Linear.Quaternion:$csinh from linear-1.19.1.3

Percentage Accurate: 99.9% → 99.9%

Time: 4.5s

Alternatives: 20

Speedup: 1.0×

• 50.0% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y)
use fmin_fmax_functions
    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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y)
use fmin_fmax_functions
    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

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

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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y)
use fmin_fmax_functions
    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 2: 98.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ t_1 := \cosh x \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(y \cdot y, -0.08333333333333333, 0.5\right) \cdot y\right) \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.08333333333333333, x \cdot x, 1\right), x \cdot x, 2\right)}{y}\\ \mathbf{elif}\;t\_1 \leq 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)) (t_1 (* (cosh x) t_0)))
   (if (<= t_1 (- INFINITY))
     (*
      (* (fma (* y y) -0.08333333333333333 0.5) y)
      (/ (fma (fma 0.08333333333333333 (* x x) 1.0) (* x x) 2.0) y))
     (if (<= t_1 1e-15)
       (* (fma (fma 0.041666666666666664 (* x x) 0.5) (* x x) 1.0) t_0)
       (* (cosh x) 1.0)))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double t_1 = cosh(x) * t_0;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (fma((y * y), -0.08333333333333333, 0.5) * y) * (fma(fma(0.08333333333333333, (x * x), 1.0), (x * x), 2.0) / y);
	} else if (t_1 <= 1e-15) {
		tmp = fma(fma(0.041666666666666664, (x * x), 0.5), (x * x), 1.0) * t_0;
	} else {
		tmp = cosh(x) * 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(y) / y)
	t_1 = Float64(cosh(x) * t_0)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(fma(Float64(y * y), -0.08333333333333333, 0.5) * y) * Float64(fma(fma(0.08333333333333333, Float64(x * x), 1.0), Float64(x * x), 2.0) / y));
	elseif (t_1 <= 1e-15)
		tmp = Float64(fma(fma(0.041666666666666664, Float64(x * x), 0.5), Float64(x * x), 1.0) * t_0);
	else
		tmp = Float64(cosh(x) * 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cosh[x], $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(N[(y * y), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision] * y), $MachinePrecision] * N[(N[(N[(0.08333333333333333 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e-15], N[(N[(N[(0.041666666666666664 * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[Cosh[x], $MachinePrecision] * 1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < -inf.0

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. rec-exp N/A

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

      8. cosh-undef N/A

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

      9. lower-*.f64 N/A

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

      10. lift-cosh.f64 100.0

          \[\leadsto \left(0.5 \cdot \sin y\right) \cdot \frac{2 \cdot \cosh x}{y} \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin y\right) \cdot \frac{2 \cdot \cosh x}{y}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin y\right) \cdot \frac{2}{y} \]
   7. Step-by-step derivation

   8. Applied rewrites 2.3%

      \[\leadsto \left(0.5 \cdot \sin y\right) \cdot \frac{2}{y} \]

   9. Taylor expanded in y around 0

      \[\leadsto \left(y \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {y}^{2}\right)\right) \cdot \frac{\color{blue}{2}}{y} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {y}^{2}\right) \cdot y\right) \cdot \frac{2}{y} \]

       2. lower-*.f64 N/A

          \[\leadsto \left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {y}^{2}\right) \cdot y\right) \cdot \frac{2}{y} \]

       3. +-commutative N/A

          \[\leadsto \left(\left(\frac{-1}{12} \cdot {y}^{2} + \frac{1}{2}\right) \cdot y\right) \cdot \frac{2}{y} \]

       4. *-commutative N/A

          \[\leadsto \left(\left({y}^{2} \cdot \frac{-1}{12} + \frac{1}{2}\right) \cdot y\right) \cdot \frac{2}{y} \]

       5. lower-fma.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left({y}^{2}, \frac{-1}{12}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{2}{y} \]

       6. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(y \cdot y, \frac{-1}{12}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{2}{y} \]

       7. lift-*.f64 53.6

          \[\leadsto \left(\mathsf{fma}\left(y \cdot y, -0.08333333333333333, 0.5\right) \cdot y\right) \cdot \frac{2}{y} \]

   11. Applied rewrites 53.6%

       \[\leadsto \left(\mathsf{fma}\left(y \cdot y, -0.08333333333333333, 0.5\right) \cdot y\right) \cdot \frac{\color{blue}{2}}{y} \]

   12. Taylor expanded in x around 0

       \[\leadsto \left(\mathsf{fma}\left(y \cdot y, \frac{-1}{12}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{2 + {x}^{2} \cdot \left(1 + \frac{1}{12} \cdot {x}^{2}\right)}{y} \]
   13. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(y \cdot y, \frac{-1}{12}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{{x}^{2} \cdot \left(1 + \frac{1}{12} \cdot {x}^{2}\right) + 2}{y} \]

       2. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(y \cdot y, \frac{-1}{12}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{\left(1 + \frac{1}{12} \cdot {x}^{2}\right) \cdot {x}^{2} + 2}{y} \]

       3. lower-fma.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(y \cdot y, \frac{-1}{12}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{\mathsf{fma}\left(1 + \frac{1}{12} \cdot {x}^{2}, {x}^{2}, 2\right)}{y} \]

       4. +-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(y \cdot y, \frac{-1}{12}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{\mathsf{fma}\left(\frac{1}{12} \cdot {x}^{2} + 1, {x}^{2}, 2\right)}{y} \]

       5. lower-fma.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(y \cdot y, \frac{-1}{12}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{12}, {x}^{2}, 1\right), {x}^{2}, 2\right)}{y} \]

       6. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(y \cdot y, \frac{-1}{12}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{12}, x \cdot x, 1\right), {x}^{2}, 2\right)}{y} \]

       7. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(y \cdot y, \frac{-1}{12}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{12}, x \cdot x, 1\right), {x}^{2}, 2\right)}{y} \]

       8. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(y \cdot y, \frac{-1}{12}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{12}, x \cdot x, 1\right), x \cdot x, 2\right)}{y} \]

       9. lift-*.f64 100.0

          \[\leadsto \left(\mathsf{fma}\left(y \cdot y, -0.08333333333333333, 0.5\right) \cdot y\right) \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.08333333333333333, x \cdot x, 1\right), x \cdot x, 2\right)}{y} \]

   14. Applied rewrites 100.0%

       \[\leadsto \left(\mathsf{fma}\left(y \cdot y, -0.08333333333333333, 0.5\right) \cdot y\right) \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.08333333333333333, x \cdot x, 1\right), x \cdot x, 2\right)}{y} \]

   if -inf.0 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < 1.0000000000000001e-15

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

         \[\leadsto \left(\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) \cdot {x}^{2} + 1\right) \cdot \frac{\sin y}{y} \]

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, {x}^{2}, \frac{1}{2}\right), {\color{blue}{x}}^{2}, 1\right) \cdot \frac{\sin y}{y} \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot \frac{\sin y}{y} \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot \frac{\sin y}{y} \]

      8. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot \color{blue}{x}, 1\right) \cdot \frac{\sin y}{y} \]

      9. lower-*.f64 99.5

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

   5. Applied rewrites 99.5%

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

   if 1.0000000000000001e-15 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 100.0%

      \[\leadsto \cosh x \cdot \color{blue}{1} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 98.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ t_1 := \cosh x \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(y \cdot y, -0.08333333333333333, 0.5\right) \cdot y\right) \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.08333333333333333, x \cdot x, 1\right), x \cdot x, 2\right)}{y}\\ \mathbf{elif}\;t\_1 \leq 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, 0.5, 1\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)) (t_1 (* (cosh x) t_0)))
   (if (<= t_1 (- INFINITY))
     (*
      (* (fma (* y y) -0.08333333333333333 0.5) y)
      (/ (fma (fma 0.08333333333333333 (* x x) 1.0) (* x x) 2.0) y))
     (if (<= t_1 1e-15) (* (fma (* x x) 0.5 1.0) t_0) (* (cosh x) 1.0)))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double t_1 = cosh(x) * t_0;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (fma((y * y), -0.08333333333333333, 0.5) * y) * (fma(fma(0.08333333333333333, (x * x), 1.0), (x * x), 2.0) / y);
	} else if (t_1 <= 1e-15) {
		tmp = fma((x * x), 0.5, 1.0) * t_0;
	} else {
		tmp = cosh(x) * 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(y) / y)
	t_1 = Float64(cosh(x) * t_0)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(fma(Float64(y * y), -0.08333333333333333, 0.5) * y) * Float64(fma(fma(0.08333333333333333, Float64(x * x), 1.0), Float64(x * x), 2.0) / y));
	elseif (t_1 <= 1e-15)
		tmp = Float64(fma(Float64(x * x), 0.5, 1.0) * t_0);
	else
		tmp = Float64(cosh(x) * 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cosh[x], $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(N[(y * y), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision] * y), $MachinePrecision] * N[(N[(N[(0.08333333333333333 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e-15], N[(N[(N[(x * x), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[Cosh[x], $MachinePrecision] * 1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < -inf.0

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. rec-exp N/A

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

      8. cosh-undef N/A

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

      9. lower-*.f64 N/A

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

      10. lift-cosh.f64 100.0

          \[\leadsto \left(0.5 \cdot \sin y\right) \cdot \frac{2 \cdot \cosh x}{y} \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin y\right) \cdot \frac{2 \cdot \cosh x}{y}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin y\right) \cdot \frac{2}{y} \]
   7. Step-by-step derivation

   8. Applied rewrites 2.3%

      \[\leadsto \left(0.5 \cdot \sin y\right) \cdot \frac{2}{y} \]

   9. Taylor expanded in y around 0

      \[\leadsto \left(y \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {y}^{2}\right)\right) \cdot \frac{\color{blue}{2}}{y} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {y}^{2}\right) \cdot y\right) \cdot \frac{2}{y} \]

       2. lower-*.f64 N/A

          \[\leadsto \left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {y}^{2}\right) \cdot y\right) \cdot \frac{2}{y} \]

       3. +-commutative N/A

          \[\leadsto \left(\left(\frac{-1}{12} \cdot {y}^{2} + \frac{1}{2}\right) \cdot y\right) \cdot \frac{2}{y} \]

       4. *-commutative N/A

          \[\leadsto \left(\left({y}^{2} \cdot \frac{-1}{12} + \frac{1}{2}\right) \cdot y\right) \cdot \frac{2}{y} \]

       5. lower-fma.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left({y}^{2}, \frac{-1}{12}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{2}{y} \]

       6. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(y \cdot y, \frac{-1}{12}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{2}{y} \]

       7. lift-*.f64 53.6

          \[\leadsto \left(\mathsf{fma}\left(y \cdot y, -0.08333333333333333, 0.5\right) \cdot y\right) \cdot \frac{2}{y} \]

   11. Applied rewrites 53.6%

       \[\leadsto \left(\mathsf{fma}\left(y \cdot y, -0.08333333333333333, 0.5\right) \cdot y\right) \cdot \frac{\color{blue}{2}}{y} \]

   12. Taylor expanded in x around 0

       \[\leadsto \left(\mathsf{fma}\left(y \cdot y, \frac{-1}{12}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{2 + {x}^{2} \cdot \left(1 + \frac{1}{12} \cdot {x}^{2}\right)}{y} \]
   13. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(y \cdot y, \frac{-1}{12}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{{x}^{2} \cdot \left(1 + \frac{1}{12} \cdot {x}^{2}\right) + 2}{y} \]

       2. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(y \cdot y, \frac{-1}{12}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{\left(1 + \frac{1}{12} \cdot {x}^{2}\right) \cdot {x}^{2} + 2}{y} \]

       3. lower-fma.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(y \cdot y, \frac{-1}{12}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{\mathsf{fma}\left(1 + \frac{1}{12} \cdot {x}^{2}, {x}^{2}, 2\right)}{y} \]

       4. +-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(y \cdot y, \frac{-1}{12}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{\mathsf{fma}\left(\frac{1}{12} \cdot {x}^{2} + 1, {x}^{2}, 2\right)}{y} \]

       5. lower-fma.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(y \cdot y, \frac{-1}{12}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{12}, {x}^{2}, 1\right), {x}^{2}, 2\right)}{y} \]

       6. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(y \cdot y, \frac{-1}{12}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{12}, x \cdot x, 1\right), {x}^{2}, 2\right)}{y} \]

       7. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(y \cdot y, \frac{-1}{12}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{12}, x \cdot x, 1\right), {x}^{2}, 2\right)}{y} \]

       8. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(y \cdot y, \frac{-1}{12}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{12}, x \cdot x, 1\right), x \cdot x, 2\right)}{y} \]

       9. lift-*.f64 100.0

          \[\leadsto \left(\mathsf{fma}\left(y \cdot y, -0.08333333333333333, 0.5\right) \cdot y\right) \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.08333333333333333, x \cdot x, 1\right), x \cdot x, 2\right)}{y} \]

   14. Applied rewrites 100.0%

       \[\leadsto \left(\mathsf{fma}\left(y \cdot y, -0.08333333333333333, 0.5\right) \cdot y\right) \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.08333333333333333, x \cdot x, 1\right), x \cdot x, 2\right)}{y} \]

   if -inf.0 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < 1.0000000000000001e-15

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 99.0

         \[\leadsto \mathsf{fma}\left(x \cdot x, 0.5, 1\right) \cdot \frac{\sin y}{y} \]

   5. Applied rewrites 99.0%

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

   if 1.0000000000000001e-15 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 100.0%

      \[\leadsto \cosh x \cdot \color{blue}{1} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 98.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ t_1 := \cosh x \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(y \cdot y, -0.08333333333333333, 0.5\right) \cdot y\right) \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.08333333333333333, x \cdot x, 1\right), x \cdot x, 2\right)}{y}\\ \mathbf{elif}\;t\_1 \leq 10^{-15}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)) (t_1 (* (cosh x) t_0)))
   (if (<= t_1 (- INFINITY))
     (*
      (* (fma (* y y) -0.08333333333333333 0.5) y)
      (/ (fma (fma 0.08333333333333333 (* x x) 1.0) (* x x) 2.0) y))
     (if (<= t_1 1e-15) t_0 (* (cosh x) 1.0)))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double t_1 = cosh(x) * t_0;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (fma((y * y), -0.08333333333333333, 0.5) * y) * (fma(fma(0.08333333333333333, (x * x), 1.0), (x * x), 2.0) / y);
	} else if (t_1 <= 1e-15) {
		tmp = t_0;
	} else {
		tmp = cosh(x) * 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(y) / y)
	t_1 = Float64(cosh(x) * t_0)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(fma(Float64(y * y), -0.08333333333333333, 0.5) * y) * Float64(fma(fma(0.08333333333333333, Float64(x * x), 1.0), Float64(x * x), 2.0) / y));
	elseif (t_1 <= 1e-15)
		tmp = t_0;
	else
		tmp = Float64(cosh(x) * 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cosh[x], $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(N[(y * y), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision] * y), $MachinePrecision] * N[(N[(N[(0.08333333333333333 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e-15], t$95$0, N[(N[Cosh[x], $MachinePrecision] * 1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < -inf.0

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. rec-exp N/A

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

      8. cosh-undef N/A

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

      9. lower-*.f64 N/A

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

      10. lift-cosh.f64 100.0

          \[\leadsto \left(0.5 \cdot \sin y\right) \cdot \frac{2 \cdot \cosh x}{y} \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin y\right) \cdot \frac{2 \cdot \cosh x}{y}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin y\right) \cdot \frac{2}{y} \]
   7. Step-by-step derivation

   8. Applied rewrites 2.3%

      \[\leadsto \left(0.5 \cdot \sin y\right) \cdot \frac{2}{y} \]

   9. Taylor expanded in y around 0

      \[\leadsto \left(y \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {y}^{2}\right)\right) \cdot \frac{\color{blue}{2}}{y} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {y}^{2}\right) \cdot y\right) \cdot \frac{2}{y} \]

       2. lower-*.f64 N/A

          \[\leadsto \left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {y}^{2}\right) \cdot y\right) \cdot \frac{2}{y} \]

       3. +-commutative N/A

          \[\leadsto \left(\left(\frac{-1}{12} \cdot {y}^{2} + \frac{1}{2}\right) \cdot y\right) \cdot \frac{2}{y} \]

       4. *-commutative N/A

          \[\leadsto \left(\left({y}^{2} \cdot \frac{-1}{12} + \frac{1}{2}\right) \cdot y\right) \cdot \frac{2}{y} \]

       5. lower-fma.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left({y}^{2}, \frac{-1}{12}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{2}{y} \]

       6. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(y \cdot y, \frac{-1}{12}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{2}{y} \]

       7. lift-*.f64 53.6

          \[\leadsto \left(\mathsf{fma}\left(y \cdot y, -0.08333333333333333, 0.5\right) \cdot y\right) \cdot \frac{2}{y} \]

   11. Applied rewrites 53.6%

       \[\leadsto \left(\mathsf{fma}\left(y \cdot y, -0.08333333333333333, 0.5\right) \cdot y\right) \cdot \frac{\color{blue}{2}}{y} \]

   12. Taylor expanded in x around 0

       \[\leadsto \left(\mathsf{fma}\left(y \cdot y, \frac{-1}{12}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{2 + {x}^{2} \cdot \left(1 + \frac{1}{12} \cdot {x}^{2}\right)}{y} \]
   13. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(y \cdot y, \frac{-1}{12}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{{x}^{2} \cdot \left(1 + \frac{1}{12} \cdot {x}^{2}\right) + 2}{y} \]

       2. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(y \cdot y, \frac{-1}{12}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{\left(1 + \frac{1}{12} \cdot {x}^{2}\right) \cdot {x}^{2} + 2}{y} \]

       3. lower-fma.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(y \cdot y, \frac{-1}{12}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{\mathsf{fma}\left(1 + \frac{1}{12} \cdot {x}^{2}, {x}^{2}, 2\right)}{y} \]

       4. +-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(y \cdot y, \frac{-1}{12}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{\mathsf{fma}\left(\frac{1}{12} \cdot {x}^{2} + 1, {x}^{2}, 2\right)}{y} \]

       5. lower-fma.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(y \cdot y, \frac{-1}{12}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{12}, {x}^{2}, 1\right), {x}^{2}, 2\right)}{y} \]

       6. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(y \cdot y, \frac{-1}{12}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{12}, x \cdot x, 1\right), {x}^{2}, 2\right)}{y} \]

       7. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(y \cdot y, \frac{-1}{12}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{12}, x \cdot x, 1\right), {x}^{2}, 2\right)}{y} \]

       8. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(y \cdot y, \frac{-1}{12}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{12}, x \cdot x, 1\right), x \cdot x, 2\right)}{y} \]

       9. lift-*.f64 100.0

          \[\leadsto \left(\mathsf{fma}\left(y \cdot y, -0.08333333333333333, 0.5\right) \cdot y\right) \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.08333333333333333, x \cdot x, 1\right), x \cdot x, 2\right)}{y} \]

   14. Applied rewrites 100.0%

       \[\leadsto \left(\mathsf{fma}\left(y \cdot y, -0.08333333333333333, 0.5\right) \cdot y\right) \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.08333333333333333, x \cdot x, 1\right), x \cdot x, 2\right)}{y} \]

   if -inf.0 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < 1.0000000000000001e-15

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0

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

      1. lift-sin.f64 N/A

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

      2. lift-/.f64 98.6

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

   5. Applied rewrites 98.6%

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

   if 1.0000000000000001e-15 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 100.0%

      \[\leadsto \cosh x \cdot \color{blue}{1} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 52.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cosh x \cdot \frac{\sin y}{y}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-142}:\\ \;\;\;\;1 \cdot \left(\left(y \cdot y\right) \cdot -0.16666666666666666\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;1 \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot 0.5\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (cosh x) (/ (sin y) y))))
   (if (<= t_0 -5e-142)
     (* 1.0 (* (* y y) -0.16666666666666666))
     (if (<= t_0 2.0) (* 1.0 1.0) (* (* (* x x) 0.5) 1.0)))))

C

double code(double x, double y) {
	double t_0 = cosh(x) * (sin(y) / y);
	double tmp;
	if (t_0 <= -5e-142) {
		tmp = 1.0 * ((y * y) * -0.16666666666666666);
	} else if (t_0 <= 2.0) {
		tmp = 1.0 * 1.0;
	} else {
		tmp = ((x * x) * 0.5) * 1.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cosh(x) * (sin(y) / y)
    if (t_0 <= (-5d-142)) then
        tmp = 1.0d0 * ((y * y) * (-0.16666666666666666d0))
    else if (t_0 <= 2.0d0) then
        tmp = 1.0d0 * 1.0d0
    else
        tmp = ((x * x) * 0.5d0) * 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.cosh(x) * (Math.sin(y) / y);
	double tmp;
	if (t_0 <= -5e-142) {
		tmp = 1.0 * ((y * y) * -0.16666666666666666);
	} else if (t_0 <= 2.0) {
		tmp = 1.0 * 1.0;
	} else {
		tmp = ((x * x) * 0.5) * 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.cosh(x) * (math.sin(y) / y)
	tmp = 0
	if t_0 <= -5e-142:
		tmp = 1.0 * ((y * y) * -0.16666666666666666)
	elif t_0 <= 2.0:
		tmp = 1.0 * 1.0
	else:
		tmp = ((x * x) * 0.5) * 1.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(cosh(x) * Float64(sin(y) / y))
	tmp = 0.0
	if (t_0 <= -5e-142)
		tmp = Float64(1.0 * Float64(Float64(y * y) * -0.16666666666666666));
	elseif (t_0 <= 2.0)
		tmp = Float64(1.0 * 1.0);
	else
		tmp = Float64(Float64(Float64(x * x) * 0.5) * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = cosh(x) * (sin(y) / y);
	tmp = 0.0;
	if (t_0 <= -5e-142)
		tmp = 1.0 * ((y * y) * -0.16666666666666666);
	elseif (t_0 <= 2.0)
		tmp = 1.0 * 1.0;
	else
		tmp = ((x * x) * 0.5) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cosh[x], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-142], N[(1.0 * N[(N[(y * y), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(1.0 * 1.0), $MachinePrecision], N[(N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision] * 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < -5.0000000000000002e-142

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 68.8

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

   5. Applied rewrites 68.8%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 63.5

         \[\leadsto \mathsf{fma}\left(x \cdot x, 0.5, 1\right) \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right) \]

   8. Applied rewrites 63.5%

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

   9. Taylor expanded in x around 0

      \[\leadsto 1 \cdot \mathsf{fma}\left(\frac{-1}{6}, y \cdot y, 1\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 28.7%

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

   12. Taylor expanded in y around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 28.7

          \[\leadsto 1 \cdot \left(\left(y \cdot y\right) \cdot -0.16666666666666666\right) \]

   14. Applied rewrites 28.7%

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

   if -5.0000000000000002e-142 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < 2

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 57.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 57.4%

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

   if 2 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 65.0

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

   5. Applied rewrites 65.0%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 34.5

         \[\leadsto \mathsf{fma}\left(x \cdot x, 0.5, 1\right) \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right) \]

   8. Applied rewrites 34.5%

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

   9. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 34.5

          \[\leadsto \left(\left(x \cdot x\right) \cdot 0.5\right) \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right) \]

   11. Applied rewrites 34.5%

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

   12. Taylor expanded in y around 0

       \[\leadsto \left(\left(x \cdot x\right) \cdot \frac{1}{2}\right) \cdot 1 \]
   13. Step-by-step derivation

   14. Applied rewrites 51.2%

       \[\leadsto \left(\left(x \cdot x\right) \cdot 0.5\right) \cdot 1 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 71.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ t_1 := \mathsf{fma}\left(y \cdot y, -0.08333333333333333, 0.5\right) \cdot y\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-307}:\\ \;\;\;\;t\_1 \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.08333333333333333, x \cdot x, 1\right), x \cdot x, 2\right)}{y}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-147}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.004166666666666667 \cdot \left(y \cdot y\right) - 0.08333333333333333, y \cdot y, 0.5\right) \cdot y\right) \cdot \frac{2}{y}\\ \mathbf{elif}\;t\_0 \leq 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, x \cdot x, 0.08333333333333333\right), x \cdot x, 1\right), x \cdot x, 2\right)}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)) (t_1 (* (fma (* y y) -0.08333333333333333 0.5) y)))
   (if (<= t_0 -2e-307)
     (* t_1 (/ (fma (fma 0.08333333333333333 (* x x) 1.0) (* x x) 2.0) y))
     (if (<= t_0 5e-147)
       (*
        (*
         (fma
          (- (* 0.004166666666666667 (* y y)) 0.08333333333333333)
          (* y y)
          0.5)
         y)
        (/ 2.0 y))
       (if (<= t_0 1e-8)
         (fma
          (fma
           (fma (* x x) 0.001388888888888889 0.041666666666666664)
           (* x x)
           0.5)
          (* x x)
          1.0)
         (*
          t_1
          (/
           (fma
            (fma
             (fma 0.002777777777777778 (* x x) 0.08333333333333333)
             (* x x)
             1.0)
            (* x x)
            2.0)
           y)))))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double t_1 = fma((y * y), -0.08333333333333333, 0.5) * y;
	double tmp;
	if (t_0 <= -2e-307) {
		tmp = t_1 * (fma(fma(0.08333333333333333, (x * x), 1.0), (x * x), 2.0) / y);
	} else if (t_0 <= 5e-147) {
		tmp = (fma(((0.004166666666666667 * (y * y)) - 0.08333333333333333), (y * y), 0.5) * y) * (2.0 / y);
	} else if (t_0 <= 1e-8) {
		tmp = fma(fma(fma((x * x), 0.001388888888888889, 0.041666666666666664), (x * x), 0.5), (x * x), 1.0);
	} else {
		tmp = t_1 * (fma(fma(fma(0.002777777777777778, (x * x), 0.08333333333333333), (x * x), 1.0), (x * x), 2.0) / y);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(y) / y)
	t_1 = Float64(fma(Float64(y * y), -0.08333333333333333, 0.5) * y)
	tmp = 0.0
	if (t_0 <= -2e-307)
		tmp = Float64(t_1 * Float64(fma(fma(0.08333333333333333, Float64(x * x), 1.0), Float64(x * x), 2.0) / y));
	elseif (t_0 <= 5e-147)
		tmp = Float64(Float64(fma(Float64(Float64(0.004166666666666667 * Float64(y * y)) - 0.08333333333333333), Float64(y * y), 0.5) * y) * Float64(2.0 / y));
	elseif (t_0 <= 1e-8)
		tmp = fma(fma(fma(Float64(x * x), 0.001388888888888889, 0.041666666666666664), Float64(x * x), 0.5), Float64(x * x), 1.0);
	else
		tmp = Float64(t_1 * Float64(fma(fma(fma(0.002777777777777778, Float64(x * x), 0.08333333333333333), Float64(x * x), 1.0), Float64(x * x), 2.0) / y));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(y * y), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-307], N[(t$95$1 * N[(N[(N[(0.08333333333333333 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e-147], N[(N[(N[(N[(N[(0.004166666666666667 * N[(y * y), $MachinePrecision]), $MachinePrecision] - 0.08333333333333333), $MachinePrecision] * N[(y * y), $MachinePrecision] + 0.5), $MachinePrecision] * y), $MachinePrecision] * N[(2.0 / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e-8], N[(N[(N[(N[(x * x), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision], N[(t$95$1 * N[(N[(N[(N[(0.002777777777777778 * N[(x * x), $MachinePrecision] + 0.08333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (sin.f64 y) y) < -1.99999999999999982e-307

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. rec-exp N/A

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

      8. cosh-undef N/A

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

      9. lower-*.f64 N/A

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

      10. lift-cosh.f64 99.7

          \[\leadsto \left(0.5 \cdot \sin y\right) \cdot \frac{2 \cdot \cosh x}{y} \]

   5. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin y\right) \cdot \frac{2 \cdot \cosh x}{y}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin y\right) \cdot \frac{2}{y} \]
   7. Step-by-step derivation

   8. Applied rewrites 47.0%

      \[\leadsto \left(0.5 \cdot \sin y\right) \cdot \frac{2}{y} \]

   9. Taylor expanded in y around 0

      \[\leadsto \left(y \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {y}^{2}\right)\right) \cdot \frac{\color{blue}{2}}{y} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {y}^{2}\right) \cdot y\right) \cdot \frac{2}{y} \]

       2. lower-*.f64 N/A

          \[\leadsto \left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {y}^{2}\right) \cdot y\right) \cdot \frac{2}{y} \]

       3. +-commutative N/A

          \[\leadsto \left(\left(\frac{-1}{12} \cdot {y}^{2} + \frac{1}{2}\right) \cdot y\right) \cdot \frac{2}{y} \]

       4. *-commutative N/A

          \[\leadsto \left(\left({y}^{2} \cdot \frac{-1}{12} + \frac{1}{2}\right) \cdot y\right) \cdot \frac{2}{y} \]

       5. lower-fma.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left({y}^{2}, \frac{-1}{12}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{2}{y} \]

       6. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(y \cdot y, \frac{-1}{12}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{2}{y} \]

       7. lift-*.f64 30.0

          \[\leadsto \left(\mathsf{fma}\left(y \cdot y, -0.08333333333333333, 0.5\right) \cdot y\right) \cdot \frac{2}{y} \]

   11. Applied rewrites 30.0%

       \[\leadsto \left(\mathsf{fma}\left(y \cdot y, -0.08333333333333333, 0.5\right) \cdot y\right) \cdot \frac{\color{blue}{2}}{y} \]

   12. Taylor expanded in x around 0

       \[\leadsto \left(\mathsf{fma}\left(y \cdot y, \frac{-1}{12}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{2 + {x}^{2} \cdot \left(1 + \frac{1}{12} \cdot {x}^{2}\right)}{y} \]
   13. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(y \cdot y, \frac{-1}{12}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{{x}^{2} \cdot \left(1 + \frac{1}{12} \cdot {x}^{2}\right) + 2}{y} \]

       2. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(y \cdot y, \frac{-1}{12}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{\left(1 + \frac{1}{12} \cdot {x}^{2}\right) \cdot {x}^{2} + 2}{y} \]

       3. lower-fma.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(y \cdot y, \frac{-1}{12}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{\mathsf{fma}\left(1 + \frac{1}{12} \cdot {x}^{2}, {x}^{2}, 2\right)}{y} \]

       4. +-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(y \cdot y, \frac{-1}{12}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{\mathsf{fma}\left(\frac{1}{12} \cdot {x}^{2} + 1, {x}^{2}, 2\right)}{y} \]

       5. lower-fma.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(y \cdot y, \frac{-1}{12}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{12}, {x}^{2}, 1\right), {x}^{2}, 2\right)}{y} \]

       6. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(y \cdot y, \frac{-1}{12}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{12}, x \cdot x, 1\right), {x}^{2}, 2\right)}{y} \]

       7. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(y \cdot y, \frac{-1}{12}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{12}, x \cdot x, 1\right), {x}^{2}, 2\right)}{y} \]

       8. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(y \cdot y, \frac{-1}{12}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{12}, x \cdot x, 1\right), x \cdot x, 2\right)}{y} \]

       9. lift-*.f64 54.6

          \[\leadsto \left(\mathsf{fma}\left(y \cdot y, -0.08333333333333333, 0.5\right) \cdot y\right) \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.08333333333333333, x \cdot x, 1\right), x \cdot x, 2\right)}{y} \]

   14. Applied rewrites 54.6%

       \[\leadsto \left(\mathsf{fma}\left(y \cdot y, -0.08333333333333333, 0.5\right) \cdot y\right) \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.08333333333333333, x \cdot x, 1\right), x \cdot x, 2\right)}{y} \]

   if -1.99999999999999982e-307 < (/.f64 (sin.f64 y) y) < 5.00000000000000013e-147

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. rec-exp N/A

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

      8. cosh-undef N/A

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

      9. lower-*.f64 N/A

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

      10. lift-cosh.f64 99.8

          \[\leadsto \left(0.5 \cdot \sin y\right) \cdot \frac{2 \cdot \cosh x}{y} \]

   5. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin y\right) \cdot \frac{2 \cdot \cosh x}{y}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin y\right) \cdot \frac{2}{y} \]
   7. Step-by-step derivation

   8. Applied rewrites 48.1%

      \[\leadsto \left(0.5 \cdot \sin y\right) \cdot \frac{2}{y} \]

   9. Taylor expanded in y around 0

      \[\leadsto \left(y \cdot \left(\frac{1}{2} + {y}^{2} \cdot \left(\frac{1}{240} \cdot {y}^{2} - \frac{1}{12}\right)\right)\right) \cdot \frac{\color{blue}{2}}{y} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(\left(\frac{1}{2} + {y}^{2} \cdot \left(\frac{1}{240} \cdot {y}^{2} - \frac{1}{12}\right)\right) \cdot y\right) \cdot \frac{2}{y} \]

       2. lower-*.f64 N/A

          \[\leadsto \left(\left(\frac{1}{2} + {y}^{2} \cdot \left(\frac{1}{240} \cdot {y}^{2} - \frac{1}{12}\right)\right) \cdot y\right) \cdot \frac{2}{y} \]

       3. +-commutative N/A

          \[\leadsto \left(\left({y}^{2} \cdot \left(\frac{1}{240} \cdot {y}^{2} - \frac{1}{12}\right) + \frac{1}{2}\right) \cdot y\right) \cdot \frac{2}{y} \]

       4. *-commutative N/A

          \[\leadsto \left(\left(\left(\frac{1}{240} \cdot {y}^{2} - \frac{1}{12}\right) \cdot {y}^{2} + \frac{1}{2}\right) \cdot y\right) \cdot \frac{2}{y} \]

       5. lower-fma.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{240} \cdot {y}^{2} - \frac{1}{12}, {y}^{2}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{2}{y} \]

       6. lower--.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{240} \cdot {y}^{2} - \frac{1}{12}, {y}^{2}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{2}{y} \]

       7. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{240} \cdot {y}^{2} - \frac{1}{12}, {y}^{2}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{2}{y} \]

       8. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{240} \cdot \left(y \cdot y\right) - \frac{1}{12}, {y}^{2}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{2}{y} \]

       9. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{240} \cdot \left(y \cdot y\right) - \frac{1}{12}, {y}^{2}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{2}{y} \]

       10. pow2 N/A

           \[\leadsto \left(\mathsf{fma}\left(\frac{1}{240} \cdot \left(y \cdot y\right) - \frac{1}{12}, y \cdot y, \frac{1}{2}\right) \cdot y\right) \cdot \frac{2}{y} \]

       11. lift-*.f64 53.7

           \[\leadsto \left(\mathsf{fma}\left(0.004166666666666667 \cdot \left(y \cdot y\right) - 0.08333333333333333, y \cdot y, 0.5\right) \cdot y\right) \cdot \frac{2}{y} \]

   11. Applied rewrites 53.7%

       \[\leadsto \left(\mathsf{fma}\left(0.004166666666666667 \cdot \left(y \cdot y\right) - 0.08333333333333333, y \cdot y, 0.5\right) \cdot y\right) \cdot \frac{\color{blue}{2}}{y} \]

   if 5.00000000000000013e-147 < (/.f64 (sin.f64 y) y) < 1e-8

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \frac{\sin y}{y} + {x}^{2} \cdot \left(\frac{1}{720} \cdot \frac{{x}^{2} \cdot \sin y}{y} + \frac{1}{24} \cdot \frac{\sin y}{y}\right)\right) \cdot {x}^{2} + \frac{\color{blue}{\sin y}}{y} \]

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 97.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.041666666666666664, \sin y, \left(0.001388888888888889 \cdot \sin y\right) \cdot \left(x \cdot x\right)\right)}{y}, x \cdot x, 0.5 \cdot \frac{\sin y}{y}\right), x \cdot x, \frac{\sin y}{y}\right)} \]

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

         \[\leadsto {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) + 1 \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) \cdot {x}^{2} + 1 \]

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left({x}^{2}, \frac{1}{720}, \frac{1}{24}\right), {x}^{2}, \frac{1}{2}\right), {x}^{2}, 1\right) \]

      10. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right), {x}^{2}, \frac{1}{2}\right), {x}^{2}, 1\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right), {x}^{2}, \frac{1}{2}\right), {x}^{2}, 1\right) \]

      12. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right), x \cdot x, \frac{1}{2}\right), {x}^{2}, 1\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right), x \cdot x, \frac{1}{2}\right), {x}^{2}, 1\right) \]

      14. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right), x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \]

      15. lift-*.f64 57.8

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right) \]

   8. Applied rewrites 57.8%

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

   if 1e-8 < (/.f64 (sin.f64 y) y)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. rec-exp N/A

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

      8. cosh-undef N/A

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

      9. lower-*.f64 N/A

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

      10. lift-cosh.f64 99.9

          \[\leadsto \left(0.5 \cdot \sin y\right) \cdot \frac{2 \cdot \cosh x}{y} \]

   5. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin y\right) \cdot \frac{2 \cdot \cosh x}{y}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin y\right) \cdot \frac{2}{y} \]
   7. Step-by-step derivation

   8. Applied rewrites 47.1%

      \[\leadsto \left(0.5 \cdot \sin y\right) \cdot \frac{2}{y} \]

   9. Taylor expanded in y around 0

      \[\leadsto \left(y \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {y}^{2}\right)\right) \cdot \frac{\color{blue}{2}}{y} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {y}^{2}\right) \cdot y\right) \cdot \frac{2}{y} \]

       2. lower-*.f64 N/A

          \[\leadsto \left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {y}^{2}\right) \cdot y\right) \cdot \frac{2}{y} \]

       3. +-commutative N/A

          \[\leadsto \left(\left(\frac{-1}{12} \cdot {y}^{2} + \frac{1}{2}\right) \cdot y\right) \cdot \frac{2}{y} \]

       4. *-commutative N/A

          \[\leadsto \left(\left({y}^{2} \cdot \frac{-1}{12} + \frac{1}{2}\right) \cdot y\right) \cdot \frac{2}{y} \]

       5. lower-fma.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left({y}^{2}, \frac{-1}{12}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{2}{y} \]

       6. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(y \cdot y, \frac{-1}{12}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{2}{y} \]

       7. lift-*.f64 47.1

          \[\leadsto \left(\mathsf{fma}\left(y \cdot y, -0.08333333333333333, 0.5\right) \cdot y\right) \cdot \frac{2}{y} \]

   11. Applied rewrites 47.1%

       \[\leadsto \left(\mathsf{fma}\left(y \cdot y, -0.08333333333333333, 0.5\right) \cdot y\right) \cdot \frac{\color{blue}{2}}{y} \]

   12. Taylor expanded in x around 0

       \[\leadsto \left(\mathsf{fma}\left(y \cdot y, \frac{-1}{12}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{2 + {x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right)}{y} \]
   13. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(y \cdot y, \frac{-1}{12}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{{x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right) + 2}{y} \]

       2. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(y \cdot y, \frac{-1}{12}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{\left(1 + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right) \cdot {x}^{2} + 2}{y} \]

       3. lower-fma.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(y \cdot y, \frac{-1}{12}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{\mathsf{fma}\left(1 + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right), {x}^{2}, 2\right)}{y} \]

       4. +-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(y \cdot y, \frac{-1}{12}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{\mathsf{fma}\left({x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right) + 1, {x}^{2}, 2\right)}{y} \]

       5. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(y \cdot y, \frac{-1}{12}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{\mathsf{fma}\left(\left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right) \cdot {x}^{2} + 1, {x}^{2}, 2\right)}{y} \]

       6. lower-fma.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(y \cdot y, \frac{-1}{12}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}, {x}^{2}, 1\right), {x}^{2}, 2\right)}{y} \]

       7. +-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(y \cdot y, \frac{-1}{12}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360} \cdot {x}^{2} + \frac{1}{12}, {x}^{2}, 1\right), {x}^{2}, 2\right)}{y} \]

       8. lower-fma.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(y \cdot y, \frac{-1}{12}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, {x}^{2}, \frac{1}{12}\right), {x}^{2}, 1\right), {x}^{2}, 2\right)}{y} \]

       9. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(y \cdot y, \frac{-1}{12}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, x \cdot x, \frac{1}{12}\right), {x}^{2}, 1\right), {x}^{2}, 2\right)}{y} \]

       10. lift-*.f64 N/A

           \[\leadsto \left(\mathsf{fma}\left(y \cdot y, \frac{-1}{12}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, x \cdot x, \frac{1}{12}\right), {x}^{2}, 1\right), {x}^{2}, 2\right)}{y} \]

       11. pow2 N/A

           \[\leadsto \left(\mathsf{fma}\left(y \cdot y, \frac{-1}{12}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, x \cdot x, \frac{1}{12}\right), x \cdot x, 1\right), {x}^{2}, 2\right)}{y} \]

       12. lift-*.f64 N/A

           \[\leadsto \left(\mathsf{fma}\left(y \cdot y, \frac{-1}{12}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, x \cdot x, \frac{1}{12}\right), x \cdot x, 1\right), {x}^{2}, 2\right)}{y} \]

       13. pow2 N/A

           \[\leadsto \left(\mathsf{fma}\left(y \cdot y, \frac{-1}{12}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, x \cdot x, \frac{1}{12}\right), x \cdot x, 1\right), x \cdot x, 2\right)}{y} \]

       14. lift-*.f64 95.2

           \[\leadsto \left(\mathsf{fma}\left(y \cdot y, -0.08333333333333333, 0.5\right) \cdot y\right) \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, x \cdot x, 0.08333333333333333\right), x \cdot x, 1\right), x \cdot x, 2\right)}{y} \]

   14. Applied rewrites 95.2%

       \[\leadsto \left(\mathsf{fma}\left(y \cdot y, -0.08333333333333333, 0.5\right) \cdot y\right) \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, x \cdot x, 0.08333333333333333\right), x \cdot x, 1\right), x \cdot x, 2\right)}{y} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 7: 70.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ t_1 := \left(\mathsf{fma}\left(y \cdot y, -0.08333333333333333, 0.5\right) \cdot y\right) \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.08333333333333333, x \cdot x, 1\right), x \cdot x, 2\right)}{y}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-307}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-147}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.004166666666666667 \cdot \left(y \cdot y\right) - 0.08333333333333333, y \cdot y, 0.5\right) \cdot y\right) \cdot \frac{2}{y}\\ \mathbf{elif}\;t\_0 \leq 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y))
        (t_1
         (*
          (* (fma (* y y) -0.08333333333333333 0.5) y)
          (/ (fma (fma 0.08333333333333333 (* x x) 1.0) (* x x) 2.0) y))))
   (if (<= t_0 -2e-307)
     t_1
     (if (<= t_0 5e-147)
       (*
        (*
         (fma
          (- (* 0.004166666666666667 (* y y)) 0.08333333333333333)
          (* y y)
          0.5)
         y)
        (/ 2.0 y))
       (if (<= t_0 1e-8)
         (fma
          (fma
           (fma (* x x) 0.001388888888888889 0.041666666666666664)
           (* x x)
           0.5)
          (* x x)
          1.0)
         t_1)))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double t_1 = (fma((y * y), -0.08333333333333333, 0.5) * y) * (fma(fma(0.08333333333333333, (x * x), 1.0), (x * x), 2.0) / y);
	double tmp;
	if (t_0 <= -2e-307) {
		tmp = t_1;
	} else if (t_0 <= 5e-147) {
		tmp = (fma(((0.004166666666666667 * (y * y)) - 0.08333333333333333), (y * y), 0.5) * y) * (2.0 / y);
	} else if (t_0 <= 1e-8) {
		tmp = fma(fma(fma((x * x), 0.001388888888888889, 0.041666666666666664), (x * x), 0.5), (x * x), 1.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(y) / y)
	t_1 = Float64(Float64(fma(Float64(y * y), -0.08333333333333333, 0.5) * y) * Float64(fma(fma(0.08333333333333333, Float64(x * x), 1.0), Float64(x * x), 2.0) / y))
	tmp = 0.0
	if (t_0 <= -2e-307)
		tmp = t_1;
	elseif (t_0 <= 5e-147)
		tmp = Float64(Float64(fma(Float64(Float64(0.004166666666666667 * Float64(y * y)) - 0.08333333333333333), Float64(y * y), 0.5) * y) * Float64(2.0 / y));
	elseif (t_0 <= 1e-8)
		tmp = fma(fma(fma(Float64(x * x), 0.001388888888888889, 0.041666666666666664), Float64(x * x), 0.5), Float64(x * x), 1.0);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(y * y), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision] * y), $MachinePrecision] * N[(N[(N[(0.08333333333333333 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-307], t$95$1, If[LessEqual[t$95$0, 5e-147], N[(N[(N[(N[(N[(0.004166666666666667 * N[(y * y), $MachinePrecision]), $MachinePrecision] - 0.08333333333333333), $MachinePrecision] * N[(y * y), $MachinePrecision] + 0.5), $MachinePrecision] * y), $MachinePrecision] * N[(2.0 / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e-8], N[(N[(N[(N[(x * x), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 y) y) < -1.99999999999999982e-307 or 1e-8 < (/.f64 (sin.f64 y) y)

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. rec-exp N/A

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

      8. cosh-undef N/A

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

      9. lower-*.f64 N/A

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

      10. lift-cosh.f64 99.8

          \[\leadsto \left(0.5 \cdot \sin y\right) \cdot \frac{2 \cdot \cosh x}{y} \]

   5. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin y\right) \cdot \frac{2 \cdot \cosh x}{y}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin y\right) \cdot \frac{2}{y} \]
   7. Step-by-step derivation

   8. Applied rewrites 47.1%

      \[\leadsto \left(0.5 \cdot \sin y\right) \cdot \frac{2}{y} \]

   9. Taylor expanded in y around 0

      \[\leadsto \left(y \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {y}^{2}\right)\right) \cdot \frac{\color{blue}{2}}{y} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {y}^{2}\right) \cdot y\right) \cdot \frac{2}{y} \]

       2. lower-*.f64 N/A

          \[\leadsto \left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {y}^{2}\right) \cdot y\right) \cdot \frac{2}{y} \]

       3. +-commutative N/A

          \[\leadsto \left(\left(\frac{-1}{12} \cdot {y}^{2} + \frac{1}{2}\right) \cdot y\right) \cdot \frac{2}{y} \]

       4. *-commutative N/A

          \[\leadsto \left(\left({y}^{2} \cdot \frac{-1}{12} + \frac{1}{2}\right) \cdot y\right) \cdot \frac{2}{y} \]

       5. lower-fma.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left({y}^{2}, \frac{-1}{12}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{2}{y} \]

       6. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(y \cdot y, \frac{-1}{12}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{2}{y} \]

       7. lift-*.f64 41.1

          \[\leadsto \left(\mathsf{fma}\left(y \cdot y, -0.08333333333333333, 0.5\right) \cdot y\right) \cdot \frac{2}{y} \]

   11. Applied rewrites 41.1%

       \[\leadsto \left(\mathsf{fma}\left(y \cdot y, -0.08333333333333333, 0.5\right) \cdot y\right) \cdot \frac{\color{blue}{2}}{y} \]

   12. Taylor expanded in x around 0

       \[\leadsto \left(\mathsf{fma}\left(y \cdot y, \frac{-1}{12}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{2 + {x}^{2} \cdot \left(1 + \frac{1}{12} \cdot {x}^{2}\right)}{y} \]
   13. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(y \cdot y, \frac{-1}{12}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{{x}^{2} \cdot \left(1 + \frac{1}{12} \cdot {x}^{2}\right) + 2}{y} \]

       2. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(y \cdot y, \frac{-1}{12}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{\left(1 + \frac{1}{12} \cdot {x}^{2}\right) \cdot {x}^{2} + 2}{y} \]

       3. lower-fma.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(y \cdot y, \frac{-1}{12}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{\mathsf{fma}\left(1 + \frac{1}{12} \cdot {x}^{2}, {x}^{2}, 2\right)}{y} \]

       4. +-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(y \cdot y, \frac{-1}{12}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{\mathsf{fma}\left(\frac{1}{12} \cdot {x}^{2} + 1, {x}^{2}, 2\right)}{y} \]

       5. lower-fma.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(y \cdot y, \frac{-1}{12}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{12}, {x}^{2}, 1\right), {x}^{2}, 2\right)}{y} \]

       6. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(y \cdot y, \frac{-1}{12}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{12}, x \cdot x, 1\right), {x}^{2}, 2\right)}{y} \]

       7. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(y \cdot y, \frac{-1}{12}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{12}, x \cdot x, 1\right), {x}^{2}, 2\right)}{y} \]

       8. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(y \cdot y, \frac{-1}{12}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{12}, x \cdot x, 1\right), x \cdot x, 2\right)}{y} \]

       9. lift-*.f64 80.0

          \[\leadsto \left(\mathsf{fma}\left(y \cdot y, -0.08333333333333333, 0.5\right) \cdot y\right) \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.08333333333333333, x \cdot x, 1\right), x \cdot x, 2\right)}{y} \]

   14. Applied rewrites 80.0%

       \[\leadsto \left(\mathsf{fma}\left(y \cdot y, -0.08333333333333333, 0.5\right) \cdot y\right) \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.08333333333333333, x \cdot x, 1\right), x \cdot x, 2\right)}{y} \]

   if -1.99999999999999982e-307 < (/.f64 (sin.f64 y) y) < 5.00000000000000013e-147

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. rec-exp N/A

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

      8. cosh-undef N/A

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

      9. lower-*.f64 N/A

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

      10. lift-cosh.f64 99.8

          \[\leadsto \left(0.5 \cdot \sin y\right) \cdot \frac{2 \cdot \cosh x}{y} \]

   5. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin y\right) \cdot \frac{2 \cdot \cosh x}{y}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin y\right) \cdot \frac{2}{y} \]
   7. Step-by-step derivation

   8. Applied rewrites 48.1%

      \[\leadsto \left(0.5 \cdot \sin y\right) \cdot \frac{2}{y} \]

   9. Taylor expanded in y around 0

      \[\leadsto \left(y \cdot \left(\frac{1}{2} + {y}^{2} \cdot \left(\frac{1}{240} \cdot {y}^{2} - \frac{1}{12}\right)\right)\right) \cdot \frac{\color{blue}{2}}{y} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(\left(\frac{1}{2} + {y}^{2} \cdot \left(\frac{1}{240} \cdot {y}^{2} - \frac{1}{12}\right)\right) \cdot y\right) \cdot \frac{2}{y} \]

       2. lower-*.f64 N/A

          \[\leadsto \left(\left(\frac{1}{2} + {y}^{2} \cdot \left(\frac{1}{240} \cdot {y}^{2} - \frac{1}{12}\right)\right) \cdot y\right) \cdot \frac{2}{y} \]

       3. +-commutative N/A

          \[\leadsto \left(\left({y}^{2} \cdot \left(\frac{1}{240} \cdot {y}^{2} - \frac{1}{12}\right) + \frac{1}{2}\right) \cdot y\right) \cdot \frac{2}{y} \]

       4. *-commutative N/A

          \[\leadsto \left(\left(\left(\frac{1}{240} \cdot {y}^{2} - \frac{1}{12}\right) \cdot {y}^{2} + \frac{1}{2}\right) \cdot y\right) \cdot \frac{2}{y} \]

       5. lower-fma.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{240} \cdot {y}^{2} - \frac{1}{12}, {y}^{2}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{2}{y} \]

       6. lower--.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{240} \cdot {y}^{2} - \frac{1}{12}, {y}^{2}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{2}{y} \]

       7. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{240} \cdot {y}^{2} - \frac{1}{12}, {y}^{2}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{2}{y} \]

       8. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{240} \cdot \left(y \cdot y\right) - \frac{1}{12}, {y}^{2}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{2}{y} \]

       9. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{240} \cdot \left(y \cdot y\right) - \frac{1}{12}, {y}^{2}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{2}{y} \]

       10. pow2 N/A

           \[\leadsto \left(\mathsf{fma}\left(\frac{1}{240} \cdot \left(y \cdot y\right) - \frac{1}{12}, y \cdot y, \frac{1}{2}\right) \cdot y\right) \cdot \frac{2}{y} \]

       11. lift-*.f64 53.7

           \[\leadsto \left(\mathsf{fma}\left(0.004166666666666667 \cdot \left(y \cdot y\right) - 0.08333333333333333, y \cdot y, 0.5\right) \cdot y\right) \cdot \frac{2}{y} \]

   11. Applied rewrites 53.7%

       \[\leadsto \left(\mathsf{fma}\left(0.004166666666666667 \cdot \left(y \cdot y\right) - 0.08333333333333333, y \cdot y, 0.5\right) \cdot y\right) \cdot \frac{\color{blue}{2}}{y} \]

   if 5.00000000000000013e-147 < (/.f64 (sin.f64 y) y) < 1e-8

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \frac{\sin y}{y} + {x}^{2} \cdot \left(\frac{1}{720} \cdot \frac{{x}^{2} \cdot \sin y}{y} + \frac{1}{24} \cdot \frac{\sin y}{y}\right)\right) \cdot {x}^{2} + \frac{\color{blue}{\sin y}}{y} \]

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 97.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.041666666666666664, \sin y, \left(0.001388888888888889 \cdot \sin y\right) \cdot \left(x \cdot x\right)\right)}{y}, x \cdot x, 0.5 \cdot \frac{\sin y}{y}\right), x \cdot x, \frac{\sin y}{y}\right)} \]

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

         \[\leadsto {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) + 1 \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) \cdot {x}^{2} + 1 \]

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left({x}^{2}, \frac{1}{720}, \frac{1}{24}\right), {x}^{2}, \frac{1}{2}\right), {x}^{2}, 1\right) \]

      10. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right), {x}^{2}, \frac{1}{2}\right), {x}^{2}, 1\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right), {x}^{2}, \frac{1}{2}\right), {x}^{2}, 1\right) \]

      12. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right), x \cdot x, \frac{1}{2}\right), {x}^{2}, 1\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right), x \cdot x, \frac{1}{2}\right), {x}^{2}, 1\right) \]

      14. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right), x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \]

      15. lift-*.f64 57.8

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right) \]

   8. Applied rewrites 57.8%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), x \cdot x, 0.5\right), \color{blue}{x \cdot x}, 1\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 97.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ \mathbf{if}\;\cosh x \cdot t\_0 \leq 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)))
   (if (<= (* (cosh x) t_0) 1e-15)
     (*
      (fma
       (fma
        (fma 0.001388888888888889 (* x x) 0.041666666666666664)
        (* x x)
        0.5)
       (* x x)
       1.0)
      t_0)
     (* (cosh x) 1.0))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double tmp;
	if ((cosh(x) * t_0) <= 1e-15) {
		tmp = fma(fma(fma(0.001388888888888889, (x * x), 0.041666666666666664), (x * x), 0.5), (x * x), 1.0) * t_0;
	} else {
		tmp = cosh(x) * 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(y) / y)
	tmp = 0.0
	if (Float64(cosh(x) * t_0) <= 1e-15)
		tmp = Float64(fma(fma(fma(0.001388888888888889, Float64(x * x), 0.041666666666666664), Float64(x * x), 0.5), Float64(x * x), 1.0) * t_0);
	else
		tmp = Float64(cosh(x) * 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[N[(N[Cosh[x], $MachinePrecision] * t$95$0), $MachinePrecision], 1e-15], N[(N[(N[(N[(0.001388888888888889 * N[(x * x), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[Cosh[x], $MachinePrecision] * 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < 1.0000000000000001e-15

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

         \[\leadsto \left({x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) + \color{blue}{1}\right) \cdot \frac{\sin y}{y} \]

      2. *-commutative N/A

         \[\leadsto \left(\left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) \cdot {x}^{2} + 1\right) \cdot \frac{\sin y}{y} \]

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right) \cdot {x}^{2} + \frac{1}{2}, {x}^{2}, 1\right) \cdot \frac{\sin y}{y} \]

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720} \cdot {x}^{2} + \frac{1}{24}, {x}^{2}, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot \frac{\sin y}{y} \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, {x}^{2}, \frac{1}{24}\right), {x}^{2}, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot \frac{\sin y}{y} \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, x \cdot x, \frac{1}{24}\right), {x}^{2}, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot \frac{\sin y}{y} \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, x \cdot x, \frac{1}{24}\right), {x}^{2}, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot \frac{\sin y}{y} \]

      11. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, x \cdot x, \frac{1}{24}\right), x \cdot x, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot \frac{\sin y}{y} \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, x \cdot x, \frac{1}{24}\right), x \cdot x, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot \frac{\sin y}{y} \]

      13. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, x \cdot x, \frac{1}{24}\right), x \cdot x, \frac{1}{2}\right), x \cdot \color{blue}{x}, 1\right) \cdot \frac{\sin y}{y} \]

      14. lower-*.f64 96.7

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot \color{blue}{x}, 1\right) \cdot \frac{\sin y}{y} \]

   5. Applied rewrites 96.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right)} \cdot \frac{\sin y}{y} \]

   if 1.0000000000000001e-15 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 100.0%

      \[\leadsto \cosh x \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 75.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -5 \cdot 10^{-142}:\\ \;\;\;\;\left(\mathsf{fma}\left(y \cdot y, -0.08333333333333333, 0.5\right) \cdot y\right) \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.08333333333333333, x \cdot x, 1\right), x \cdot x, 2\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cosh x) (/ (sin y) y)) -5e-142)
   (*
    (* (fma (* y y) -0.08333333333333333 0.5) y)
    (/ (fma (fma 0.08333333333333333 (* x x) 1.0) (* x x) 2.0) y))
   (* (cosh x) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((cosh(x) * (sin(y) / y)) <= -5e-142) {
		tmp = (fma((y * y), -0.08333333333333333, 0.5) * y) * (fma(fma(0.08333333333333333, (x * x), 1.0), (x * x), 2.0) / y);
	} else {
		tmp = cosh(x) * 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(cosh(x) * Float64(sin(y) / y)) <= -5e-142)
		tmp = Float64(Float64(fma(Float64(y * y), -0.08333333333333333, 0.5) * y) * Float64(fma(fma(0.08333333333333333, Float64(x * x), 1.0), Float64(x * x), 2.0) / y));
	else
		tmp = Float64(cosh(x) * 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Cosh[x], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], -5e-142], N[(N[(N[(N[(y * y), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision] * y), $MachinePrecision] * N[(N[(N[(0.08333333333333333 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(N[Cosh[x], $MachinePrecision] * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < -5.0000000000000002e-142

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. rec-exp N/A

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

      8. cosh-undef N/A

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

      9. lower-*.f64 N/A

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

      10. lift-cosh.f64 99.9

          \[\leadsto \left(0.5 \cdot \sin y\right) \cdot \frac{2 \cdot \cosh x}{y} \]

   5. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin y\right) \cdot \frac{2 \cdot \cosh x}{y}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin y\right) \cdot \frac{2}{y} \]
   7. Step-by-step derivation

   8. Applied rewrites 33.0%

      \[\leadsto \left(0.5 \cdot \sin y\right) \cdot \frac{2}{y} \]

   9. Taylor expanded in y around 0

      \[\leadsto \left(y \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {y}^{2}\right)\right) \cdot \frac{\color{blue}{2}}{y} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {y}^{2}\right) \cdot y\right) \cdot \frac{2}{y} \]

       2. lower-*.f64 N/A

          \[\leadsto \left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {y}^{2}\right) \cdot y\right) \cdot \frac{2}{y} \]

       3. +-commutative N/A

          \[\leadsto \left(\left(\frac{-1}{12} \cdot {y}^{2} + \frac{1}{2}\right) \cdot y\right) \cdot \frac{2}{y} \]

       4. *-commutative N/A

          \[\leadsto \left(\left({y}^{2} \cdot \frac{-1}{12} + \frac{1}{2}\right) \cdot y\right) \cdot \frac{2}{y} \]

       5. lower-fma.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left({y}^{2}, \frac{-1}{12}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{2}{y} \]

       6. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(y \cdot y, \frac{-1}{12}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{2}{y} \]

       7. lift-*.f64 37.6

          \[\leadsto \left(\mathsf{fma}\left(y \cdot y, -0.08333333333333333, 0.5\right) \cdot y\right) \cdot \frac{2}{y} \]

   11. Applied rewrites 37.6%

       \[\leadsto \left(\mathsf{fma}\left(y \cdot y, -0.08333333333333333, 0.5\right) \cdot y\right) \cdot \frac{\color{blue}{2}}{y} \]

   12. Taylor expanded in x around 0

       \[\leadsto \left(\mathsf{fma}\left(y \cdot y, \frac{-1}{12}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{2 + {x}^{2} \cdot \left(1 + \frac{1}{12} \cdot {x}^{2}\right)}{y} \]
   13. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(y \cdot y, \frac{-1}{12}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{{x}^{2} \cdot \left(1 + \frac{1}{12} \cdot {x}^{2}\right) + 2}{y} \]

       2. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(y \cdot y, \frac{-1}{12}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{\left(1 + \frac{1}{12} \cdot {x}^{2}\right) \cdot {x}^{2} + 2}{y} \]

       3. lower-fma.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(y \cdot y, \frac{-1}{12}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{\mathsf{fma}\left(1 + \frac{1}{12} \cdot {x}^{2}, {x}^{2}, 2\right)}{y} \]

       4. +-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(y \cdot y, \frac{-1}{12}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{\mathsf{fma}\left(\frac{1}{12} \cdot {x}^{2} + 1, {x}^{2}, 2\right)}{y} \]

       5. lower-fma.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(y \cdot y, \frac{-1}{12}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{12}, {x}^{2}, 1\right), {x}^{2}, 2\right)}{y} \]

       6. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(y \cdot y, \frac{-1}{12}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{12}, x \cdot x, 1\right), {x}^{2}, 2\right)}{y} \]

       7. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(y \cdot y, \frac{-1}{12}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{12}, x \cdot x, 1\right), {x}^{2}, 2\right)}{y} \]

       8. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(y \cdot y, \frac{-1}{12}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{12}, x \cdot x, 1\right), x \cdot x, 2\right)}{y} \]

       9. lift-*.f64 68.8

          \[\leadsto \left(\mathsf{fma}\left(y \cdot y, -0.08333333333333333, 0.5\right) \cdot y\right) \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.08333333333333333, x \cdot x, 1\right), x \cdot x, 2\right)}{y} \]

   14. Applied rewrites 68.8%

       \[\leadsto \left(\mathsf{fma}\left(y \cdot y, -0.08333333333333333, 0.5\right) \cdot y\right) \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.08333333333333333, x \cdot x, 1\right), x \cdot x, 2\right)}{y} \]

   if -5.0000000000000002e-142 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y))

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 78.9%

      \[\leadsto \cosh x \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 72.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ t_1 := \mathsf{fma}\left(y \cdot y, -0.08333333333333333, 0.5\right) \cdot y\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-307}:\\ \;\;\;\;t\_1 \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.08333333333333333, x \cdot x, 1\right), x \cdot x, 2\right)}{y}\\ \mathbf{elif}\;t\_0 \leq 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot \frac{\mathsf{fma}\left(0.008333333333333333 \cdot \left(y \cdot y\right) - 0.16666666666666666, y \cdot y, 1\right) \cdot y}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, x \cdot x, 0.08333333333333333\right), x \cdot x, 1\right), x \cdot x, 2\right)}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)) (t_1 (* (fma (* y y) -0.08333333333333333 0.5) y)))
   (if (<= t_0 -2e-307)
     (* t_1 (/ (fma (fma 0.08333333333333333 (* x x) 1.0) (* x x) 2.0) y))
     (if (<= t_0 1e-8)
       (*
        (fma (fma 0.041666666666666664 (* x x) 0.5) (* x x) 1.0)
        (/
         (*
          (fma
           (- (* 0.008333333333333333 (* y y)) 0.16666666666666666)
           (* y y)
           1.0)
          y)
         y))
       (*
        t_1
        (/
         (fma
          (fma
           (fma 0.002777777777777778 (* x x) 0.08333333333333333)
           (* x x)
           1.0)
          (* x x)
          2.0)
         y))))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double t_1 = fma((y * y), -0.08333333333333333, 0.5) * y;
	double tmp;
	if (t_0 <= -2e-307) {
		tmp = t_1 * (fma(fma(0.08333333333333333, (x * x), 1.0), (x * x), 2.0) / y);
	} else if (t_0 <= 1e-8) {
		tmp = fma(fma(0.041666666666666664, (x * x), 0.5), (x * x), 1.0) * ((fma(((0.008333333333333333 * (y * y)) - 0.16666666666666666), (y * y), 1.0) * y) / y);
	} else {
		tmp = t_1 * (fma(fma(fma(0.002777777777777778, (x * x), 0.08333333333333333), (x * x), 1.0), (x * x), 2.0) / y);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(y) / y)
	t_1 = Float64(fma(Float64(y * y), -0.08333333333333333, 0.5) * y)
	tmp = 0.0
	if (t_0 <= -2e-307)
		tmp = Float64(t_1 * Float64(fma(fma(0.08333333333333333, Float64(x * x), 1.0), Float64(x * x), 2.0) / y));
	elseif (t_0 <= 1e-8)
		tmp = Float64(fma(fma(0.041666666666666664, Float64(x * x), 0.5), Float64(x * x), 1.0) * Float64(Float64(fma(Float64(Float64(0.008333333333333333 * Float64(y * y)) - 0.16666666666666666), Float64(y * y), 1.0) * y) / y));
	else
		tmp = Float64(t_1 * Float64(fma(fma(fma(0.002777777777777778, Float64(x * x), 0.08333333333333333), Float64(x * x), 1.0), Float64(x * x), 2.0) / y));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(y * y), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-307], N[(t$95$1 * N[(N[(N[(0.08333333333333333 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e-8], N[(N[(N[(0.041666666666666664 * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(N[(N[(N[(0.008333333333333333 * N[(y * y), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(N[(N[(N[(0.002777777777777778 * N[(x * x), $MachinePrecision] + 0.08333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 y) y) < -1.99999999999999982e-307

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. rec-exp N/A

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

      8. cosh-undef N/A

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

      9. lower-*.f64 N/A

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

      10. lift-cosh.f64 99.7

          \[\leadsto \left(0.5 \cdot \sin y\right) \cdot \frac{2 \cdot \cosh x}{y} \]

   5. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin y\right) \cdot \frac{2 \cdot \cosh x}{y}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin y\right) \cdot \frac{2}{y} \]
   7. Step-by-step derivation

   8. Applied rewrites 47.0%

      \[\leadsto \left(0.5 \cdot \sin y\right) \cdot \frac{2}{y} \]

   9. Taylor expanded in y around 0

      \[\leadsto \left(y \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {y}^{2}\right)\right) \cdot \frac{\color{blue}{2}}{y} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {y}^{2}\right) \cdot y\right) \cdot \frac{2}{y} \]

       2. lower-*.f64 N/A

          \[\leadsto \left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {y}^{2}\right) \cdot y\right) \cdot \frac{2}{y} \]

       3. +-commutative N/A

          \[\leadsto \left(\left(\frac{-1}{12} \cdot {y}^{2} + \frac{1}{2}\right) \cdot y\right) \cdot \frac{2}{y} \]

       4. *-commutative N/A

          \[\leadsto \left(\left({y}^{2} \cdot \frac{-1}{12} + \frac{1}{2}\right) \cdot y\right) \cdot \frac{2}{y} \]

       5. lower-fma.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left({y}^{2}, \frac{-1}{12}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{2}{y} \]

       6. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(y \cdot y, \frac{-1}{12}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{2}{y} \]

       7. lift-*.f64 30.0

          \[\leadsto \left(\mathsf{fma}\left(y \cdot y, -0.08333333333333333, 0.5\right) \cdot y\right) \cdot \frac{2}{y} \]

   11. Applied rewrites 30.0%

       \[\leadsto \left(\mathsf{fma}\left(y \cdot y, -0.08333333333333333, 0.5\right) \cdot y\right) \cdot \frac{\color{blue}{2}}{y} \]

   12. Taylor expanded in x around 0

       \[\leadsto \left(\mathsf{fma}\left(y \cdot y, \frac{-1}{12}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{2 + {x}^{2} \cdot \left(1 + \frac{1}{12} \cdot {x}^{2}\right)}{y} \]
   13. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(y \cdot y, \frac{-1}{12}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{{x}^{2} \cdot \left(1 + \frac{1}{12} \cdot {x}^{2}\right) + 2}{y} \]

       2. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(y \cdot y, \frac{-1}{12}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{\left(1 + \frac{1}{12} \cdot {x}^{2}\right) \cdot {x}^{2} + 2}{y} \]

       3. lower-fma.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(y \cdot y, \frac{-1}{12}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{\mathsf{fma}\left(1 + \frac{1}{12} \cdot {x}^{2}, {x}^{2}, 2\right)}{y} \]

       4. +-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(y \cdot y, \frac{-1}{12}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{\mathsf{fma}\left(\frac{1}{12} \cdot {x}^{2} + 1, {x}^{2}, 2\right)}{y} \]

       5. lower-fma.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(y \cdot y, \frac{-1}{12}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{12}, {x}^{2}, 1\right), {x}^{2}, 2\right)}{y} \]

       6. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(y \cdot y, \frac{-1}{12}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{12}, x \cdot x, 1\right), {x}^{2}, 2\right)}{y} \]

       7. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(y \cdot y, \frac{-1}{12}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{12}, x \cdot x, 1\right), {x}^{2}, 2\right)}{y} \]

       8. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(y \cdot y, \frac{-1}{12}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{12}, x \cdot x, 1\right), x \cdot x, 2\right)}{y} \]

       9. lift-*.f64 54.6

          \[\leadsto \left(\mathsf{fma}\left(y \cdot y, -0.08333333333333333, 0.5\right) \cdot y\right) \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.08333333333333333, x \cdot x, 1\right), x \cdot x, 2\right)}{y} \]

   14. Applied rewrites 54.6%

       \[\leadsto \left(\mathsf{fma}\left(y \cdot y, -0.08333333333333333, 0.5\right) \cdot y\right) \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.08333333333333333, x \cdot x, 1\right), x \cdot x, 2\right)}{y} \]

   if -1.99999999999999982e-307 < (/.f64 (sin.f64 y) y) < 1e-8

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

         \[\leadsto \left(\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) \cdot {x}^{2} + 1\right) \cdot \frac{\sin y}{y} \]

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, {x}^{2}, \frac{1}{2}\right), {\color{blue}{x}}^{2}, 1\right) \cdot \frac{\sin y}{y} \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot \frac{\sin y}{y} \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot \frac{\sin y}{y} \]

      8. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot \color{blue}{x}, 1\right) \cdot \frac{\sin y}{y} \]

      9. lower-*.f64 83.9

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

   5. Applied rewrites 83.9%

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

   6. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot \frac{\color{blue}{y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)\right)}}{y} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot \frac{\left(1 + {y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)\right) \cdot \color{blue}{y}}{y} \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot \frac{\left(1 + {y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)\right) \cdot \color{blue}{y}}{y} \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot \frac{\left({y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right) + 1\right) \cdot y}{y} \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot \frac{\left(\left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right) \cdot {y}^{2} + 1\right) \cdot y}{y} \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot \frac{\mathsf{fma}\left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}, {y}^{2}, 1\right) \cdot y}{y} \]

      6. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot \frac{\mathsf{fma}\left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}, {y}^{2}, 1\right) \cdot y}{y} \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot \frac{\mathsf{fma}\left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}, {y}^{2}, 1\right) \cdot y}{y} \]

      8. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot \frac{\mathsf{fma}\left(\frac{1}{120} \cdot \left(y \cdot y\right) - \frac{1}{6}, {y}^{2}, 1\right) \cdot y}{y} \]

      9. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot \frac{\mathsf{fma}\left(\frac{1}{120} \cdot \left(y \cdot y\right) - \frac{1}{6}, {y}^{2}, 1\right) \cdot y}{y} \]

      10. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot \frac{\mathsf{fma}\left(\frac{1}{120} \cdot \left(y \cdot y\right) - \frac{1}{6}, y \cdot y, 1\right) \cdot y}{y} \]

      11. lift-*.f64 54.9

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot \frac{\mathsf{fma}\left(0.008333333333333333 \cdot \left(y \cdot y\right) - 0.16666666666666666, y \cdot y, 1\right) \cdot y}{y} \]

   8. Applied rewrites 54.9%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot \frac{\color{blue}{\mathsf{fma}\left(0.008333333333333333 \cdot \left(y \cdot y\right) - 0.16666666666666666, y \cdot y, 1\right) \cdot y}}{y} \]

   if 1e-8 < (/.f64 (sin.f64 y) y)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. rec-exp N/A

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

      8. cosh-undef N/A

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

      9. lower-*.f64 N/A

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

      10. lift-cosh.f64 99.9

          \[\leadsto \left(0.5 \cdot \sin y\right) \cdot \frac{2 \cdot \cosh x}{y} \]

   5. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin y\right) \cdot \frac{2 \cdot \cosh x}{y}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin y\right) \cdot \frac{2}{y} \]
   7. Step-by-step derivation

   8. Applied rewrites 47.1%

      \[\leadsto \left(0.5 \cdot \sin y\right) \cdot \frac{2}{y} \]

   9. Taylor expanded in y around 0

      \[\leadsto \left(y \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {y}^{2}\right)\right) \cdot \frac{\color{blue}{2}}{y} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {y}^{2}\right) \cdot y\right) \cdot \frac{2}{y} \]

       2. lower-*.f64 N/A

          \[\leadsto \left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {y}^{2}\right) \cdot y\right) \cdot \frac{2}{y} \]

       3. +-commutative N/A

          \[\leadsto \left(\left(\frac{-1}{12} \cdot {y}^{2} + \frac{1}{2}\right) \cdot y\right) \cdot \frac{2}{y} \]

       4. *-commutative N/A

          \[\leadsto \left(\left({y}^{2} \cdot \frac{-1}{12} + \frac{1}{2}\right) \cdot y\right) \cdot \frac{2}{y} \]

       5. lower-fma.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left({y}^{2}, \frac{-1}{12}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{2}{y} \]

       6. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(y \cdot y, \frac{-1}{12}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{2}{y} \]

       7. lift-*.f64 47.1

          \[\leadsto \left(\mathsf{fma}\left(y \cdot y, -0.08333333333333333, 0.5\right) \cdot y\right) \cdot \frac{2}{y} \]

   11. Applied rewrites 47.1%

       \[\leadsto \left(\mathsf{fma}\left(y \cdot y, -0.08333333333333333, 0.5\right) \cdot y\right) \cdot \frac{\color{blue}{2}}{y} \]

   12. Taylor expanded in x around 0

       \[\leadsto \left(\mathsf{fma}\left(y \cdot y, \frac{-1}{12}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{2 + {x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right)}{y} \]
   13. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(y \cdot y, \frac{-1}{12}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{{x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right) + 2}{y} \]

       2. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(y \cdot y, \frac{-1}{12}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{\left(1 + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right) \cdot {x}^{2} + 2}{y} \]

       3. lower-fma.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(y \cdot y, \frac{-1}{12}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{\mathsf{fma}\left(1 + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right), {x}^{2}, 2\right)}{y} \]

       4. +-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(y \cdot y, \frac{-1}{12}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{\mathsf{fma}\left({x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right) + 1, {x}^{2}, 2\right)}{y} \]

       5. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(y \cdot y, \frac{-1}{12}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{\mathsf{fma}\left(\left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right) \cdot {x}^{2} + 1, {x}^{2}, 2\right)}{y} \]

       6. lower-fma.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(y \cdot y, \frac{-1}{12}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}, {x}^{2}, 1\right), {x}^{2}, 2\right)}{y} \]

       7. +-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(y \cdot y, \frac{-1}{12}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360} \cdot {x}^{2} + \frac{1}{12}, {x}^{2}, 1\right), {x}^{2}, 2\right)}{y} \]

       8. lower-fma.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(y \cdot y, \frac{-1}{12}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, {x}^{2}, \frac{1}{12}\right), {x}^{2}, 1\right), {x}^{2}, 2\right)}{y} \]

       9. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(y \cdot y, \frac{-1}{12}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, x \cdot x, \frac{1}{12}\right), {x}^{2}, 1\right), {x}^{2}, 2\right)}{y} \]

       10. lift-*.f64 N/A

           \[\leadsto \left(\mathsf{fma}\left(y \cdot y, \frac{-1}{12}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, x \cdot x, \frac{1}{12}\right), {x}^{2}, 1\right), {x}^{2}, 2\right)}{y} \]

       11. pow2 N/A

           \[\leadsto \left(\mathsf{fma}\left(y \cdot y, \frac{-1}{12}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, x \cdot x, \frac{1}{12}\right), x \cdot x, 1\right), {x}^{2}, 2\right)}{y} \]

       12. lift-*.f64 N/A

           \[\leadsto \left(\mathsf{fma}\left(y \cdot y, \frac{-1}{12}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, x \cdot x, \frac{1}{12}\right), x \cdot x, 1\right), {x}^{2}, 2\right)}{y} \]

       13. pow2 N/A

           \[\leadsto \left(\mathsf{fma}\left(y \cdot y, \frac{-1}{12}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, x \cdot x, \frac{1}{12}\right), x \cdot x, 1\right), x \cdot x, 2\right)}{y} \]

       14. lift-*.f64 95.2

           \[\leadsto \left(\mathsf{fma}\left(y \cdot y, -0.08333333333333333, 0.5\right) \cdot y\right) \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, x \cdot x, 0.08333333333333333\right), x \cdot x, 1\right), x \cdot x, 2\right)}{y} \]

   14. Applied rewrites 95.2%

       \[\leadsto \left(\mathsf{fma}\left(y \cdot y, -0.08333333333333333, 0.5\right) \cdot y\right) \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, x \cdot x, 0.08333333333333333\right), x \cdot x, 1\right), x \cdot x, 2\right)}{y} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 70.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ t_1 := \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), x \cdot x, 0.5\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-307}:\\ \;\;\;\;\mathsf{fma}\left(t\_1 \cdot x, x, 1\right) \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-147}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.004166666666666667 \cdot \left(y \cdot y\right) - 0.08333333333333333, y \cdot y, 0.5\right) \cdot y\right) \cdot \frac{2}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, x \cdot x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y))
        (t_1
         (fma
          (fma (* x x) 0.001388888888888889 0.041666666666666664)
          (* x x)
          0.5)))
   (if (<= t_0 -2e-307)
     (* (fma (* t_1 x) x 1.0) (fma -0.16666666666666666 (* y y) 1.0))
     (if (<= t_0 5e-147)
       (*
        (*
         (fma
          (- (* 0.004166666666666667 (* y y)) 0.08333333333333333)
          (* y y)
          0.5)
         y)
        (/ 2.0 y))
       (fma t_1 (* x x) 1.0)))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double t_1 = fma(fma((x * x), 0.001388888888888889, 0.041666666666666664), (x * x), 0.5);
	double tmp;
	if (t_0 <= -2e-307) {
		tmp = fma((t_1 * x), x, 1.0) * fma(-0.16666666666666666, (y * y), 1.0);
	} else if (t_0 <= 5e-147) {
		tmp = (fma(((0.004166666666666667 * (y * y)) - 0.08333333333333333), (y * y), 0.5) * y) * (2.0 / y);
	} else {
		tmp = fma(t_1, (x * x), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(y) / y)
	t_1 = fma(fma(Float64(x * x), 0.001388888888888889, 0.041666666666666664), Float64(x * x), 0.5)
	tmp = 0.0
	if (t_0 <= -2e-307)
		tmp = Float64(fma(Float64(t_1 * x), x, 1.0) * fma(-0.16666666666666666, Float64(y * y), 1.0));
	elseif (t_0 <= 5e-147)
		tmp = Float64(Float64(fma(Float64(Float64(0.004166666666666667 * Float64(y * y)) - 0.08333333333333333), Float64(y * y), 0.5) * y) * Float64(2.0 / y));
	else
		tmp = fma(t_1, Float64(x * x), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(x * x), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-307], N[(N[(N[(t$95$1 * x), $MachinePrecision] * x + 1.0), $MachinePrecision] * N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e-147], N[(N[(N[(N[(N[(0.004166666666666667 * N[(y * y), $MachinePrecision]), $MachinePrecision] - 0.08333333333333333), $MachinePrecision] * N[(y * y), $MachinePrecision] + 0.5), $MachinePrecision] * y), $MachinePrecision] * N[(2.0 / y), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 y) y) < -1.99999999999999982e-307

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 54.6

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

   5. Applied rewrites 54.6%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 53.2

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

   8. Applied rewrites 53.2%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-fma.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-fma.f64 N/A

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

      13. lift-*.f64 53.2

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), x \cdot x, 0.5\right) \cdot x, x, 1\right) \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right) \]

   10. Applied rewrites 53.2%

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

   if -1.99999999999999982e-307 < (/.f64 (sin.f64 y) y) < 5.00000000000000013e-147

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. rec-exp N/A

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

      8. cosh-undef N/A

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

      9. lower-*.f64 N/A

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

      10. lift-cosh.f64 99.8

          \[\leadsto \left(0.5 \cdot \sin y\right) \cdot \frac{2 \cdot \cosh x}{y} \]

   5. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin y\right) \cdot \frac{2 \cdot \cosh x}{y}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin y\right) \cdot \frac{2}{y} \]
   7. Step-by-step derivation

   8. Applied rewrites 48.1%

      \[\leadsto \left(0.5 \cdot \sin y\right) \cdot \frac{2}{y} \]

   9. Taylor expanded in y around 0

      \[\leadsto \left(y \cdot \left(\frac{1}{2} + {y}^{2} \cdot \left(\frac{1}{240} \cdot {y}^{2} - \frac{1}{12}\right)\right)\right) \cdot \frac{\color{blue}{2}}{y} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(\left(\frac{1}{2} + {y}^{2} \cdot \left(\frac{1}{240} \cdot {y}^{2} - \frac{1}{12}\right)\right) \cdot y\right) \cdot \frac{2}{y} \]

       2. lower-*.f64 N/A

          \[\leadsto \left(\left(\frac{1}{2} + {y}^{2} \cdot \left(\frac{1}{240} \cdot {y}^{2} - \frac{1}{12}\right)\right) \cdot y\right) \cdot \frac{2}{y} \]

       3. +-commutative N/A

          \[\leadsto \left(\left({y}^{2} \cdot \left(\frac{1}{240} \cdot {y}^{2} - \frac{1}{12}\right) + \frac{1}{2}\right) \cdot y\right) \cdot \frac{2}{y} \]

       4. *-commutative N/A

          \[\leadsto \left(\left(\left(\frac{1}{240} \cdot {y}^{2} - \frac{1}{12}\right) \cdot {y}^{2} + \frac{1}{2}\right) \cdot y\right) \cdot \frac{2}{y} \]

       5. lower-fma.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{240} \cdot {y}^{2} - \frac{1}{12}, {y}^{2}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{2}{y} \]

       6. lower--.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{240} \cdot {y}^{2} - \frac{1}{12}, {y}^{2}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{2}{y} \]

       7. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{240} \cdot {y}^{2} - \frac{1}{12}, {y}^{2}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{2}{y} \]

       8. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{240} \cdot \left(y \cdot y\right) - \frac{1}{12}, {y}^{2}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{2}{y} \]

       9. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{240} \cdot \left(y \cdot y\right) - \frac{1}{12}, {y}^{2}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{2}{y} \]

       10. pow2 N/A

           \[\leadsto \left(\mathsf{fma}\left(\frac{1}{240} \cdot \left(y \cdot y\right) - \frac{1}{12}, y \cdot y, \frac{1}{2}\right) \cdot y\right) \cdot \frac{2}{y} \]

       11. lift-*.f64 53.7

           \[\leadsto \left(\mathsf{fma}\left(0.004166666666666667 \cdot \left(y \cdot y\right) - 0.08333333333333333, y \cdot y, 0.5\right) \cdot y\right) \cdot \frac{2}{y} \]

   11. Applied rewrites 53.7%

       \[\leadsto \left(\mathsf{fma}\left(0.004166666666666667 \cdot \left(y \cdot y\right) - 0.08333333333333333, y \cdot y, 0.5\right) \cdot y\right) \cdot \frac{\color{blue}{2}}{y} \]

   if 5.00000000000000013e-147 < (/.f64 (sin.f64 y) y)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \frac{\sin y}{y} + {x}^{2} \cdot \left(\frac{1}{720} \cdot \frac{{x}^{2} \cdot \sin y}{y} + \frac{1}{24} \cdot \frac{\sin y}{y}\right)\right) \cdot {x}^{2} + \frac{\color{blue}{\sin y}}{y} \]

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 90.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.041666666666666664, \sin y, \left(0.001388888888888889 \cdot \sin y\right) \cdot \left(x \cdot x\right)\right)}{y}, x \cdot x, 0.5 \cdot \frac{\sin y}{y}\right), x \cdot x, \frac{\sin y}{y}\right)} \]

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

         \[\leadsto {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) + 1 \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) \cdot {x}^{2} + 1 \]

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left({x}^{2}, \frac{1}{720}, \frac{1}{24}\right), {x}^{2}, \frac{1}{2}\right), {x}^{2}, 1\right) \]

      10. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right), {x}^{2}, \frac{1}{2}\right), {x}^{2}, 1\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right), {x}^{2}, \frac{1}{2}\right), {x}^{2}, 1\right) \]

      12. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right), x \cdot x, \frac{1}{2}\right), {x}^{2}, 1\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right), x \cdot x, \frac{1}{2}\right), {x}^{2}, 1\right) \]

      14. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right), x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \]

      15. lift-*.f64 82.4

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right) \]

   8. Applied rewrites 82.4%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), x \cdot x, 0.5\right), \color{blue}{x \cdot x}, 1\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 70.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-307}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right), x \cdot x, 1\right) \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-147}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.004166666666666667 \cdot \left(y \cdot y\right) - 0.08333333333333333, y \cdot y, 0.5\right) \cdot y\right) \cdot \frac{2}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)))
   (if (<= t_0 -2e-307)
     (*
      (fma (fma (* x x) 0.041666666666666664 0.5) (* x x) 1.0)
      (fma -0.16666666666666666 (* y y) 1.0))
     (if (<= t_0 5e-147)
       (*
        (*
         (fma
          (- (* 0.004166666666666667 (* y y)) 0.08333333333333333)
          (* y y)
          0.5)
         y)
        (/ 2.0 y))
       (fma
        (fma
         (fma (* x x) 0.001388888888888889 0.041666666666666664)
         (* x x)
         0.5)
        (* x x)
        1.0)))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double tmp;
	if (t_0 <= -2e-307) {
		tmp = fma(fma((x * x), 0.041666666666666664, 0.5), (x * x), 1.0) * fma(-0.16666666666666666, (y * y), 1.0);
	} else if (t_0 <= 5e-147) {
		tmp = (fma(((0.004166666666666667 * (y * y)) - 0.08333333333333333), (y * y), 0.5) * y) * (2.0 / y);
	} else {
		tmp = fma(fma(fma((x * x), 0.001388888888888889, 0.041666666666666664), (x * x), 0.5), (x * x), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(y) / y)
	tmp = 0.0
	if (t_0 <= -2e-307)
		tmp = Float64(fma(fma(Float64(x * x), 0.041666666666666664, 0.5), Float64(x * x), 1.0) * fma(-0.16666666666666666, Float64(y * y), 1.0));
	elseif (t_0 <= 5e-147)
		tmp = Float64(Float64(fma(Float64(Float64(0.004166666666666667 * Float64(y * y)) - 0.08333333333333333), Float64(y * y), 0.5) * y) * Float64(2.0 / y));
	else
		tmp = fma(fma(fma(Float64(x * x), 0.001388888888888889, 0.041666666666666664), Float64(x * x), 0.5), Float64(x * x), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-307], N[(N[(N[(N[(x * x), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e-147], N[(N[(N[(N[(N[(0.004166666666666667 * N[(y * y), $MachinePrecision]), $MachinePrecision] - 0.08333333333333333), $MachinePrecision] * N[(y * y), $MachinePrecision] + 0.5), $MachinePrecision] * y), $MachinePrecision] * N[(2.0 / y), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(x * x), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 y) y) < -1.99999999999999982e-307

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 54.6

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

   5. Applied rewrites 54.6%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 50.4

         \[\leadsto \mathsf{fma}\left(x \cdot x, 0.5, 1\right) \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right) \]

   8. Applied rewrites 50.4%

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

   9. Taylor expanded in x around 0

      \[\leadsto 1 \cdot \mathsf{fma}\left(\frac{-1}{6}, y \cdot y, 1\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 23.0%

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

   12. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. +-commutative N/A

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

       5. *-commutative N/A

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

       6. lower-fma.f64 N/A

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

       7. pow2 N/A

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

       8. lift-*.f64 N/A

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

       9. pow2 N/A

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

       10. lift-*.f64 53.2

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

   14. Applied rewrites 53.2%

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

   if -1.99999999999999982e-307 < (/.f64 (sin.f64 y) y) < 5.00000000000000013e-147

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. rec-exp N/A

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

      8. cosh-undef N/A

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

      9. lower-*.f64 N/A

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

      10. lift-cosh.f64 99.8

          \[\leadsto \left(0.5 \cdot \sin y\right) \cdot \frac{2 \cdot \cosh x}{y} \]

   5. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin y\right) \cdot \frac{2 \cdot \cosh x}{y}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin y\right) \cdot \frac{2}{y} \]
   7. Step-by-step derivation

   8. Applied rewrites 48.1%

      \[\leadsto \left(0.5 \cdot \sin y\right) \cdot \frac{2}{y} \]

   9. Taylor expanded in y around 0

      \[\leadsto \left(y \cdot \left(\frac{1}{2} + {y}^{2} \cdot \left(\frac{1}{240} \cdot {y}^{2} - \frac{1}{12}\right)\right)\right) \cdot \frac{\color{blue}{2}}{y} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(\left(\frac{1}{2} + {y}^{2} \cdot \left(\frac{1}{240} \cdot {y}^{2} - \frac{1}{12}\right)\right) \cdot y\right) \cdot \frac{2}{y} \]

       2. lower-*.f64 N/A

          \[\leadsto \left(\left(\frac{1}{2} + {y}^{2} \cdot \left(\frac{1}{240} \cdot {y}^{2} - \frac{1}{12}\right)\right) \cdot y\right) \cdot \frac{2}{y} \]

       3. +-commutative N/A

          \[\leadsto \left(\left({y}^{2} \cdot \left(\frac{1}{240} \cdot {y}^{2} - \frac{1}{12}\right) + \frac{1}{2}\right) \cdot y\right) \cdot \frac{2}{y} \]

       4. *-commutative N/A

          \[\leadsto \left(\left(\left(\frac{1}{240} \cdot {y}^{2} - \frac{1}{12}\right) \cdot {y}^{2} + \frac{1}{2}\right) \cdot y\right) \cdot \frac{2}{y} \]

       5. lower-fma.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{240} \cdot {y}^{2} - \frac{1}{12}, {y}^{2}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{2}{y} \]

       6. lower--.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{240} \cdot {y}^{2} - \frac{1}{12}, {y}^{2}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{2}{y} \]

       7. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{240} \cdot {y}^{2} - \frac{1}{12}, {y}^{2}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{2}{y} \]

       8. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{240} \cdot \left(y \cdot y\right) - \frac{1}{12}, {y}^{2}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{2}{y} \]

       9. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{240} \cdot \left(y \cdot y\right) - \frac{1}{12}, {y}^{2}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{2}{y} \]

       10. pow2 N/A

           \[\leadsto \left(\mathsf{fma}\left(\frac{1}{240} \cdot \left(y \cdot y\right) - \frac{1}{12}, y \cdot y, \frac{1}{2}\right) \cdot y\right) \cdot \frac{2}{y} \]

       11. lift-*.f64 53.7

           \[\leadsto \left(\mathsf{fma}\left(0.004166666666666667 \cdot \left(y \cdot y\right) - 0.08333333333333333, y \cdot y, 0.5\right) \cdot y\right) \cdot \frac{2}{y} \]

   11. Applied rewrites 53.7%

       \[\leadsto \left(\mathsf{fma}\left(0.004166666666666667 \cdot \left(y \cdot y\right) - 0.08333333333333333, y \cdot y, 0.5\right) \cdot y\right) \cdot \frac{\color{blue}{2}}{y} \]

   if 5.00000000000000013e-147 < (/.f64 (sin.f64 y) y)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \frac{\sin y}{y} + {x}^{2} \cdot \left(\frac{1}{720} \cdot \frac{{x}^{2} \cdot \sin y}{y} + \frac{1}{24} \cdot \frac{\sin y}{y}\right)\right) \cdot {x}^{2} + \frac{\color{blue}{\sin y}}{y} \]

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 90.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.041666666666666664, \sin y, \left(0.001388888888888889 \cdot \sin y\right) \cdot \left(x \cdot x\right)\right)}{y}, x \cdot x, 0.5 \cdot \frac{\sin y}{y}\right), x \cdot x, \frac{\sin y}{y}\right)} \]

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

         \[\leadsto {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) + 1 \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) \cdot {x}^{2} + 1 \]

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left({x}^{2}, \frac{1}{720}, \frac{1}{24}\right), {x}^{2}, \frac{1}{2}\right), {x}^{2}, 1\right) \]

      10. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right), {x}^{2}, \frac{1}{2}\right), {x}^{2}, 1\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right), {x}^{2}, \frac{1}{2}\right), {x}^{2}, 1\right) \]

      12. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right), x \cdot x, \frac{1}{2}\right), {x}^{2}, 1\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right), x \cdot x, \frac{1}{2}\right), {x}^{2}, 1\right) \]

      14. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right), x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \]

      15. lift-*.f64 82.4

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right) \]

   8. Applied rewrites 82.4%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), x \cdot x, 0.5\right), \color{blue}{x \cdot x}, 1\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 69.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -5 \cdot 10^{-142}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right), x \cdot x, 1\right) \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cosh x) (/ (sin y) y)) -5e-142)
   (*
    (fma (fma (* x x) 0.041666666666666664 0.5) (* x x) 1.0)
    (fma -0.16666666666666666 (* y y) 1.0))
   (fma
    (fma (fma (* x x) 0.001388888888888889 0.041666666666666664) (* x x) 0.5)
    (* x x)
    1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((cosh(x) * (sin(y) / y)) <= -5e-142) {
		tmp = fma(fma((x * x), 0.041666666666666664, 0.5), (x * x), 1.0) * fma(-0.16666666666666666, (y * y), 1.0);
	} else {
		tmp = fma(fma(fma((x * x), 0.001388888888888889, 0.041666666666666664), (x * x), 0.5), (x * x), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(cosh(x) * Float64(sin(y) / y)) <= -5e-142)
		tmp = Float64(fma(fma(Float64(x * x), 0.041666666666666664, 0.5), Float64(x * x), 1.0) * fma(-0.16666666666666666, Float64(y * y), 1.0));
	else
		tmp = fma(fma(fma(Float64(x * x), 0.001388888888888889, 0.041666666666666664), Float64(x * x), 0.5), Float64(x * x), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Cosh[x], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], -5e-142], N[(N[(N[(N[(x * x), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(x * x), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < -5.0000000000000002e-142

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 68.8

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

   5. Applied rewrites 68.8%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 63.5

         \[\leadsto \mathsf{fma}\left(x \cdot x, 0.5, 1\right) \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right) \]

   8. Applied rewrites 63.5%

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

   9. Taylor expanded in x around 0

      \[\leadsto 1 \cdot \mathsf{fma}\left(\frac{-1}{6}, y \cdot y, 1\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 28.7%

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

   12. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. +-commutative N/A

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

       5. *-commutative N/A

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

       6. lower-fma.f64 N/A

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

       7. pow2 N/A

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

       8. lift-*.f64 N/A

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

       9. pow2 N/A

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

       10. lift-*.f64 67.0

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

   14. Applied rewrites 67.0%

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

   if -5.0000000000000002e-142 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y))

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \frac{\sin y}{y} + {x}^{2} \cdot \left(\frac{1}{720} \cdot \frac{{x}^{2} \cdot \sin y}{y} + \frac{1}{24} \cdot \frac{\sin y}{y}\right)\right) \cdot {x}^{2} + \frac{\color{blue}{\sin y}}{y} \]

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 88.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.041666666666666664, \sin y, \left(0.001388888888888889 \cdot \sin y\right) \cdot \left(x \cdot x\right)\right)}{y}, x \cdot x, 0.5 \cdot \frac{\sin y}{y}\right), x \cdot x, \frac{\sin y}{y}\right)} \]

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

         \[\leadsto {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) + 1 \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) \cdot {x}^{2} + 1 \]

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left({x}^{2}, \frac{1}{720}, \frac{1}{24}\right), {x}^{2}, \frac{1}{2}\right), {x}^{2}, 1\right) \]

      10. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right), {x}^{2}, \frac{1}{2}\right), {x}^{2}, 1\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right), {x}^{2}, \frac{1}{2}\right), {x}^{2}, 1\right) \]

      12. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right), x \cdot x, \frac{1}{2}\right), {x}^{2}, 1\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right), x \cdot x, \frac{1}{2}\right), {x}^{2}, 1\right) \]

      14. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right), x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \]

      15. lift-*.f64 68.8

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right) \]

   8. Applied rewrites 68.8%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), x \cdot x, 0.5\right), \color{blue}{x \cdot x}, 1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 69.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -5 \cdot 10^{-142}:\\ \;\;\;\;\left(\mathsf{fma}\left(y \cdot y, -0.08333333333333333, 0.5\right) \cdot y\right) \cdot \frac{\mathsf{fma}\left(x, x, 2\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cosh x) (/ (sin y) y)) -5e-142)
   (* (* (fma (* y y) -0.08333333333333333 0.5) y) (/ (fma x x 2.0) y))
   (fma
    (fma (fma (* x x) 0.001388888888888889 0.041666666666666664) (* x x) 0.5)
    (* x x)
    1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((cosh(x) * (sin(y) / y)) <= -5e-142) {
		tmp = (fma((y * y), -0.08333333333333333, 0.5) * y) * (fma(x, x, 2.0) / y);
	} else {
		tmp = fma(fma(fma((x * x), 0.001388888888888889, 0.041666666666666664), (x * x), 0.5), (x * x), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(cosh(x) * Float64(sin(y) / y)) <= -5e-142)
		tmp = Float64(Float64(fma(Float64(y * y), -0.08333333333333333, 0.5) * y) * Float64(fma(x, x, 2.0) / y));
	else
		tmp = fma(fma(fma(Float64(x * x), 0.001388888888888889, 0.041666666666666664), Float64(x * x), 0.5), Float64(x * x), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Cosh[x], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], -5e-142], N[(N[(N[(N[(y * y), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision] * y), $MachinePrecision] * N[(N[(x * x + 2.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(x * x), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < -5.0000000000000002e-142

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. rec-exp N/A

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

      8. cosh-undef N/A

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

      9. lower-*.f64 N/A

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

      10. lift-cosh.f64 99.9

          \[\leadsto \left(0.5 \cdot \sin y\right) \cdot \frac{2 \cdot \cosh x}{y} \]

   5. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin y\right) \cdot \frac{2 \cdot \cosh x}{y}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin y\right) \cdot \frac{2}{y} \]
   7. Step-by-step derivation

   8. Applied rewrites 33.0%

      \[\leadsto \left(0.5 \cdot \sin y\right) \cdot \frac{2}{y} \]

   9. Taylor expanded in y around 0

      \[\leadsto \left(y \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {y}^{2}\right)\right) \cdot \frac{\color{blue}{2}}{y} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {y}^{2}\right) \cdot y\right) \cdot \frac{2}{y} \]

       2. lower-*.f64 N/A

          \[\leadsto \left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {y}^{2}\right) \cdot y\right) \cdot \frac{2}{y} \]

       3. +-commutative N/A

          \[\leadsto \left(\left(\frac{-1}{12} \cdot {y}^{2} + \frac{1}{2}\right) \cdot y\right) \cdot \frac{2}{y} \]

       4. *-commutative N/A

          \[\leadsto \left(\left({y}^{2} \cdot \frac{-1}{12} + \frac{1}{2}\right) \cdot y\right) \cdot \frac{2}{y} \]

       5. lower-fma.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left({y}^{2}, \frac{-1}{12}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{2}{y} \]

       6. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(y \cdot y, \frac{-1}{12}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{2}{y} \]

       7. lift-*.f64 37.6

          \[\leadsto \left(\mathsf{fma}\left(y \cdot y, -0.08333333333333333, 0.5\right) \cdot y\right) \cdot \frac{2}{y} \]

   11. Applied rewrites 37.6%

       \[\leadsto \left(\mathsf{fma}\left(y \cdot y, -0.08333333333333333, 0.5\right) \cdot y\right) \cdot \frac{\color{blue}{2}}{y} \]

   12. Taylor expanded in x around 0

       \[\leadsto \left(\mathsf{fma}\left(y \cdot y, \frac{-1}{12}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{2 + {x}^{2}}{y} \]
   13. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(y \cdot y, \frac{-1}{12}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{{x}^{2} + 2}{y} \]

       2. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(y \cdot y, \frac{-1}{12}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{x \cdot x + 2}{y} \]

       3. lower-fma.f64 65.2

          \[\leadsto \left(\mathsf{fma}\left(y \cdot y, -0.08333333333333333, 0.5\right) \cdot y\right) \cdot \frac{\mathsf{fma}\left(x, x, 2\right)}{y} \]

   14. Applied rewrites 65.2%

       \[\leadsto \left(\mathsf{fma}\left(y \cdot y, -0.08333333333333333, 0.5\right) \cdot y\right) \cdot \frac{\mathsf{fma}\left(x, x, 2\right)}{y} \]

   if -5.0000000000000002e-142 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y))

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \frac{\sin y}{y} + {x}^{2} \cdot \left(\frac{1}{720} \cdot \frac{{x}^{2} \cdot \sin y}{y} + \frac{1}{24} \cdot \frac{\sin y}{y}\right)\right) \cdot {x}^{2} + \frac{\color{blue}{\sin y}}{y} \]

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 88.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.041666666666666664, \sin y, \left(0.001388888888888889 \cdot \sin y\right) \cdot \left(x \cdot x\right)\right)}{y}, x \cdot x, 0.5 \cdot \frac{\sin y}{y}\right), x \cdot x, \frac{\sin y}{y}\right)} \]

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

         \[\leadsto {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) + 1 \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) \cdot {x}^{2} + 1 \]

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left({x}^{2}, \frac{1}{720}, \frac{1}{24}\right), {x}^{2}, \frac{1}{2}\right), {x}^{2}, 1\right) \]

      10. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right), {x}^{2}, \frac{1}{2}\right), {x}^{2}, 1\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right), {x}^{2}, \frac{1}{2}\right), {x}^{2}, 1\right) \]

      12. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right), x \cdot x, \frac{1}{2}\right), {x}^{2}, 1\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right), x \cdot x, \frac{1}{2}\right), {x}^{2}, 1\right) \]

      14. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right), x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \]

      15. lift-*.f64 68.8

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right) \]

   8. Applied rewrites 68.8%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), x \cdot x, 0.5\right), \color{blue}{x \cdot x}, 1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 69.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -5 \cdot 10^{-142}:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot 0.5\right) \cdot \left(\left(y \cdot y\right) \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cosh x) (/ (sin y) y)) -5e-142)
   (* (* (* x x) 0.5) (* (* y y) -0.16666666666666666))
   (fma
    (fma (fma (* x x) 0.001388888888888889 0.041666666666666664) (* x x) 0.5)
    (* x x)
    1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((cosh(x) * (sin(y) / y)) <= -5e-142) {
		tmp = ((x * x) * 0.5) * ((y * y) * -0.16666666666666666);
	} else {
		tmp = fma(fma(fma((x * x), 0.001388888888888889, 0.041666666666666664), (x * x), 0.5), (x * x), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(cosh(x) * Float64(sin(y) / y)) <= -5e-142)
		tmp = Float64(Float64(Float64(x * x) * 0.5) * Float64(Float64(y * y) * -0.16666666666666666));
	else
		tmp = fma(fma(fma(Float64(x * x), 0.001388888888888889, 0.041666666666666664), Float64(x * x), 0.5), Float64(x * x), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Cosh[x], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], -5e-142], N[(N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(x * x), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < -5.0000000000000002e-142

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 68.8

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

   5. Applied rewrites 68.8%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 63.5

         \[\leadsto \mathsf{fma}\left(x \cdot x, 0.5, 1\right) \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right) \]

   8. Applied rewrites 63.5%

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

   9. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 63.9

          \[\leadsto \left(\left(x \cdot x\right) \cdot 0.5\right) \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right) \]

   11. Applied rewrites 63.9%

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

   12. Taylor expanded in y around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 63.9

          \[\leadsto \left(\left(x \cdot x\right) \cdot 0.5\right) \cdot \left(\left(y \cdot y\right) \cdot -0.16666666666666666\right) \]

   14. Applied rewrites 63.9%

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

   if -5.0000000000000002e-142 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y))

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \frac{\sin y}{y} + {x}^{2} \cdot \left(\frac{1}{720} \cdot \frac{{x}^{2} \cdot \sin y}{y} + \frac{1}{24} \cdot \frac{\sin y}{y}\right)\right) \cdot {x}^{2} + \frac{\color{blue}{\sin y}}{y} \]

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 88.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.041666666666666664, \sin y, \left(0.001388888888888889 \cdot \sin y\right) \cdot \left(x \cdot x\right)\right)}{y}, x \cdot x, 0.5 \cdot \frac{\sin y}{y}\right), x \cdot x, \frac{\sin y}{y}\right)} \]

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

         \[\leadsto {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) + 1 \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) \cdot {x}^{2} + 1 \]

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left({x}^{2}, \frac{1}{720}, \frac{1}{24}\right), {x}^{2}, \frac{1}{2}\right), {x}^{2}, 1\right) \]

      10. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right), {x}^{2}, \frac{1}{2}\right), {x}^{2}, 1\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right), {x}^{2}, \frac{1}{2}\right), {x}^{2}, 1\right) \]

      12. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right), x \cdot x, \frac{1}{2}\right), {x}^{2}, 1\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right), x \cdot x, \frac{1}{2}\right), {x}^{2}, 1\right) \]

      14. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right), x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \]

      15. lift-*.f64 68.8

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right) \]

   8. Applied rewrites 68.8%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), x \cdot x, 0.5\right), \color{blue}{x \cdot x}, 1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 57.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -5 \cdot 10^{-142}:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot 0.5\right) \cdot \left(\left(y \cdot y\right) \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, 0.5, 1\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cosh x) (/ (sin y) y)) -5e-142)
   (* (* (* x x) 0.5) (* (* y y) -0.16666666666666666))
   (* (fma (* x x) 0.5 1.0) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((cosh(x) * (sin(y) / y)) <= -5e-142) {
		tmp = ((x * x) * 0.5) * ((y * y) * -0.16666666666666666);
	} else {
		tmp = fma((x * x), 0.5, 1.0) * 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(cosh(x) * Float64(sin(y) / y)) <= -5e-142)
		tmp = Float64(Float64(Float64(x * x) * 0.5) * Float64(Float64(y * y) * -0.16666666666666666));
	else
		tmp = Float64(fma(Float64(x * x), 0.5, 1.0) * 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Cosh[x], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], -5e-142], N[(N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * x), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < -5.0000000000000002e-142

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 68.8

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

   5. Applied rewrites 68.8%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 63.5

         \[\leadsto \mathsf{fma}\left(x \cdot x, 0.5, 1\right) \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right) \]

   8. Applied rewrites 63.5%

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

   9. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 63.9

          \[\leadsto \left(\left(x \cdot x\right) \cdot 0.5\right) \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right) \]

   11. Applied rewrites 63.9%

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

   12. Taylor expanded in y around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 63.9

          \[\leadsto \left(\left(x \cdot x\right) \cdot 0.5\right) \cdot \left(\left(y \cdot y\right) \cdot -0.16666666666666666\right) \]

   14. Applied rewrites 63.9%

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

   if -5.0000000000000002e-142 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y))

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 78.9%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot 1 \]

      5. lift-*.f64 54.3

         \[\leadsto \mathsf{fma}\left(x \cdot x, 0.5, 1\right) \cdot 1 \]

   8. Applied rewrites 54.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, 0.5, 1\right)} \cdot 1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 52.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -5 \cdot 10^{-142}:\\ \;\;\;\;1 \cdot \left(\left(y \cdot y\right) \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, 0.5, 1\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cosh x) (/ (sin y) y)) -5e-142)
   (* 1.0 (* (* y y) -0.16666666666666666))
   (* (fma (* x x) 0.5 1.0) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((cosh(x) * (sin(y) / y)) <= -5e-142) {
		tmp = 1.0 * ((y * y) * -0.16666666666666666);
	} else {
		tmp = fma((x * x), 0.5, 1.0) * 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(cosh(x) * Float64(sin(y) / y)) <= -5e-142)
		tmp = Float64(1.0 * Float64(Float64(y * y) * -0.16666666666666666));
	else
		tmp = Float64(fma(Float64(x * x), 0.5, 1.0) * 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Cosh[x], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], -5e-142], N[(1.0 * N[(N[(y * y), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * x), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < -5.0000000000000002e-142

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 68.8

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

   5. Applied rewrites 68.8%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 63.5

         \[\leadsto \mathsf{fma}\left(x \cdot x, 0.5, 1\right) \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right) \]

   8. Applied rewrites 63.5%

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

   9. Taylor expanded in x around 0

      \[\leadsto 1 \cdot \mathsf{fma}\left(\frac{-1}{6}, y \cdot y, 1\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 28.7%

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

   12. Taylor expanded in y around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 28.7

          \[\leadsto 1 \cdot \left(\left(y \cdot y\right) \cdot -0.16666666666666666\right) \]

   14. Applied rewrites 28.7%

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

   if -5.0000000000000002e-142 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y))

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 78.9%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot 1 \]

      5. lift-*.f64 54.3

         \[\leadsto \mathsf{fma}\left(x \cdot x, 0.5, 1\right) \cdot 1 \]

   8. Applied rewrites 54.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, 0.5, 1\right)} \cdot 1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 52.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq 2:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot 0.5\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cosh x) (/ (sin y) y)) 2.0)
   (* 1.0 (fma -0.16666666666666666 (* y y) 1.0))
   (* (* (* x x) 0.5) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((cosh(x) * (sin(y) / y)) <= 2.0) {
		tmp = 1.0 * fma(-0.16666666666666666, (y * y), 1.0);
	} else {
		tmp = ((x * x) * 0.5) * 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(cosh(x) * Float64(sin(y) / y)) <= 2.0)
		tmp = Float64(1.0 * fma(-0.16666666666666666, Float64(y * y), 1.0));
	else
		tmp = Float64(Float64(Float64(x * x) * 0.5) * 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Cosh[x], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], 2.0], N[(1.0 * N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision] * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < 2

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 60.4

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

   5. Applied rewrites 60.4%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \cdot \mathsf{fma}\left(\frac{-1}{6}, y \cdot y, 1\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 46.8%

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

   if 2 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 65.0

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

   5. Applied rewrites 65.0%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 34.5

         \[\leadsto \mathsf{fma}\left(x \cdot x, 0.5, 1\right) \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right) \]

   8. Applied rewrites 34.5%

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

   9. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 34.5

          \[\leadsto \left(\left(x \cdot x\right) \cdot 0.5\right) \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right) \]

   11. Applied rewrites 34.5%

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

   12. Taylor expanded in y around 0

       \[\leadsto \left(\left(x \cdot x\right) \cdot \frac{1}{2}\right) \cdot 1 \]
   13. Step-by-step derivation

   14. Applied rewrites 51.2%

       \[\leadsto \left(\left(x \cdot x\right) \cdot 0.5\right) \cdot 1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 33.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -5 \cdot 10^{-142}:\\ \;\;\;\;1 \cdot \left(\left(y \cdot y\right) \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cosh x) (/ (sin y) y)) -5e-142)
   (* 1.0 (* (* y y) -0.16666666666666666))
   (* 1.0 1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((cosh(x) * (sin(y) / y)) <= -5e-142) {
		tmp = 1.0 * ((y * y) * -0.16666666666666666);
	} else {
		tmp = 1.0 * 1.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((cosh(x) * (sin(y) / y)) <= (-5d-142)) then
        tmp = 1.0d0 * ((y * y) * (-0.16666666666666666d0))
    else
        tmp = 1.0d0 * 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((Math.cosh(x) * (Math.sin(y) / y)) <= -5e-142) {
		tmp = 1.0 * ((y * y) * -0.16666666666666666);
	} else {
		tmp = 1.0 * 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (math.cosh(x) * (math.sin(y) / y)) <= -5e-142:
		tmp = 1.0 * ((y * y) * -0.16666666666666666)
	else:
		tmp = 1.0 * 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(cosh(x) * Float64(sin(y) / y)) <= -5e-142)
		tmp = Float64(1.0 * Float64(Float64(y * y) * -0.16666666666666666));
	else
		tmp = Float64(1.0 * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((cosh(x) * (sin(y) / y)) <= -5e-142)
		tmp = 1.0 * ((y * y) * -0.16666666666666666);
	else
		tmp = 1.0 * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Cosh[x], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], -5e-142], N[(1.0 * N[(N[(y * y), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision], N[(1.0 * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < -5.0000000000000002e-142

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 68.8

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

   5. Applied rewrites 68.8%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 63.5

         \[\leadsto \mathsf{fma}\left(x \cdot x, 0.5, 1\right) \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right) \]

   8. Applied rewrites 63.5%

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

   9. Taylor expanded in x around 0

      \[\leadsto 1 \cdot \mathsf{fma}\left(\frac{-1}{6}, y \cdot y, 1\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 28.7%

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

   12. Taylor expanded in y around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 28.7

          \[\leadsto 1 \cdot \left(\left(y \cdot y\right) \cdot -0.16666666666666666\right) \]

   14. Applied rewrites 28.7%

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

   if -5.0000000000000002e-142 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y))

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 78.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 30.0%

      \[\leadsto \color{blue}{1} \cdot 1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 26.8% accurate, 36.2× speedup

Math

\[\begin{array}{l} \\ 1 \cdot 1 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* 1.0 1.0))

C

double code(double x, double y) {
	return 1.0 * 1.0;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 * 1.0d0
end function

Java

public static double code(double x, double y) {
	return 1.0 * 1.0;
}

Python

def code(x, y):
	return 1.0 * 1.0

Julia

function code(x, y)
	return Float64(1.0 * 1.0)
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0 * 1.0;
end

Wolfram

code[x_, y_] := N[(1.0 * 1.0), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in y around 0

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

5. Applied rewrites 63.0%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 24.1%

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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y)
use fmin_fmax_functions
    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 2025037 
(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)))