Linear.Quaternion:$csinh from linear-1.19.1.3

Percentage Accurate: 99.9% → 99.9%

Time: 4.8s

Alternatives: 18

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: 99.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ t_1 := \cosh x \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(y \cdot y, -0.08333333333333333, 0.5\right) \cdot y\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, x \cdot x, 0.08333333333333333\right), x \cdot x, 1\right), x \cdot x, 2\right)}{y}\\ \mathbf{elif}\;t\_1 \leq 0.9999999999999999:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)) (t_1 (* (cosh x) t_0)))
   (if (<= t_1 (- INFINITY))
     (/
      (*
       (* (fma (* y y) -0.08333333333333333 0.5) y)
       (fma
        (fma
         (fma 0.002777777777777778 (* x x) 0.08333333333333333)
         (* x x)
         1.0)
        (* x x)
        2.0))
      y)
     (if (<= t_1 0.9999999999999999)
       (*
        (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 t_1 = cosh(x) * t_0;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = ((fma((y * y), -0.08333333333333333, 0.5) * y) * fma(fma(fma(0.002777777777777778, (x * x), 0.08333333333333333), (x * x), 1.0), (x * x), 2.0)) / y;
	} else if (t_1 <= 0.9999999999999999) {
		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)
	t_1 = Float64(cosh(x) * t_0)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(fma(Float64(y * y), -0.08333333333333333, 0.5) * y) * fma(fma(fma(0.002777777777777778, Float64(x * x), 0.08333333333333333), Float64(x * x), 1.0), Float64(x * x), 2.0)) / y);
	elseif (t_1 <= 0.9999999999999999)
		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]}, Block[{t$95$1 = N[(N[Cosh[x], $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(N[(N[(y * y), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision] * y), $MachinePrecision] * 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]), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[t$95$1, 0.9999999999999999], 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 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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 85.5

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

   8. Applied rewrites 85.5%

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

   9. Taylor expanded in y around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. pow2 N/A

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

       7. lift-*.f64 97.7

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

   11. Applied rewrites 97.7%

       \[\leadsto \left(\mathsf{fma}\left(y \cdot y, -0.08333333333333333, 0.5\right) \cdot y\right) \cdot \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, x \cdot x, 0.08333333333333333\right), x \cdot x, 1\right), x \cdot x, 2\right)}}{y} \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-*r/ N/A

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

       4. lower-/.f64 N/A

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

       5. lower-*.f64 100.0

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

   13. Applied rewrites 100.0%

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

   if -inf.0 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < 0.999999999999999889

   1. Initial program 99.6%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   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 98.6

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

   5. Applied rewrites 98.6%

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

   if 0.999999999999999889 < (*.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: 99.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ t_1 := \cosh x \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(y \cdot y, -0.08333333333333333, 0.5\right) \cdot y\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, x \cdot x, 0.08333333333333333\right), x \cdot x, 1\right), x \cdot x, 2\right)}{y}\\ \mathbf{elif}\;t\_1 \leq 0.9999999999999999:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right) \cdot x, x, 1\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot 1\\ \end{array} \end{array} \]

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
         (fma 0.002777777777777778 (* x x) 0.08333333333333333)
         (* x x)
         1.0)
        (* x x)
        2.0))
      y)
     (if (<= t_1 0.9999999999999999)
       (* (fma (* (fma (* x x) 0.041666666666666664 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(fma(0.002777777777777778, (x * x), 0.08333333333333333), (x * x), 1.0), (x * x), 2.0)) / y;
	} else if (t_1 <= 0.9999999999999999) {
		tmp = fma((fma((x * x), 0.041666666666666664, 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(Float64(fma(Float64(y * y), -0.08333333333333333, 0.5) * y) * fma(fma(fma(0.002777777777777778, Float64(x * x), 0.08333333333333333), Float64(x * x), 1.0), Float64(x * x), 2.0)) / y);
	elseif (t_1 <= 0.9999999999999999)
		tmp = Float64(fma(Float64(fma(Float64(x * x), 0.041666666666666664, 0.5) * 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[(N[(y * y), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision] * y), $MachinePrecision] * 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]), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[t$95$1, 0.9999999999999999], N[(N[(N[(N[(N[(x * x), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision] * x), $MachinePrecision] * x + 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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 85.5

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

   8. Applied rewrites 85.5%

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

   9. Taylor expanded in y around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. pow2 N/A

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

       7. lift-*.f64 97.7

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

   11. Applied rewrites 97.7%

       \[\leadsto \left(\mathsf{fma}\left(y \cdot y, -0.08333333333333333, 0.5\right) \cdot y\right) \cdot \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, x \cdot x, 0.08333333333333333\right), x \cdot x, 1\right), x \cdot x, 2\right)}}{y} \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-*r/ N/A

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

       4. lower-/.f64 N/A

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

       5. lower-*.f64 100.0

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

   13. Applied rewrites 100.0%

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

   if -inf.0 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < 0.999999999999999889

   1. Initial program 99.6%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   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 98.5

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

   5. Applied rewrites 98.5%

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

      1. lift-fma.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 98.5

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

   7. Applied rewrites 98.5%

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

   if 0.999999999999999889 < (*.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: 99.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ t_1 := \cosh x \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(y \cdot y, -0.08333333333333333, 0.5\right) \cdot y\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, x \cdot x, 0.08333333333333333\right), x \cdot x, 1\right), x \cdot x, 2\right)}{y}\\ \mathbf{elif}\;t\_1 \leq 0.9999999999999999:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, 0.5, 1\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot 1\\ \end{array} \end{array} \]

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
         (fma 0.002777777777777778 (* x x) 0.08333333333333333)
         (* x x)
         1.0)
        (* x x)
        2.0))
      y)
     (if (<= t_1 0.9999999999999999)
       (* (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(fma(0.002777777777777778, (x * x), 0.08333333333333333), (x * x), 1.0), (x * x), 2.0)) / y;
	} else if (t_1 <= 0.9999999999999999) {
		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(Float64(fma(Float64(y * y), -0.08333333333333333, 0.5) * y) * fma(fma(fma(0.002777777777777778, Float64(x * x), 0.08333333333333333), Float64(x * x), 1.0), Float64(x * x), 2.0)) / y);
	elseif (t_1 <= 0.9999999999999999)
		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[(N[(y * y), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision] * y), $MachinePrecision] * 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]), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[t$95$1, 0.9999999999999999], 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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 85.5

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

   8. Applied rewrites 85.5%

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

   9. Taylor expanded in y around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. pow2 N/A

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

       7. lift-*.f64 97.7

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

   11. Applied rewrites 97.7%

       \[\leadsto \left(\mathsf{fma}\left(y \cdot y, -0.08333333333333333, 0.5\right) \cdot y\right) \cdot \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, x \cdot x, 0.08333333333333333\right), x \cdot x, 1\right), x \cdot x, 2\right)}}{y} \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-*r/ N/A

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

       4. lower-/.f64 N/A

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

       5. lower-*.f64 100.0

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

   13. Applied rewrites 100.0%

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

   if -inf.0 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < 0.999999999999999889

   1. Initial program 99.6%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   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 98.0

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

   5. Applied rewrites 98.0%

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

   if 0.999999999999999889 < (*.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: 99.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ t_1 := \cosh x \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(y \cdot y, -0.08333333333333333, 0.5\right) \cdot y\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, x \cdot x, 0.08333333333333333\right), x \cdot x, 1\right), x \cdot x, 2\right)}{y}\\ \mathbf{elif}\;t\_1 \leq 0.9999999999999999:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot 1\\ \end{array} \end{array} \]

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
         (fma 0.002777777777777778 (* x x) 0.08333333333333333)
         (* x x)
         1.0)
        (* x x)
        2.0))
      y)
     (if (<= t_1 0.9999999999999999) 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(fma(0.002777777777777778, (x * x), 0.08333333333333333), (x * x), 1.0), (x * x), 2.0)) / y;
	} else if (t_1 <= 0.9999999999999999) {
		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(Float64(fma(Float64(y * y), -0.08333333333333333, 0.5) * y) * fma(fma(fma(0.002777777777777778, Float64(x * x), 0.08333333333333333), Float64(x * x), 1.0), Float64(x * x), 2.0)) / y);
	elseif (t_1 <= 0.9999999999999999)
		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[(N[(y * y), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision] * y), $MachinePrecision] * 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]), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[t$95$1, 0.9999999999999999], 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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 85.5

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

   8. Applied rewrites 85.5%

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

   9. Taylor expanded in y around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. pow2 N/A

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

       7. lift-*.f64 97.7

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

   11. Applied rewrites 97.7%

       \[\leadsto \left(\mathsf{fma}\left(y \cdot y, -0.08333333333333333, 0.5\right) \cdot y\right) \cdot \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, x \cdot x, 0.08333333333333333\right), x \cdot x, 1\right), x \cdot x, 2\right)}}{y} \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-*r/ N/A

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

       4. lower-/.f64 N/A

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

       5. lower-*.f64 100.0

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

   13. Applied rewrites 100.0%

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

   if -inf.0 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < 0.999999999999999889

   1. Initial program 99.6%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   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 96.9

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

   5. Applied rewrites 96.9%

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

   if 0.999999999999999889 < (*.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 6: 75.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -5 \cdot 10^{-154}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(y \cdot y, -0.08333333333333333, 0.5\right) \cdot y\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, x \cdot x, 0.08333333333333333\right), x \cdot x, 1\right), x \cdot x, 2\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cosh x) (/ (sin y) y)) -5e-154)
   (/
    (*
     (* (fma (* y y) -0.08333333333333333 0.5) y)
     (fma
      (fma (fma 0.002777777777777778 (* x x) 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-154) {
		tmp = ((fma((y * y), -0.08333333333333333, 0.5) * y) * fma(fma(fma(0.002777777777777778, (x * x), 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-154)
		tmp = Float64(Float64(Float64(fma(Float64(y * y), -0.08333333333333333, 0.5) * y) * fma(fma(fma(0.002777777777777778, Float64(x * x), 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-154], N[(N[(N[(N[(N[(y * y), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision] * y), $MachinePrecision] * 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]), $MachinePrecision] / y), $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-154

   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 99.9

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

   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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 89.5

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

   8. Applied rewrites 89.5%

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

   9. Taylor expanded in y around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. pow2 N/A

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

       7. lift-*.f64 71.0

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

   11. Applied rewrites 71.0%

       \[\leadsto \left(\mathsf{fma}\left(y \cdot y, -0.08333333333333333, 0.5\right) \cdot y\right) \cdot \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, x \cdot x, 0.08333333333333333\right), x \cdot x, 1\right), x \cdot x, 2\right)}}{y} \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-*r/ N/A

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

       4. lower-/.f64 N/A

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

       5. lower-*.f64 72.6

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

   13. Applied rewrites 72.6%

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

   if -5.0000000000000002e-154 < (*.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 74.1%

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

Alternative 7: 72.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, x \cdot x, 0.08333333333333333\right), x \cdot x, 1\right), x \cdot x, 2\right)\\ \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -5 \cdot 10^{-302}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(y \cdot y, -0.08333333333333333, 0.5\right) \cdot y\right) \cdot t\_0}{y}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.004166666666666667 \cdot \left(y \cdot y\right) - 0.08333333333333333, y \cdot y, 0.5\right) \cdot y\right) \cdot \frac{t\_0}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (fma
          (fma
           (fma 0.002777777777777778 (* x x) 0.08333333333333333)
           (* x x)
           1.0)
          (* x x)
          2.0)))
   (if (<= (* (cosh x) (/ (sin y) y)) -5e-302)
     (/ (* (* (fma (* y y) -0.08333333333333333 0.5) y) t_0) y)
     (*
      (*
       (fma
        (- (* 0.004166666666666667 (* y y)) 0.08333333333333333)
        (* y y)
        0.5)
       y)
      (/ t_0 y)))))

C

double code(double x, double y) {
	double t_0 = fma(fma(fma(0.002777777777777778, (x * x), 0.08333333333333333), (x * x), 1.0), (x * x), 2.0);
	double tmp;
	if ((cosh(x) * (sin(y) / y)) <= -5e-302) {
		tmp = ((fma((y * y), -0.08333333333333333, 0.5) * y) * t_0) / y;
	} else {
		tmp = (fma(((0.004166666666666667 * (y * y)) - 0.08333333333333333), (y * y), 0.5) * y) * (t_0 / y);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(fma(fma(0.002777777777777778, Float64(x * x), 0.08333333333333333), Float64(x * x), 1.0), Float64(x * x), 2.0)
	tmp = 0.0
	if (Float64(cosh(x) * Float64(sin(y) / y)) <= -5e-302)
		tmp = Float64(Float64(Float64(fma(Float64(y * y), -0.08333333333333333, 0.5) * y) * t_0) / y);
	else
		tmp = Float64(Float64(fma(Float64(Float64(0.004166666666666667 * Float64(y * y)) - 0.08333333333333333), Float64(y * y), 0.5) * y) * Float64(t_0 / y));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = 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]}, If[LessEqual[N[(N[Cosh[x], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], -5e-302], N[(N[(N[(N[(N[(y * y), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision] * y), $MachinePrecision] * t$95$0), $MachinePrecision] / y), $MachinePrecision], 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[(t$95$0 / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < -5.00000000000000033e-302

   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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 91.8

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

   8. Applied rewrites 91.8%

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

   9. Taylor expanded in y around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. pow2 N/A

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

       7. lift-*.f64 54.9

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

   11. Applied rewrites 54.9%

       \[\leadsto \left(\mathsf{fma}\left(y \cdot y, -0.08333333333333333, 0.5\right) \cdot y\right) \cdot \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, x \cdot x, 0.08333333333333333\right), x \cdot x, 1\right), x \cdot x, 2\right)}}{y} \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-*r/ N/A

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

       4. lower-/.f64 N/A

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

       5. lower-*.f64 56.2

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

   13. Applied rewrites 56.2%

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

   if -5.00000000000000033e-302 < (*.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 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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 93.3

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

   8. Applied rewrites 93.3%

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

   9. Taylor expanded in y around 0

      \[\leadsto \left(y \cdot \left(\frac{1}{2} + {y}^{2} \cdot \left(\frac{1}{240} \cdot {y}^{2} - \frac{1}{12}\right)\right)\right) \cdot \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, x \cdot x, \frac{1}{12}\right), x \cdot x, 1\right), x \cdot x, 2\right)}}{y} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(\left(\frac{1}{2} + {y}^{2} \cdot \left(\frac{1}{240} \cdot {y}^{2} - \frac{1}{12}\right)\right) \cdot y\right) \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, x \cdot x, \frac{1}{12}\right), x \cdot x, 1\right), \color{blue}{x \cdot x}, 2\right)}{y} \]

       2. lower-*.f64 N/A

          \[\leadsto \left(\left(\frac{1}{2} + {y}^{2} \cdot \left(\frac{1}{240} \cdot {y}^{2} - \frac{1}{12}\right)\right) \cdot y\right) \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, x \cdot x, \frac{1}{12}\right), x \cdot x, 1\right), \color{blue}{x \cdot x}, 2\right)}{y} \]

       3. +-commutative N/A

          \[\leadsto \left(\left({y}^{2} \cdot \left(\frac{1}{240} \cdot {y}^{2} - \frac{1}{12}\right) + \frac{1}{2}\right) \cdot y\right) \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, x \cdot x, \frac{1}{12}\right), x \cdot x, 1\right), \color{blue}{x} \cdot x, 2\right)}{y} \]

       4. *-commutative N/A

          \[\leadsto \left(\left(\left(\frac{1}{240} \cdot {y}^{2} - \frac{1}{12}\right) \cdot {y}^{2} + \frac{1}{2}\right) \cdot y\right) \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, x \cdot x, \frac{1}{12}\right), x \cdot x, 1\right), x \cdot x, 2\right)}{y} \]

       5. lower-fma.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{240} \cdot {y}^{2} - \frac{1}{12}, {y}^{2}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, x \cdot x, \frac{1}{12}\right), x \cdot x, 1\right), \color{blue}{x} \cdot x, 2\right)}{y} \]

       6. lower--.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{240} \cdot {y}^{2} - \frac{1}{12}, {y}^{2}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, x \cdot x, \frac{1}{12}\right), x \cdot x, 1\right), x \cdot x, 2\right)}{y} \]

       7. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{240} \cdot {y}^{2} - \frac{1}{12}, {y}^{2}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, x \cdot x, \frac{1}{12}\right), x \cdot x, 1\right), x \cdot x, 2\right)}{y} \]

       8. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{240} \cdot \left(y \cdot y\right) - \frac{1}{12}, {y}^{2}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, x \cdot x, \frac{1}{12}\right), x \cdot x, 1\right), x \cdot x, 2\right)}{y} \]

       9. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{240} \cdot \left(y \cdot y\right) - \frac{1}{12}, {y}^{2}, \frac{1}{2}\right) \cdot y\right) \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, x \cdot x, \frac{1}{12}\right), x \cdot x, 1\right), x \cdot x, 2\right)}{y} \]

       10. pow2 N/A

           \[\leadsto \left(\mathsf{fma}\left(\frac{1}{240} \cdot \left(y \cdot y\right) - \frac{1}{12}, y \cdot y, \frac{1}{2}\right) \cdot y\right) \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, x \cdot x, \frac{1}{12}\right), x \cdot x, 1\right), x \cdot x, 2\right)}{y} \]

       11. lift-*.f64 75.6

           \[\leadsto \left(\mathsf{fma}\left(0.004166666666666667 \cdot \left(y \cdot y\right) - 0.08333333333333333, y \cdot y, 0.5\right) \cdot y\right) \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, x \cdot x, 0.08333333333333333\right), x \cdot x, 1\right), x \cdot x, 2\right)}{y} \]

   11. Applied rewrites 75.6%

       \[\leadsto \left(\mathsf{fma}\left(0.004166666666666667 \cdot \left(y \cdot y\right) - 0.08333333333333333, y \cdot y, 0.5\right) \cdot y\right) \cdot \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, x \cdot x, 0.08333333333333333\right), x \cdot x, 1\right), x \cdot x, 2\right)}}{y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 70.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, x \cdot x, 0.08333333333333333\right), x \cdot x, 1\right), x \cdot x, 2\right)\\ \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -5 \cdot 10^{-154}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(y \cdot y, -0.08333333333333333, 0.5\right) \cdot y\right) \cdot t\_0}{y}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot y\right) \cdot \frac{t\_0}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (fma
          (fma
           (fma 0.002777777777777778 (* x x) 0.08333333333333333)
           (* x x)
           1.0)
          (* x x)
          2.0)))
   (if (<= (* (cosh x) (/ (sin y) y)) -5e-154)
     (/ (* (* (fma (* y y) -0.08333333333333333 0.5) y) t_0) y)
     (* (* 0.5 y) (/ t_0 y)))))

C

double code(double x, double y) {
	double t_0 = fma(fma(fma(0.002777777777777778, (x * x), 0.08333333333333333), (x * x), 1.0), (x * x), 2.0);
	double tmp;
	if ((cosh(x) * (sin(y) / y)) <= -5e-154) {
		tmp = ((fma((y * y), -0.08333333333333333, 0.5) * y) * t_0) / y;
	} else {
		tmp = (0.5 * y) * (t_0 / y);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(fma(fma(0.002777777777777778, Float64(x * x), 0.08333333333333333), Float64(x * x), 1.0), Float64(x * x), 2.0)
	tmp = 0.0
	if (Float64(cosh(x) * Float64(sin(y) / y)) <= -5e-154)
		tmp = Float64(Float64(Float64(fma(Float64(y * y), -0.08333333333333333, 0.5) * y) * t_0) / y);
	else
		tmp = Float64(Float64(0.5 * y) * Float64(t_0 / y));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = 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]}, If[LessEqual[N[(N[Cosh[x], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], -5e-154], N[(N[(N[(N[(N[(y * y), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision] * y), $MachinePrecision] * t$95$0), $MachinePrecision] / y), $MachinePrecision], N[(N[(0.5 * y), $MachinePrecision] * N[(t$95$0 / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < -5.0000000000000002e-154

   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 99.9

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

   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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 89.5

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

   8. Applied rewrites 89.5%

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

   9. Taylor expanded in y around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. pow2 N/A

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

       7. lift-*.f64 71.0

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

   11. Applied rewrites 71.0%

       \[\leadsto \left(\mathsf{fma}\left(y \cdot y, -0.08333333333333333, 0.5\right) \cdot y\right) \cdot \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, x \cdot x, 0.08333333333333333\right), x \cdot x, 1\right), x \cdot x, 2\right)}}{y} \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-*r/ N/A

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

       4. lower-/.f64 N/A

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

       5. lower-*.f64 72.6

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

   13. Applied rewrites 72.6%

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

   if -5.0000000000000002e-154 < (*.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 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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 93.8

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

   8. Applied rewrites 93.8%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 68.4%

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

Alternative 9: 70.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -5 \cdot 10^{-154}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot y\right) \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, x \cdot x, 0.08333333333333333\right), x \cdot x, 1\right), x \cdot x, 2\right)}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cosh x) (/ (sin y) y)) -5e-154)
   (*
    (fma
     (fma (fma 0.001388888888888889 (* x x) 0.041666666666666664) (* x x) 0.5)
     (* x x)
     1.0)
    (fma -0.16666666666666666 (* y y) 1.0))
   (*
    (* 0.5 y)
    (/
     (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 tmp;
	if ((cosh(x) * (sin(y) / y)) <= -5e-154) {
		tmp = fma(fma(fma(0.001388888888888889, (x * x), 0.041666666666666664), (x * x), 0.5), (x * x), 1.0) * fma(-0.16666666666666666, (y * y), 1.0);
	} else {
		tmp = (0.5 * y) * (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)
	tmp = 0.0
	if (Float64(cosh(x) * Float64(sin(y) / y)) <= -5e-154)
		tmp = Float64(fma(fma(fma(0.001388888888888889, Float64(x * x), 0.041666666666666664), Float64(x * x), 0.5), Float64(x * x), 1.0) * fma(-0.16666666666666666, Float64(y * y), 1.0));
	else
		tmp = Float64(Float64(0.5 * y) * 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_] := If[LessEqual[N[(N[Cosh[x], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], -5e-154], 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] * N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * y), $MachinePrecision] * 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 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < -5.0000000000000002e-154

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

      \[\leadsto \cosh x \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {y}^{2}\right)} \]
   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 72.7

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

   5. Applied rewrites 72.7%

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

   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. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 71.0

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

   8. Applied rewrites 71.0%

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

   if -5.0000000000000002e-154 < (*.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 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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 93.8

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

   8. Applied rewrites 93.8%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 68.4%

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

Alternative 10: 68.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -5 \cdot 10^{-154}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right) \cdot x, x, 0.5\right), x \cdot x, 1\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cosh x) (/ (sin y) y)) -5e-154)
   (*
    (fma
     (fma (fma 0.001388888888888889 (* x x) 0.041666666666666664) (* x x) 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)
    1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((cosh(x) * (sin(y) / y)) <= -5e-154) {
		tmp = fma(fma(fma(0.001388888888888889, (x * x), 0.041666666666666664), (x * x), 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) * 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(cosh(x) * Float64(sin(y) / y)) <= -5e-154)
		tmp = Float64(fma(fma(fma(0.001388888888888889, Float64(x * x), 0.041666666666666664), Float64(x * x), 0.5), Float64(x * x), 1.0) * fma(-0.16666666666666666, Float64(y * y), 1.0));
	else
		tmp = Float64(fma(fma(Float64(fma(Float64(x * x), 0.001388888888888889, 0.041666666666666664) * x), x, 0.5), Float64(x * x), 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-154], 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] * N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(x * x), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision] * x), $MachinePrecision] * x + 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 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-154

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

      \[\leadsto \cosh x \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {y}^{2}\right)} \]
   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 72.7

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

   5. Applied rewrites 72.7%

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

   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. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 71.0

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

   8. Applied rewrites 71.0%

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

   if -5.0000000000000002e-154 < (*.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 74.1%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 66.7

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

   8. Applied rewrites 66.7%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-*.f64 66.7

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

   10. Applied rewrites 66.7%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right) \cdot x, x, 0.5\right), \color{blue}{x} \cdot x, 1\right) \cdot 1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 67.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -5 \cdot 10^{-154}:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot 0.5\right) \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right) \cdot x, x, 0.5\right), x \cdot x, 1\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cosh x) (/ (sin y) y)) -5e-154)
   (* (* (* x x) 0.5) (fma -0.16666666666666666 (* y y) 1.0))
   (*
    (fma
     (fma (* (fma (* x x) 0.001388888888888889 0.041666666666666664) x) x 0.5)
     (* x x)
     1.0)
    1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((cosh(x) * (sin(y) / y)) <= -5e-154) {
		tmp = ((x * x) * 0.5) * 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) * 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(cosh(x) * Float64(sin(y) / y)) <= -5e-154)
		tmp = Float64(Float64(Float64(x * x) * 0.5) * fma(-0.16666666666666666, Float64(y * y), 1.0));
	else
		tmp = Float64(fma(fma(Float64(fma(Float64(x * x), 0.001388888888888889, 0.041666666666666664) * x), x, 0.5), Float64(x * x), 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-154], N[(N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision] * N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(x * x), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision] * x), $MachinePrecision] * x + 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 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-154

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

      \[\leadsto \cosh x \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {y}^{2}\right)} \]
   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 72.7

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

   5. Applied rewrites 72.7%

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

   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 67.6

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

   8. Applied rewrites 67.6%

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

   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 67.6

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

   11. Applied rewrites 67.6%

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

   if -5.0000000000000002e-154 < (*.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 74.1%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 66.7

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

   8. Applied rewrites 66.7%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-*.f64 66.7

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

   10. Applied rewrites 66.7%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right) \cdot x, x, 0.5\right), \color{blue}{x} \cdot x, 1\right) \cdot 1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 67.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -5 \cdot 10^{-154}:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot 0.5\right) \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left(x \cdot x\right) \cdot 0.001388888888888889, x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cosh x) (/ (sin y) y)) -5e-154)
   (* (* (* x x) 0.5) (fma -0.16666666666666666 (* y y) 1.0))
   (*
    (fma (fma (* (* x x) 0.001388888888888889) (* x x) 0.5) (* x x) 1.0)
    1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((cosh(x) * (sin(y) / y)) <= -5e-154) {
		tmp = ((x * x) * 0.5) * fma(-0.16666666666666666, (y * y), 1.0);
	} else {
		tmp = fma(fma(((x * x) * 0.001388888888888889), (x * x), 0.5), (x * x), 1.0) * 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(cosh(x) * Float64(sin(y) / y)) <= -5e-154)
		tmp = Float64(Float64(Float64(x * x) * 0.5) * fma(-0.16666666666666666, Float64(y * y), 1.0));
	else
		tmp = Float64(fma(fma(Float64(Float64(x * x) * 0.001388888888888889), Float64(x * x), 0.5), Float64(x * x), 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-154], N[(N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision] * N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(x * x), $MachinePrecision] * 0.001388888888888889), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 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-154

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

      \[\leadsto \cosh x \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {y}^{2}\right)} \]
   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 72.7

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

   5. Applied rewrites 72.7%

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

   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 67.6

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

   8. Applied rewrites 67.6%

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

   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 67.6

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

   11. Applied rewrites 67.6%

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

   if -5.0000000000000002e-154 < (*.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 74.1%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 66.7

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

   8. Applied rewrites 66.7%

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

   9. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720} \cdot {x}^{2}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot 1 \]
   10. Step-by-step derivation

       1. pow2 N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 N/A

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

       4. lift-*.f64 66.6

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

   11. Applied rewrites 66.6%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(x \cdot x\right) \cdot 0.001388888888888889, x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot 1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 64.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -5 \cdot 10^{-154}:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot 0.5\right) \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right), x \cdot x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cosh x) (/ (sin y) y)) -5e-154)
   (* (* (* x x) 0.5) (fma -0.16666666666666666 (* y y) 1.0))
   (fma (fma (* x x) 0.041666666666666664 0.5) (* x x) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((cosh(x) * (sin(y) / y)) <= -5e-154) {
		tmp = ((x * x) * 0.5) * fma(-0.16666666666666666, (y * y), 1.0);
	} else {
		tmp = fma(fma((x * x), 0.041666666666666664, 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-154)
		tmp = Float64(Float64(Float64(x * x) * 0.5) * fma(-0.16666666666666666, Float64(y * y), 1.0));
	else
		tmp = fma(fma(Float64(x * x), 0.041666666666666664, 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-154], N[(N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision] * N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * x), $MachinePrecision] * 0.041666666666666664 + 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-154

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

      \[\leadsto \cosh x \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {y}^{2}\right)} \]
   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 72.7

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

   5. Applied rewrites 72.7%

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

   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 67.6

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

   8. Applied rewrites 67.6%

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

   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 67.6

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

   11. Applied rewrites 67.6%

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

   if -5.0000000000000002e-154 < (*.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

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 88.6%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 64.2

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

   8. Applied rewrites 64.2%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right), \color{blue}{x \cdot x}, 1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 60.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -5 \cdot 10^{-154}:\\ \;\;\;\;1 \cdot \left(\left(y \cdot y\right) \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right), x \cdot x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cosh x) (/ (sin y) y)) -5e-154)
   (* 1.0 (* (* y y) -0.16666666666666666))
   (fma (fma (* x x) 0.041666666666666664 0.5) (* x x) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((cosh(x) * (sin(y) / y)) <= -5e-154) {
		tmp = 1.0 * ((y * y) * -0.16666666666666666);
	} else {
		tmp = fma(fma((x * x), 0.041666666666666664, 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-154)
		tmp = Float64(1.0 * Float64(Float64(y * y) * -0.16666666666666666));
	else
		tmp = fma(fma(Float64(x * x), 0.041666666666666664, 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-154], N[(1.0 * N[(N[(y * y), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * x), $MachinePrecision] * 0.041666666666666664 + 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-154

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

      \[\leadsto \cosh x \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {y}^{2}\right)} \]
   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 72.7

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

   5. Applied rewrites 72.7%

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

   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 67.6

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

   8. Applied rewrites 67.6%

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

   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 40.9%

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

   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 40.9

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

   14. Applied rewrites 40.9%

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

   if -5.0000000000000002e-154 < (*.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

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 88.6%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 64.2

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

   8. Applied rewrites 64.2%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right), \color{blue}{x \cdot x}, 1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 51.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -5 \cdot 10^{-154}:\\ \;\;\;\;1 \cdot \left(\left(y \cdot y\right) \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, 0.5, 1\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cosh x) (/ (sin y) y)) -5e-154)
   (* 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-154) {
		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-154)
		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-154], 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-154

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

      \[\leadsto \cosh x \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {y}^{2}\right)} \]
   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 72.7

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

   5. Applied rewrites 72.7%

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

   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 67.6

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

   8. Applied rewrites 67.6%

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

   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 40.9%

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

   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 40.9

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

   14. Applied rewrites 40.9%

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

   if -5.0000000000000002e-154 < (*.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 74.1%

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

   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 58.5

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

   8. Applied rewrites 58.5%

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

Alternative 16: 32.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -5 \cdot 10^{-154}:\\ \;\;\;\;1 \cdot \left(\left(y \cdot y\right) \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cosh x) (/ (sin y) y)) -5e-154)
   (* 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-154) {
		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-154)) 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-154) {
		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-154:
		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-154)
		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-154)
		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-154], 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-154

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

      \[\leadsto \cosh x \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {y}^{2}\right)} \]
   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 72.7

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

   5. Applied rewrites 72.7%

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

   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 67.6

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

   8. Applied rewrites 67.6%

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

   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 40.9%

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

   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 40.9

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

   14. Applied rewrites 40.9%

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

   if -5.0000000000000002e-154 < (*.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 74.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 34.9%

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

Alternative 17: 31.7% accurate, 12.8× speedup

Math

\[\begin{array}{l} \\ 1 \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right) \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (* 1.0 (fma -0.16666666666666666 (* y y) 1.0)))

C

double code(double x, double y) {
	return 1.0 * fma(-0.16666666666666666, (y * y), 1.0);
}

Julia

function code(x, y)
	return Float64(1.0 * fma(-0.16666666666666666, Float64(y * y), 1.0))
end

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in y around 0

   \[\leadsto \cosh x \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {y}^{2}\right)} \]
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 61.5

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

5. Applied rewrites 61.5%

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

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 35.3%

   \[\leadsto \color{blue}{1} \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right) \]
9. Add Preprocessing

Alternative 18: 26.0% 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 58.0%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 27.5%

   \[\leadsto \color{blue}{1} \cdot 1 \]
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 2025084 
(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)))