Linear.Quaternion:$csinh from linear-1.19.1.3

Percentage Accurate: 99.9% → 99.9%

Time: 4.0s

Alternatives: 19

Speedup: 1.0×

• 49.4% 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: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ t_1 := \cosh x \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\cosh x \cdot \left(\left(y \cdot y\right) \cdot -0.16666666666666666\right)\\ \mathbf{elif}\;t\_1 \leq 0.9999999999999956:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right) \cdot x, x, 0.5\right), x \cdot x, 1\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)) (t_1 (* (cosh x) t_0)))
   (if (<= t_1 (- INFINITY))
     (* (cosh x) (* (* y y) -0.16666666666666666))
     (if (<= t_1 0.9999999999999956)
       (*
        (fma
         (fma
          (* (fma (* x x) 0.001388888888888889 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 = cosh(x) * ((y * y) * -0.16666666666666666);
	} else if (t_1 <= 0.9999999999999956) {
		tmp = fma(fma((fma((x * x), 0.001388888888888889, 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(cosh(x) * Float64(Float64(y * y) * -0.16666666666666666));
	elseif (t_1 <= 0.9999999999999956)
		tmp = Float64(fma(fma(Float64(fma(Float64(x * x), 0.001388888888888889, 0.041666666666666664) * 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[Cosh[x], $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.9999999999999956], 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] * 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 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 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 100.0

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

   8. Applied rewrites 100.0%

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

   if -inf.0 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < 0.99999999999999556

   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 98.2

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

   5. Applied rewrites 98.2%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 98.2

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

   7. Applied rewrites 98.2%

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

   if 0.99999999999999556 < (*.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: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ t_1 := \cosh x \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\cosh x \cdot \left(\left(y \cdot y\right) \cdot -0.16666666666666666\right)\\ \mathbf{elif}\;t\_1 \leq 0.9999999999999956:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left(x \cdot x\right) \cdot 0.001388888888888889, x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)) (t_1 (* (cosh x) t_0)))
   (if (<= t_1 (- INFINITY))
     (* (cosh x) (* (* y y) -0.16666666666666666))
     (if (<= t_1 0.9999999999999956)
       (*
        (fma (fma (* (* x x) 0.001388888888888889) (* 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 = cosh(x) * ((y * y) * -0.16666666666666666);
	} else if (t_1 <= 0.9999999999999956) {
		tmp = fma(fma(((x * x) * 0.001388888888888889), (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(cosh(x) * Float64(Float64(y * y) * -0.16666666666666666));
	elseif (t_1 <= 0.9999999999999956)
		tmp = Float64(fma(fma(Float64(Float64(x * x) * 0.001388888888888889), 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[Cosh[x], $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.9999999999999956], 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] * 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 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 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 100.0

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

   8. Applied rewrites 100.0%

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

   if -inf.0 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < 0.99999999999999556

   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 98.2

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

   5. Applied rewrites 98.2%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 98.2

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

   8. Applied rewrites 98.2%

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

   if 0.99999999999999556 < (*.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: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ t_1 := \cosh x \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\cosh x \cdot \left(\left(y \cdot y\right) \cdot -0.16666666666666666\right)\\ \mathbf{elif}\;t\_1 \leq 0.9999999999999956:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)) (t_1 (* (cosh x) t_0)))
   (if (<= t_1 (- INFINITY))
     (* (cosh x) (* (* y y) -0.16666666666666666))
     (if (<= t_1 0.9999999999999956)
       (* (fma (fma 0.041666666666666664 (* x x) 0.5) (* x x) 1.0) t_0)
       (* (cosh x) 1.0)))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double t_1 = cosh(x) * t_0;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = cosh(x) * ((y * y) * -0.16666666666666666);
	} else if (t_1 <= 0.9999999999999956) {
		tmp = fma(fma(0.041666666666666664, (x * x), 0.5), (x * x), 1.0) * t_0;
	} else {
		tmp = cosh(x) * 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(y) / y)
	t_1 = Float64(cosh(x) * t_0)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(cosh(x) * Float64(Float64(y * y) * -0.16666666666666666));
	elseif (t_1 <= 0.9999999999999956)
		tmp = Float64(fma(fma(0.041666666666666664, Float64(x * x), 0.5), Float64(x * x), 1.0) * t_0);
	else
		tmp = Float64(cosh(x) * 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cosh[x], $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[Cosh[x], $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.9999999999999956], N[(N[(N[(0.041666666666666664 * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[Cosh[x], $MachinePrecision] * 1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in 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 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 100.0

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

   8. Applied rewrites 100.0%

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

   if -inf.0 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < 0.99999999999999556

   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 98.2

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

   5. Applied rewrites 98.2%

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

   if 0.99999999999999556 < (*.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: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ t_1 := \cosh x \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\cosh x \cdot \left(\left(y \cdot y\right) \cdot -0.16666666666666666\right)\\ \mathbf{elif}\;t\_1 \leq 0.9999999999999956:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, 0.5, 1\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)) (t_1 (* (cosh x) t_0)))
   (if (<= t_1 (- INFINITY))
     (* (cosh x) (* (* y y) -0.16666666666666666))
     (if (<= t_1 0.9999999999999956)
       (* (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 = cosh(x) * ((y * y) * -0.16666666666666666);
	} else if (t_1 <= 0.9999999999999956) {
		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(cosh(x) * Float64(Float64(y * y) * -0.16666666666666666));
	elseif (t_1 <= 0.9999999999999956)
		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[Cosh[x], $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.9999999999999956], 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 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 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 100.0

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

   8. Applied rewrites 100.0%

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

   if -inf.0 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < 0.99999999999999556

   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 98.2

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

   5. Applied rewrites 98.2%

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

   if 0.99999999999999556 < (*.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: 99.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cosh x \cdot \frac{\sin y}{y}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\cosh x \cdot \left(\left(y \cdot y\right) \cdot -0.16666666666666666\right)\\ \mathbf{elif}\;t\_0 \leq 0.9999999999999956:\\ \;\;\;\;\left(0.5 \cdot \sin y\right) \cdot \frac{2}{y}\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (cosh x) (/ (sin y) y))))
   (if (<= t_0 (- INFINITY))
     (* (cosh x) (* (* y y) -0.16666666666666666))
     (if (<= t_0 0.9999999999999956)
       (* (* 0.5 (sin y)) (/ 2.0 y))
       (* (cosh x) 1.0)))))

C

double code(double x, double y) {
	double t_0 = cosh(x) * (sin(y) / y);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = cosh(x) * ((y * y) * -0.16666666666666666);
	} else if (t_0 <= 0.9999999999999956) {
		tmp = (0.5 * sin(y)) * (2.0 / y);
	} else {
		tmp = cosh(x) * 1.0;
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double t_0 = Math.cosh(x) * (Math.sin(y) / y);
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = Math.cosh(x) * ((y * y) * -0.16666666666666666);
	} else if (t_0 <= 0.9999999999999956) {
		tmp = (0.5 * Math.sin(y)) * (2.0 / y);
	} else {
		tmp = Math.cosh(x) * 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.cosh(x) * (math.sin(y) / y)
	tmp = 0
	if t_0 <= -math.inf:
		tmp = math.cosh(x) * ((y * y) * -0.16666666666666666)
	elif t_0 <= 0.9999999999999956:
		tmp = (0.5 * math.sin(y)) * (2.0 / y)
	else:
		tmp = math.cosh(x) * 1.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(cosh(x) * Float64(sin(y) / y))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(cosh(x) * Float64(Float64(y * y) * -0.16666666666666666));
	elseif (t_0 <= 0.9999999999999956)
		tmp = Float64(Float64(0.5 * sin(y)) * Float64(2.0 / y));
	else
		tmp = Float64(cosh(x) * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = cosh(x) * (sin(y) / y);
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = cosh(x) * ((y * y) * -0.16666666666666666);
	elseif (t_0 <= 0.9999999999999956)
		tmp = (0.5 * sin(y)) * (2.0 / y);
	else
		tmp = cosh(x) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cosh[x], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[Cosh[x], $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.9999999999999956], N[(N[(0.5 * N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(2.0 / y), $MachinePrecision]), $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 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 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 100.0

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

   8. Applied rewrites 100.0%

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

   if -inf.0 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < 0.99999999999999556

   1. Initial program 99.6%

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

   3. Taylor expanded in x around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. rec-exp N/A

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

      8. cosh-undef N/A

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

      9. lower-*.f64 N/A

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

      10. lift-cosh.f64 99.7

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 97.8%

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

   if 0.99999999999999556 < (*.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 7: 99.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ t_1 := \cosh x \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\cosh x \cdot \left(\left(y \cdot y\right) \cdot -0.16666666666666666\right)\\ \mathbf{elif}\;t\_1 \leq 0.9999999999999956:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)) (t_1 (* (cosh x) t_0)))
   (if (<= t_1 (- INFINITY))
     (* (cosh x) (* (* y y) -0.16666666666666666))
     (if (<= t_1 0.9999999999999956) 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 = cosh(x) * ((y * y) * -0.16666666666666666);
	} else if (t_1 <= 0.9999999999999956) {
		tmp = t_0;
	} else {
		tmp = cosh(x) * 1.0;
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double t_0 = Math.sin(y) / y;
	double t_1 = Math.cosh(x) * t_0;
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = Math.cosh(x) * ((y * y) * -0.16666666666666666);
	} else if (t_1 <= 0.9999999999999956) {
		tmp = t_0;
	} else {
		tmp = Math.cosh(x) * 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sin(y) / y
	t_1 = math.cosh(x) * t_0
	tmp = 0
	if t_1 <= -math.inf:
		tmp = math.cosh(x) * ((y * y) * -0.16666666666666666)
	elif t_1 <= 0.9999999999999956:
		tmp = t_0
	else:
		tmp = math.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(cosh(x) * Float64(Float64(y * y) * -0.16666666666666666));
	elseif (t_1 <= 0.9999999999999956)
		tmp = t_0;
	else
		tmp = Float64(cosh(x) * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sin(y) / y;
	t_1 = cosh(x) * t_0;
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = cosh(x) * ((y * y) * -0.16666666666666666);
	elseif (t_1 <= 0.9999999999999956)
		tmp = t_0;
	else
		tmp = cosh(x) * 1.0;
	end
	tmp_2 = 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[Cosh[x], $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.9999999999999956], 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 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 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 100.0

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

   8. Applied rewrites 100.0%

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

   if -inf.0 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < 0.99999999999999556

   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 97.7

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

   5. Applied rewrites 97.7%

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

   if 0.99999999999999556 < (*.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 8: 98.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ t_1 := \cosh x \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{elif}\;t\_1 \leq 0.9999999999999956:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)) (t_1 (* (cosh x) t_0)))
   (if (<= t_1 (- INFINITY))
     (*
      (fma
       (fma
        (fma 0.001388888888888889 (* x x) 0.041666666666666664)
        (* x x)
        0.5)
       (* x x)
       1.0)
      (fma -0.16666666666666666 (* y y) 1.0))
     (if (<= t_1 0.9999999999999956) 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(fma(fma(0.001388888888888889, (x * x), 0.041666666666666664), (x * x), 0.5), (x * x), 1.0) * fma(-0.16666666666666666, (y * y), 1.0);
	} else if (t_1 <= 0.9999999999999956) {
		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(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));
	elseif (t_1 <= 0.9999999999999956)
		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[(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], If[LessEqual[t$95$1, 0.9999999999999956], 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 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 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)} \cdot \mathsf{fma}\left(\frac{-1}{6}, y \cdot y, 1\right) \]
   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 97.5

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

   8. Applied rewrites 97.5%

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

   if -inf.0 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < 0.99999999999999556

   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 97.7

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

   5. Applied rewrites 97.7%

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

   if 0.99999999999999556 < (*.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 9: 74.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -2 \cdot 10^{-152}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cosh x) (/ (sin y) y)) -2e-152)
   (*
    (fma
     (fma (fma 0.001388888888888889 (* x x) 0.041666666666666664) (* x x) 0.5)
     (* x x)
     1.0)
    (fma -0.16666666666666666 (* y y) 1.0))
   (* (cosh x) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((cosh(x) * (sin(y) / y)) <= -2e-152) {
		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 = cosh(x) * 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(cosh(x) * Float64(sin(y) / y)) <= -2e-152)
		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(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], -2e-152], 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[Cosh[x], $MachinePrecision] * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < -2.00000000000000013e-152

   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 69.2

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

   5. Applied rewrites 69.2%

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

   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 67.5

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

   8. Applied rewrites 67.5%

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

   if -2.00000000000000013e-152 < (*.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 79.2%

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

Alternative 10: 69.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-301}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{-95}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(y \cdot y\right) \cdot 0.008333333333333333, y \cdot y, 1\right) \cdot y}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right) \cdot x, x, 0.5\right), x \cdot x, 1\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)))
   (if (<= t_0 -5e-301)
     (*
      (fma
       (fma
        (fma 0.001388888888888889 (* x x) 0.041666666666666664)
        (* x x)
        0.5)
       (* x x)
       1.0)
      (fma -0.16666666666666666 (* y y) 1.0))
     (if (<= t_0 4e-95)
       (/ (* (fma (* (* y y) 0.008333333333333333) (* y y) 1.0) y) y)
       (*
        (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 t_0 = sin(y) / y;
	double tmp;
	if (t_0 <= -5e-301) {
		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 if (t_0 <= 4e-95) {
		tmp = (fma(((y * y) * 0.008333333333333333), (y * y), 1.0) * y) / y;
	} 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)
	t_0 = Float64(sin(y) / y)
	tmp = 0.0
	if (t_0 <= -5e-301)
		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));
	elseif (t_0 <= 4e-95)
		tmp = Float64(Float64(fma(Float64(Float64(y * y) * 0.008333333333333333), Float64(y * y), 1.0) * y) / y);
	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_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-301], 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], If[LessEqual[t$95$0, 4e-95], N[(N[(N[(N[(N[(y * y), $MachinePrecision] * 0.008333333333333333), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision] / y), $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 3 regimes

2. if (/.f64 (sin.f64 y) y) < -5.00000000000000013e-301

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 53.6

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

   5. Applied rewrites 53.6%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. 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 52.3

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

   8. Applied rewrites 52.3%

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

   if -5.00000000000000013e-301 < (/.f64 (sin.f64 y) y) < 3.99999999999999996e-95

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. lift-sin.f64 N/A

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

      2. lift-/.f64 50.9

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

   5. Applied rewrites 50.9%

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

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 51.0

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

   8. Applied rewrites 51.0%

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

   9. Taylor expanded in y around inf

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{120} \cdot {y}^{2}, y \cdot y, 1\right) \cdot y}{y} \]
   10. Step-by-step derivation

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(y \cdot y\right) \cdot \frac{1}{120}, y \cdot y, 1\right) \cdot y}{y} \]

       4. lift-*.f64 51.0

          \[\leadsto \frac{\mathsf{fma}\left(\left(y \cdot y\right) \cdot 0.008333333333333333, y \cdot y, 1\right) \cdot y}{y} \]

   11. Applied rewrites 51.0%

       \[\leadsto \frac{\mathsf{fma}\left(\left(y \cdot y\right) \cdot 0.008333333333333333, y \cdot y, 1\right) \cdot y}{y} \]

   if 3.99999999999999996e-95 < (/.f64 (sin.f64 y) y)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)} \cdot \frac{\sin y}{y} \]
   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 91.9

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

   5. Applied rewrites 91.9%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 91.9

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

   7. Applied rewrites 91.9%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 89.2%

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

Alternative 11: 69.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-301}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{-95}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(y \cdot y\right) \cdot 0.008333333333333333, y \cdot y, 1\right) \cdot y}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right) \cdot x, x, 0.5\right), x \cdot x, 1\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)))
   (if (<= t_0 -5e-301)
     (*
      (fma (fma 0.041666666666666664 (* x x) 0.5) (* x x) 1.0)
      (fma -0.16666666666666666 (* y y) 1.0))
     (if (<= t_0 4e-95)
       (/ (* (fma (* (* y y) 0.008333333333333333) (* y y) 1.0) y) y)
       (*
        (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 t_0 = sin(y) / y;
	double tmp;
	if (t_0 <= -5e-301) {
		tmp = fma(fma(0.041666666666666664, (x * x), 0.5), (x * x), 1.0) * fma(-0.16666666666666666, (y * y), 1.0);
	} else if (t_0 <= 4e-95) {
		tmp = (fma(((y * y) * 0.008333333333333333), (y * y), 1.0) * y) / y;
	} 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)
	t_0 = Float64(sin(y) / y)
	tmp = 0.0
	if (t_0 <= -5e-301)
		tmp = Float64(fma(fma(0.041666666666666664, Float64(x * x), 0.5), Float64(x * x), 1.0) * fma(-0.16666666666666666, Float64(y * y), 1.0));
	elseif (t_0 <= 4e-95)
		tmp = Float64(Float64(fma(Float64(Float64(y * y) * 0.008333333333333333), Float64(y * y), 1.0) * y) / y);
	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_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-301], N[(N[(N[(0.041666666666666664 * 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], If[LessEqual[t$95$0, 4e-95], N[(N[(N[(N[(N[(y * y), $MachinePrecision] * 0.008333333333333333), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision] / y), $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 3 regimes

2. if (/.f64 (sin.f64 y) y) < -5.00000000000000013e-301

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 53.6

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

   5. Applied rewrites 53.6%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 52.2

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

   8. Applied rewrites 52.2%

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

   if -5.00000000000000013e-301 < (/.f64 (sin.f64 y) y) < 3.99999999999999996e-95

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. lift-sin.f64 N/A

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

      2. lift-/.f64 50.9

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

   5. Applied rewrites 50.9%

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

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 51.0

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

   8. Applied rewrites 51.0%

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

   9. Taylor expanded in y around inf

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{120} \cdot {y}^{2}, y \cdot y, 1\right) \cdot y}{y} \]
   10. Step-by-step derivation

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(y \cdot y\right) \cdot \frac{1}{120}, y \cdot y, 1\right) \cdot y}{y} \]

       4. lift-*.f64 51.0

          \[\leadsto \frac{\mathsf{fma}\left(\left(y \cdot y\right) \cdot 0.008333333333333333, y \cdot y, 1\right) \cdot y}{y} \]

   11. Applied rewrites 51.0%

       \[\leadsto \frac{\mathsf{fma}\left(\left(y \cdot y\right) \cdot 0.008333333333333333, y \cdot y, 1\right) \cdot y}{y} \]

   if 3.99999999999999996e-95 < (/.f64 (sin.f64 y) y)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)} \cdot \frac{\sin y}{y} \]
   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 91.9

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

   5. Applied rewrites 91.9%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 91.9

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

   7. Applied rewrites 91.9%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 89.2%

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

Alternative 12: 66.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-301}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{-95}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(y \cdot y\right) \cdot 0.008333333333333333, y \cdot y, 1\right) \cdot y}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right) \cdot x, x, 1\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)))
   (if (<= t_0 -5e-301)
     (*
      (fma (fma 0.041666666666666664 (* x x) 0.5) (* x x) 1.0)
      (fma -0.16666666666666666 (* y y) 1.0))
     (if (<= t_0 4e-95)
       (/ (* (fma (* (* y y) 0.008333333333333333) (* y y) 1.0) y) y)
       (* (fma (* (fma (* x x) 0.041666666666666664 0.5) x) x 1.0) 1.0)))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double tmp;
	if (t_0 <= -5e-301) {
		tmp = fma(fma(0.041666666666666664, (x * x), 0.5), (x * x), 1.0) * fma(-0.16666666666666666, (y * y), 1.0);
	} else if (t_0 <= 4e-95) {
		tmp = (fma(((y * y) * 0.008333333333333333), (y * y), 1.0) * y) / y;
	} else {
		tmp = fma((fma((x * x), 0.041666666666666664, 0.5) * x), x, 1.0) * 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(y) / y)
	tmp = 0.0
	if (t_0 <= -5e-301)
		tmp = Float64(fma(fma(0.041666666666666664, Float64(x * x), 0.5), Float64(x * x), 1.0) * fma(-0.16666666666666666, Float64(y * y), 1.0));
	elseif (t_0 <= 4e-95)
		tmp = Float64(Float64(fma(Float64(Float64(y * y) * 0.008333333333333333), Float64(y * y), 1.0) * y) / y);
	else
		tmp = Float64(fma(Float64(fma(Float64(x * x), 0.041666666666666664, 0.5) * x), x, 1.0) * 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-301], N[(N[(N[(0.041666666666666664 * 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], If[LessEqual[t$95$0, 4e-95], N[(N[(N[(N[(N[(y * y), $MachinePrecision] * 0.008333333333333333), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision] / y), $MachinePrecision], N[(N[(N[(N[(N[(x * x), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision] * x), $MachinePrecision] * x + 1.0), $MachinePrecision] * 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 y) y) < -5.00000000000000013e-301

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 53.6

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

   5. Applied rewrites 53.6%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 52.2

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

   8. Applied rewrites 52.2%

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

   if -5.00000000000000013e-301 < (/.f64 (sin.f64 y) y) < 3.99999999999999996e-95

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. lift-sin.f64 N/A

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

      2. lift-/.f64 50.9

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

   5. Applied rewrites 50.9%

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

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 51.0

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

   8. Applied rewrites 51.0%

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

   9. Taylor expanded in y around inf

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{120} \cdot {y}^{2}, y \cdot y, 1\right) \cdot y}{y} \]
   10. Step-by-step derivation

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(y \cdot y\right) \cdot \frac{1}{120}, y \cdot y, 1\right) \cdot y}{y} \]

       4. lift-*.f64 51.0

          \[\leadsto \frac{\mathsf{fma}\left(\left(y \cdot y\right) \cdot 0.008333333333333333, y \cdot y, 1\right) \cdot y}{y} \]

   11. Applied rewrites 51.0%

       \[\leadsto \frac{\mathsf{fma}\left(\left(y \cdot y\right) \cdot 0.008333333333333333, y \cdot y, 1\right) \cdot y}{y} \]

   if 3.99999999999999996e-95 < (/.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 97.3%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 84.4

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

   8. Applied rewrites 84.4%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 84.4

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

   10. Applied rewrites 84.4%

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

Alternative 13: 63.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-301}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right) \cdot y}{y}\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{-95}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(y \cdot y\right) \cdot 0.008333333333333333, y \cdot y, 1\right) \cdot y}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right) \cdot x, x, 1\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)))
   (if (<= t_0 -5e-301)
     (/ (* (fma (* y y) -0.16666666666666666 1.0) y) y)
     (if (<= t_0 4e-95)
       (/ (* (fma (* (* y y) 0.008333333333333333) (* y y) 1.0) y) y)
       (* (fma (* (fma (* x x) 0.041666666666666664 0.5) x) x 1.0) 1.0)))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double tmp;
	if (t_0 <= -5e-301) {
		tmp = (fma((y * y), -0.16666666666666666, 1.0) * y) / y;
	} else if (t_0 <= 4e-95) {
		tmp = (fma(((y * y) * 0.008333333333333333), (y * y), 1.0) * y) / y;
	} else {
		tmp = fma((fma((x * x), 0.041666666666666664, 0.5) * x), x, 1.0) * 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(y) / y)
	tmp = 0.0
	if (t_0 <= -5e-301)
		tmp = Float64(Float64(fma(Float64(y * y), -0.16666666666666666, 1.0) * y) / y);
	elseif (t_0 <= 4e-95)
		tmp = Float64(Float64(fma(Float64(Float64(y * y) * 0.008333333333333333), Float64(y * y), 1.0) * y) / y);
	else
		tmp = Float64(fma(Float64(fma(Float64(x * x), 0.041666666666666664, 0.5) * x), x, 1.0) * 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-301], N[(N[(N[(N[(y * y), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * y), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[t$95$0, 4e-95], N[(N[(N[(N[(N[(y * y), $MachinePrecision] * 0.008333333333333333), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision] / y), $MachinePrecision], N[(N[(N[(N[(N[(x * x), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision] * x), $MachinePrecision] * x + 1.0), $MachinePrecision] * 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 y) y) < -5.00000000000000013e-301

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. lift-sin.f64 N/A

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

      2. lift-/.f64 47.5

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

   5. Applied rewrites 47.5%

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

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. pow2 N/A

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

      3. +-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 35.5

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

   8. Applied rewrites 35.5%

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

   if -5.00000000000000013e-301 < (/.f64 (sin.f64 y) y) < 3.99999999999999996e-95

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. lift-sin.f64 N/A

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

      2. lift-/.f64 50.9

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

   5. Applied rewrites 50.9%

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

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 51.0

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

   8. Applied rewrites 51.0%

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

   9. Taylor expanded in y around inf

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{120} \cdot {y}^{2}, y \cdot y, 1\right) \cdot y}{y} \]
   10. Step-by-step derivation

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(y \cdot y\right) \cdot \frac{1}{120}, y \cdot y, 1\right) \cdot y}{y} \]

       4. lift-*.f64 51.0

          \[\leadsto \frac{\mathsf{fma}\left(\left(y \cdot y\right) \cdot 0.008333333333333333, y \cdot y, 1\right) \cdot y}{y} \]

   11. Applied rewrites 51.0%

       \[\leadsto \frac{\mathsf{fma}\left(\left(y \cdot y\right) \cdot 0.008333333333333333, y \cdot y, 1\right) \cdot y}{y} \]

   if 3.99999999999999996e-95 < (/.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 97.3%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 84.4

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

   8. Applied rewrites 84.4%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 84.4

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

   10. Applied rewrites 84.4%

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

Alternative 14: 62.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -2 \cdot 10^{-152}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right) \cdot y}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right) \cdot x, x, 1\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cosh x) (/ (sin y) y)) -2e-152)
   (/ (* (fma (* y y) -0.16666666666666666 1.0) y) y)
   (* (fma (* (fma (* x x) 0.041666666666666664 0.5) x) x 1.0) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((cosh(x) * (sin(y) / y)) <= -2e-152) {
		tmp = (fma((y * y), -0.16666666666666666, 1.0) * y) / y;
	} else {
		tmp = fma((fma((x * x), 0.041666666666666664, 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)) <= -2e-152)
		tmp = Float64(Float64(fma(Float64(y * y), -0.16666666666666666, 1.0) * y) / y);
	else
		tmp = Float64(fma(Float64(fma(Float64(x * x), 0.041666666666666664, 0.5) * 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], -2e-152], N[(N[(N[(N[(y * y), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * y), $MachinePrecision] / y), $MachinePrecision], N[(N[(N[(N[(N[(x * x), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision] * x), $MachinePrecision] * x + 1.0), $MachinePrecision] * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < -2.00000000000000013e-152

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. lift-sin.f64 N/A

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

      2. lift-/.f64 31.8

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

   5. Applied rewrites 31.8%

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

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. pow2 N/A

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

      3. +-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 45.6

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

   8. Applied rewrites 45.6%

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

   if -2.00000000000000013e-152 < (*.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 79.2%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 66.1

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

   8. Applied rewrites 66.1%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 66.1

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

   10. Applied rewrites 66.1%

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

Alternative 15: 62.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -2 \cdot 10^{-152}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right) \cdot y}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(x \cdot x\right) \cdot 0.041666666666666664, x \cdot x, 1\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cosh x) (/ (sin y) y)) -2e-152)
   (/ (* (fma (* y y) -0.16666666666666666 1.0) y) y)
   (* (fma (* (* x x) 0.041666666666666664) (* x x) 1.0) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((cosh(x) * (sin(y) / y)) <= -2e-152) {
		tmp = (fma((y * y), -0.16666666666666666, 1.0) * y) / y;
	} else {
		tmp = fma(((x * x) * 0.041666666666666664), (x * x), 1.0) * 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(cosh(x) * Float64(sin(y) / y)) <= -2e-152)
		tmp = Float64(Float64(fma(Float64(y * y), -0.16666666666666666, 1.0) * y) / y);
	else
		tmp = Float64(fma(Float64(Float64(x * x) * 0.041666666666666664), 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], -2e-152], N[(N[(N[(N[(y * y), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * y), $MachinePrecision] / y), $MachinePrecision], N[(N[(N[(N[(x * x), $MachinePrecision] * 0.041666666666666664), $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)) < -2.00000000000000013e-152

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. lift-sin.f64 N/A

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

      2. lift-/.f64 31.8

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

   5. Applied rewrites 31.8%

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

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. pow2 N/A

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

      3. +-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 45.6

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

   8. Applied rewrites 45.6%

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

   if -2.00000000000000013e-152 < (*.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 79.2%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 66.1

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

   8. Applied rewrites 66.1%

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

   9. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 65.7

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

   11. Applied rewrites 65.7%

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

Alternative 16: 60.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -2 \cdot 10^{-152}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(x \cdot x\right) \cdot 0.041666666666666664, x \cdot x, 1\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cosh x) (/ (sin y) y)) -2e-152)
   (* 1.0 (fma -0.16666666666666666 (* y y) 1.0))
   (* (fma (* (* x x) 0.041666666666666664) (* x x) 1.0) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((cosh(x) * (sin(y) / y)) <= -2e-152) {
		tmp = 1.0 * fma(-0.16666666666666666, (y * y), 1.0);
	} else {
		tmp = fma(((x * x) * 0.041666666666666664), (x * x), 1.0) * 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(cosh(x) * Float64(sin(y) / y)) <= -2e-152)
		tmp = Float64(1.0 * fma(-0.16666666666666666, Float64(y * y), 1.0));
	else
		tmp = Float64(fma(Float64(Float64(x * x) * 0.041666666666666664), 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], -2e-152], N[(1.0 * N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(x * x), $MachinePrecision] * 0.041666666666666664), $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)) < -2.00000000000000013e-152

   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 69.2

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

   5. Applied rewrites 69.2%

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

   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 35.2%

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

   if -2.00000000000000013e-152 < (*.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 79.2%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 66.1

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

   8. Applied rewrites 66.1%

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

   9. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 65.7

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

   11. Applied rewrites 65.7%

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

Alternative 17: 51.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -2 \cdot 10^{-152}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, x \cdot x, 1\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cosh x) (/ (sin y) y)) -2e-152)
   (* 1.0 (fma -0.16666666666666666 (* y y) 1.0))
   (* (fma 0.5 (* x x) 1.0) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((cosh(x) * (sin(y) / y)) <= -2e-152) {
		tmp = 1.0 * fma(-0.16666666666666666, (y * y), 1.0);
	} else {
		tmp = fma(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)) <= -2e-152)
		tmp = Float64(1.0 * fma(-0.16666666666666666, Float64(y * y), 1.0));
	else
		tmp = Float64(fma(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], -2e-152], N[(1.0 * N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * 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)) < -2.00000000000000013e-152

   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 69.2

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

   5. Applied rewrites 69.2%

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

   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 35.2%

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

   if -2.00000000000000013e-152 < (*.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 79.2%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 66.1

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

   8. Applied rewrites 66.1%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 54.0%

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

Alternative 18: 31.9% 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 63.8

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

5. Applied rewrites 63.8%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 35.2%

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

Alternative 19: 26.2% 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 62.9%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 29.0%

   \[\leadsto \color{blue}{1} \cdot 1 \]
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 2025028 
(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)))