + ((y * y) * -0.16666666666666666));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (1.0d0 + ((x * x) * (0.5d0 + (x * (x * 0.041666666666666664d0))))) * (1.0d0 + ((y * y) * (-0.16666666666666666d0)))
end function

Java

public static double code(double x, double y) {
	return (1.0 + ((x * x) * (0.5 + (x * (x * 0.041666666666666664))))) * (1.0 + ((y * y) * -0.16666666666666666));
}

Python

def code(x, y):
	return (1.0 + ((x * x) * (0.5 + (x * (x * 0.041666666666666664))))) * (1.0 + ((y * y) * -0.16666666666666666))

Julia

function code(x, y)
	return Float64(Float64(1.0 + Float64(Float64(x * x) * Float64(0.5 + Float64(x * Float64(x * 0.041666666666666664))))) * Float64(1.0 + Float64(Float64(y * y) * -0.16666666666666666)))
end

MATLAB

function tmp = code(x, y)
	tmp = (1.0 + ((x * x) * (0.5 + (x * (x * 0.041666666666666664))))) * (1.0 + ((y * y) * -0.16666666666666666));
end

Wolfram

code[x_, y_] := N[(N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(0.5 + N[(x * N[(x * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(y * y), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in x around 0

   \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)}, \mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), y\right)\right) \]
4. Step-by-step derivation

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left({x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)\right), \mathsf{/.f64}\left(\color{blue}{\mathsf{sin.f64}\left(y\right)}, y\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), y\right)\right) \]

   3. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), y\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), y\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{24} \cdot {x}^{2}\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), y\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left({x}^{2} \cdot \frac{1}{24}\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), y\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(x \cdot x\right) \cdot \frac{1}{24}\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), y\right)\right) \]

   8. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(x \cdot \left(x \cdot \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), y\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \left(x \cdot \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), y\right)\right) \]

   10. *-lowering-*.f64 86.8%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), y\right)\right) \]

5. Simplified 86.8%

   \[\leadsto \color{blue}{\left(1 + \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot 0.041666666666666664\right)\right)\right)} \cdot \frac{\sin y}{y} \]

6. Taylor expanded in y around 0

   \[\leadsto \color{blue}{1 + \left(\frac{-1}{6} \cdot \left({y}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)\right) + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto 1 + \left({x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) + \color{blue}{\frac{-1}{6} \cdot \left({y}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)\right)}\right) \]

   2. associate-+r+ N/A

      \[\leadsto \left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right) + \color{blue}{\frac{-1}{6} \cdot \left({y}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)\right)} \]

   3. associate-*r* N/A

      \[\leadsto \left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right) + \left(\frac{-1}{6} \cdot {y}^{2}\right) \cdot \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)} \]

   4. distribute-rgt1-in N/A

      \[\leadsto \left(\frac{-1}{6} \cdot {y}^{2} + 1\right) \cdot \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)} \]

   5. +-commutative N/A

      \[\leadsto \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) \cdot \left(\color{blue}{1} + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\left(1 + \frac{-1}{6} \cdot {y}^{2}\right), \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot {y}^{2}\right)\right), \left(\color{blue}{1} + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left({y}^{2} \cdot \frac{-1}{6}\right)\right), \left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({y}^{2}\right), \frac{-1}{6}\right)\right), \left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)\right) \]

   10. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(y \cdot y\right), \frac{-1}{6}\right)\right), \left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right), \left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)}\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)}\right)\right)\right) \]

8. Simplified 58.2%

   \[\leadsto \color{blue}{\left(1 + \left(y \cdot y\right) \cdot -0.16666666666666666\right) \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot 0.041666666666666664\right)\right)\right)} \]

9. Final simplification 58.2%

   \[\leadsto \left(1 + \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot 0.041666666666666664\right)\right)\right) \cdot \left(1 + \left(y \cdot y\right) \cdot -0.16666666666666666\right) \]
10. Add Preprocessing

Alternative 15: 55.3% accurate, 11.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 380:\\ \;\;\;\;1 + \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot 0.041666666666666664\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(0.5 + \left(y \cdot y\right) \cdot -0.08333333333333333\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 380.0)
   (+ 1.0 (* (* x x) (+ 0.5 (* x (* x 0.041666666666666664)))))
   (* x (* x (+ 0.5 (* (* y y) -0.08333333333333333))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 380.0) {
		tmp = 1.0 + ((x * x) * (0.5 + (x * (x * 0.041666666666666664))));
	} else {
		tmp = x * (x * (0.5 + ((y * y) * -0.08333333333333333)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 380.0d0) then
        tmp = 1.0d0 + ((x * x) * (0.5d0 + (x * (x * 0.041666666666666664d0))))
    else
        tmp = x * (x * (0.5d0 + ((y * y) * (-0.08333333333333333d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 380.0) {
		tmp = 1.0 + ((x * x) * (0.5 + (x * (x * 0.041666666666666664))));
	} else {
		tmp = x * (x * (0.5 + ((y * y) * -0.08333333333333333)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 380.0:
		tmp = 1.0 + ((x * x) * (0.5 + (x * (x * 0.041666666666666664))))
	else:
		tmp = x * (x * (0.5 + ((y * y) * -0.08333333333333333)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 380.0)
		tmp = Float64(1.0 + Float64(Float64(x * x) * Float64(0.5 + Float64(x * Float64(x * 0.041666666666666664)))));
	else
		tmp = Float64(x * Float64(x * Float64(0.5 + Float64(Float64(y * y) * -0.08333333333333333))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 380.0)
		tmp = 1.0 + ((x * x) * (0.5 + (x * (x * 0.041666666666666664))));
	else
		tmp = x * (x * (0.5 + ((y * y) * -0.08333333333333333)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 380.0], N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(0.5 + N[(x * N[(x * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(x * N[(0.5 + N[(N[(y * y), $MachinePrecision] * -0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 380

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(x\right), \color{blue}{1}\right) \]
   4. Step-by-step derivation

   5. Simplified 73.4%

      \[\leadsto \cosh x \cdot \color{blue}{1} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)} \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{1}{2}} + \frac{1}{24} \cdot {x}^{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{1}{2}} + \frac{1}{24} \cdot {x}^{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{24} \cdot {x}^{2}\right)}\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left({x}^{2} \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(x \cdot x\right) \cdot \frac{1}{24}\right)\right)\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(x \cdot \color{blue}{\left(x \cdot \frac{1}{24}\right)}\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(x \cdot \left(\frac{1}{24} \cdot \color{blue}{x}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{24} \cdot x\right)}\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 62.9%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right) \]

   8. Simplified 62.9%

      \[\leadsto \color{blue}{1 + \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot 0.041666666666666664\right)\right)} \]

   if 380 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

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

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

      2. *-commutative N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 33.3%

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

   5. Simplified 33.3%

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

   6. Taylor expanded in x around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 30.5%

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

   8. Simplified 30.5%

      \[\leadsto \color{blue}{\left(1 + 0.5 \cdot \left(x \cdot x\right)\right)} \cdot \left(1 + \left(y \cdot y\right) \cdot -0.16666666666666666\right) \]

   9. Taylor expanded in x around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. unpow2 N/A

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

       5. associate-*l* N/A

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

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

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

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

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

       8. distribute-lft-in N/A

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

       9. metadata-eval N/A

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

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

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

       11. associate-*r* N/A

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

       12. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{-1}{12} \cdot {\color{blue}{y}}^{2}\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{-1}{12}, \color{blue}{\left({y}^{2}\right)}\right)\right)\right)\right) \]

       14. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{-1}{12}, \left(y \cdot \color{blue}{y}\right)\right)\right)\right)\right) \]

       15. *-lowering-*.f64 30.8%

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{-1}{12}, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right)\right)\right)\right) \]

   11. Simplified 30.8%

       \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(0.5 + -0.08333333333333333 \cdot \left(y \cdot y\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 380:\\ \;\;\;\;1 + \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot 0.041666666666666664\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(0.5 + \left(y \cdot y\right) \cdot -0.08333333333333333\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 55.1% accurate, 12.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 380:\\ \;\;\;\;1 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.041666666666666664\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(0.5 + \left(y \cdot y\right) \cdot -0.08333333333333333\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 380.0)
   (+ 1.0 (* (* x x) (* (* x x) 0.041666666666666664)))
   (* x (* x (+ 0.5 (* (* y y) -0.08333333333333333))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 380.0) {
		tmp = 1.0 + ((x * x) * ((x * x) * 0.041666666666666664));
	} else {
		tmp = x * (x * (0.5 + ((y * y) * -0.08333333333333333)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 380.0d0) then
        tmp = 1.0d0 + ((x * x) * ((x * x) * 0.041666666666666664d0))
    else
        tmp = x * (x * (0.5d0 + ((y * y) * (-0.08333333333333333d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 380.0) {
		tmp = 1.0 + ((x * x) * ((x * x) * 0.041666666666666664));
	} else {
		tmp = x * (x * (0.5 + ((y * y) * -0.08333333333333333)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 380.0:
		tmp = 1.0 + ((x * x) * ((x * x) * 0.041666666666666664))
	else:
		tmp = x * (x * (0.5 + ((y * y) * -0.08333333333333333)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 380.0)
		tmp = Float64(1.0 + Float64(Float64(x * x) * Float64(Float64(x * x) * 0.041666666666666664)));
	else
		tmp = Float64(x * Float64(x * Float64(0.5 + Float64(Float64(y * y) * -0.08333333333333333))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 380.0)
		tmp = 1.0 + ((x * x) * ((x * x) * 0.041666666666666664));
	else
		tmp = x * (x * (0.5 + ((y * y) * -0.08333333333333333)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 380.0], N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(x * N[(0.5 + N[(N[(y * y), $MachinePrecision] * -0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 380

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(x\right), \color{blue}{1}\right) \]
   4. Step-by-step derivation

   5. Simplified 73.4%

      \[\leadsto \cosh x \cdot \color{blue}{1} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)} \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{1}{2}} + \frac{1}{24} \cdot {x}^{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{1}{2}} + \frac{1}{24} \cdot {x}^{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{24} \cdot {x}^{2}\right)}\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left({x}^{2} \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(x \cdot x\right) \cdot \frac{1}{24}\right)\right)\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(x \cdot \color{blue}{\left(x \cdot \frac{1}{24}\right)}\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(x \cdot \left(\frac{1}{24} \cdot \color{blue}{x}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{24} \cdot x\right)}\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 62.9%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right) \]

   8. Simplified 62.9%

      \[\leadsto \color{blue}{1 + \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot 0.041666666666666664\right)\right)} \]

   9. Taylor expanded in x around inf

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \color{blue}{\left(\frac{1}{24} \cdot {x}^{2}\right)}\right)\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({x}^{2} \cdot \color{blue}{\frac{1}{24}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{24}}\right)\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{24}\right)\right)\right) \]

       4. *-lowering-*.f64 62.5%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right) \]

   11. Simplified 62.5%

       \[\leadsto 1 + \left(x \cdot x\right) \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot 0.041666666666666664\right)} \]

   if 380 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

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

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

      2. *-commutative N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 33.3%

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

   5. Simplified 33.3%

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

   6. Taylor expanded in x around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 30.5%

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

   8. Simplified 30.5%

      \[\leadsto \color{blue}{\left(1 + 0.5 \cdot \left(x \cdot x\right)\right)} \cdot \left(1 + \left(y \cdot y\right) \cdot -0.16666666666666666\right) \]

   9. Taylor expanded in x around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. unpow2 N/A

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

       5. associate-*l* N/A

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

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

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

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

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

       8. distribute-lft-in N/A

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

       9. metadata-eval N/A

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

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

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

       11. associate-*r* N/A

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

       12. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{-1}{12} \cdot {\color{blue}{y}}^{2}\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{-1}{12}, \color{blue}{\left({y}^{2}\right)}\right)\right)\right)\right) \]

       14. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{-1}{12}, \left(y \cdot \color{blue}{y}\right)\right)\right)\right)\right) \]

       15. *-lowering-*.f64 30.8%

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{-1}{12}, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right)\right)\right)\right) \]

   11. Simplified 30.8%

       \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(0.5 + -0.08333333333333333 \cdot \left(y \cdot y\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 380:\\ \;\;\;\;1 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.041666666666666664\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(0.5 + \left(y \cdot y\right) \cdot -0.08333333333333333\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 44.5% accurate, 14.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.95 \cdot 10^{+77}:\\ \;\;\;\;1 + \left(y \cdot y\right) \cdot -0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1.95e+77)
   (+ 1.0 (* (* y y) -0.16666666666666666))
   (* x (* x (* (* x x) 0.041666666666666664)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.95e+77) {
		tmp = 1.0 + ((y * y) * -0.16666666666666666);
	} else {
		tmp = x * (x * ((x * x) * 0.041666666666666664));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 1.95d+77) then
        tmp = 1.0d0 + ((y * y) * (-0.16666666666666666d0))
    else
        tmp = x * (x * ((x * x) * 0.041666666666666664d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 1.95e+77) {
		tmp = 1.0 + ((y * y) * -0.16666666666666666);
	} else {
		tmp = x * (x * ((x * x) * 0.041666666666666664));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 1.95e+77:
		tmp = 1.0 + ((y * y) * -0.16666666666666666)
	else:
		tmp = x * (x * ((x * x) * 0.041666666666666664))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.95e+77)
		tmp = Float64(1.0 + Float64(Float64(y * y) * -0.16666666666666666));
	else
		tmp = Float64(x * Float64(x * Float64(Float64(x * x) * 0.041666666666666664)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 1.95e+77)
		tmp = 1.0 + ((y * y) * -0.16666666666666666);
	else
		tmp = x * (x * ((x * x) * 0.041666666666666664));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.95e+77], N[(1.0 + N[(N[(y * y), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision], N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.9499999999999999e77

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

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

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

      2. *-commutative N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 58.4%

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

   5. Simplified 58.4%

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

   6. Taylor expanded in x around 0

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

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

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

      2. *-commutative N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 32.4%

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

   8. Simplified 32.4%

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

   if 1.9499999999999999e77 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(x\right), \color{blue}{1}\right) \]
   4. Step-by-step derivation

   5. Simplified 64.4%

      \[\leadsto \cosh x \cdot \color{blue}{1} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)} \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{1}{2}} + \frac{1}{24} \cdot {x}^{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{1}{2}} + \frac{1}{24} \cdot {x}^{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{24} \cdot {x}^{2}\right)}\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left({x}^{2} \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(x \cdot x\right) \cdot \frac{1}{24}\right)\right)\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(x \cdot \color{blue}{\left(x \cdot \frac{1}{24}\right)}\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(x \cdot \left(\frac{1}{24} \cdot \color{blue}{x}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{24} \cdot x\right)}\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 64.4%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right) \]

   8. Simplified 64.4%

      \[\leadsto \color{blue}{1 + \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot 0.041666666666666664\right)\right)} \]

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{24} \cdot {x}^{4}} \]
   10. Step-by-step derivation

       1. metadata-eval N/A

          \[\leadsto \frac{1}{24} \cdot {x}^{\left(2 \cdot \color{blue}{2}\right)} \]

       2. pow-sqr N/A

          \[\leadsto \frac{1}{24} \cdot \left({x}^{2} \cdot \color{blue}{{x}^{2}}\right) \]

       3. associate-*l* N/A

          \[\leadsto \left(\frac{1}{24} \cdot {x}^{2}\right) \cdot \color{blue}{{x}^{2}} \]

       4. *-commutative N/A

          \[\leadsto {x}^{2} \cdot \color{blue}{\left(\frac{1}{24} \cdot {x}^{2}\right)} \]

       5. unpow2 N/A

          \[\leadsto \left(x \cdot x\right) \cdot \left(\color{blue}{\frac{1}{24}} \cdot {x}^{2}\right) \]

       6. associate-*l* N/A

          \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{24} \cdot {x}^{2}\right)\right)} \]

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

          \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{1}{24} \cdot {x}^{2}\right)\right)}\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{24} \cdot {x}^{2}\right)}\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \color{blue}{\frac{1}{24}}\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{24}}\right)\right)\right) \]

       11. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{24}\right)\right)\right) \]

       12. *-lowering-*.f64 64.4%

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right) \]

   11. Simplified 64.4%

       \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 46.4% accurate, 17.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 6.6 \cdot 10^{+174}:\\ \;\;\;\;1 + \left(x \cdot x\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;1 + \left(y \cdot y\right) \cdot -0.16666666666666666\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 6.6e+174)
   (+ 1.0 (* (* x x) 0.5))
   (+ 1.0 (* (* y y) -0.16666666666666666))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 6.6e+174) {
		tmp = 1.0 + ((x * x) * 0.5);
	} else {
		tmp = 1.0 + ((y * y) * -0.16666666666666666);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 6.6d+174) then
        tmp = 1.0d0 + ((x * x) * 0.5d0)
    else
        tmp = 1.0d0 + ((y * y) * (-0.16666666666666666d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 6.6e+174) {
		tmp = 1.0 + ((x * x) * 0.5);
	} else {
		tmp = 1.0 + ((y * y) * -0.16666666666666666);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 6.6e+174:
		tmp = 1.0 + ((x * x) * 0.5)
	else:
		tmp = 1.0 + ((y * y) * -0.16666666666666666)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 6.6e+174)
		tmp = Float64(1.0 + Float64(Float64(x * x) * 0.5));
	else
		tmp = Float64(1.0 + Float64(Float64(y * y) * -0.16666666666666666));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 6.6e+174)
		tmp = 1.0 + ((x * x) * 0.5);
	else
		tmp = 1.0 + ((y * y) * -0.16666666666666666);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 6.6e+174], N[(1.0 + N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(y * y), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6.6000000000000001e174

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(x\right), \color{blue}{1}\right) \]
   4. Step-by-step derivation

   5. Simplified 65.7%

      \[\leadsto \cosh x \cdot \color{blue}{1} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + \frac{1}{2} \cdot {x}^{2}} \]
   7. Step-by-step derivation

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

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 47.1%

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

   8. Simplified 47.1%

      \[\leadsto \color{blue}{1 + 0.5 \cdot \left(x \cdot x\right)} \]

   if 6.6000000000000001e174 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

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

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

      2. *-commutative N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 33.1%

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

   5. Simplified 33.1%

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

   6. Taylor expanded in x around 0

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

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

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

      2. *-commutative N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 33.1%

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

   8. Simplified 33.1%

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

4. Final simplification 45.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6.6 \cdot 10^{+174}:\\ \;\;\;\;1 + \left(x \cdot x\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;1 + \left(y \cdot y\right) \cdot -0.16666666666666666\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 36.4% accurate, 20.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.4:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x 1.4) 1.0 (* (* x x) 0.5)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.4) {
		tmp = 1.0;
	} else {
		tmp = (x * x) * 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 1.4d0) then
        tmp = 1.0d0
    else
        tmp = (x * x) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 1.4) {
		tmp = 1.0;
	} else {
		tmp = (x * x) * 0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 1.4:
		tmp = 1.0
	else:
		tmp = (x * x) * 0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.4)
		tmp = 1.0;
	else
		tmp = Float64(Float64(x * x) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 1.4)
		tmp = 1.0;
	else
		tmp = (x * x) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.4], 1.0, N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.3999999999999999

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(x\right), \color{blue}{1}\right) \]
   4. Step-by-step derivation

   5. Simplified 57.3%

      \[\leadsto \cosh x \cdot \color{blue}{1} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \]
   7. Step-by-step derivation

   8. Simplified 28.7%

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

   if 1.3999999999999999 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(x\right), \color{blue}{1}\right) \]
   4. Step-by-step derivation

   5. Simplified 68.3%

      \[\leadsto \cosh x \cdot \color{blue}{1} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + \frac{1}{2} \cdot {x}^{2}} \]
   7. Step-by-step derivation

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

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 35.6%

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

   8. Simplified 35.6%

      \[\leadsto \color{blue}{1 + 0.5 \cdot \left(x \cdot x\right)} \]

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot {x}^{2}} \]
   10. Step-by-step derivation

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

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

       2. unpow2 N/A

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

       3. *-lowering-*.f64 35.6%

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right) \]

   11. Simplified 35.6%

       \[\leadsto \color{blue}{0.5 \cdot \left(x \cdot x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 30.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.4:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 45.2% accurate, 29.3× speedup

Math

\[\begin{array}{l} \\ 1 + \left(x \cdot x\right) \cdot 0.5 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ 1.0 (* (* x x) 0.5)))

C

double code(double x, double y) {
	return 1.0 + ((x * x) * 0.5);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 + ((x * x) * 0.5d0)
end function

Java

public static double code(double x, double y) {
	return 1.0 + ((x * x) * 0.5);
}

Python

def code(x, y):
	return 1.0 + ((x * x) * 0.5)

Julia

function code(x, y)
	return Float64(1.0 + Float64(Float64(x * x) * 0.5))
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0 + ((x * x) * 0.5);
end

Wolfram

code[x_, y_] := N[(1.0 + N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in y around 0

   \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(x\right), \color{blue}{1}\right) \]
4. Step-by-step derivation

5. Simplified 59.9%

   \[\leadsto \cosh x \cdot \color{blue}{1} \]

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1 + \frac{1}{2} \cdot {x}^{2}} \]
7. Step-by-step derivation

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

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

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

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

   3. unpow2 N/A

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

   4. *-lowering-*.f64 42.4%

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

8. Simplified 42.4%

   \[\leadsto \color{blue}{1 + 0.5 \cdot \left(x \cdot x\right)} \]

9. Final simplification 42.4%

   \[\leadsto 1 + \left(x \cdot x\right) \cdot 0.5 \]
10. Add Preprocessing

Alternative 21: 27.3% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 1.0)

C

double code(double x, double y) {
	return 1.0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0
end function

Java

public static double code(double x, double y) {
	return 1.0;
}

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0;
end

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in y around 0

   \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(x\right), \color{blue}{1}\right) \]
4. Step-by-step derivation

5. Simplified 59.9%

   \[\leadsto \cosh x \cdot \color{blue}{1} \]

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1} \]
7. Step-by-step derivation

8. Simplified 22.6%

   \[\leadsto \color{blue}{1} \]
9. Add Preprocessing

Developer Target 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024138 
(FPCore (x y)
  :name "Linear.Quaternion:$csinh from linear-1.19.1.3"
  :precision binary64

  :alt
  (! :herbie-platform default (/ (* (cosh x) (sin y)) y))

  (* (cosh x) (/ (sin y) y)))