Linear.Quaternion:$csinh from linear-1.19.1.3

Percentage Accurate: 99.9% → 99.9%

Time: 4.2s

Alternatives: 21

Speedup: 1.0×

• 48.7% 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, 0.7× speedup

Math

\[\begin{array}{l} \\ \frac{e^{-x} + e^{x}}{2} \cdot \frac{\sin y}{y} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (* (/ (+ (exp (- x)) (exp x)) 2.0) (/ (sin y) y)))

C

double code(double x, double y) {
	return ((exp(-x) + exp(x)) / 2.0) * (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 = ((exp(-x) + exp(x)) / 2.0d0) * (sin(y) / y)
end function

Java

public static double code(double x, double y) {
	return ((Math.exp(-x) + Math.exp(x)) / 2.0) * (Math.sin(y) / y);
}

Python

def code(x, y):
	return ((math.exp(-x) + math.exp(x)) / 2.0) * (math.sin(y) / y)

Julia

function code(x, y)
	return Float64(Float64(Float64(exp(Float64(-x)) + exp(x)) / 2.0) * Float64(sin(y) / y))
end

MATLAB

function tmp = code(x, y)
	tmp = ((exp(-x) + exp(x)) / 2.0) * (sin(y) / y);
end

Wolfram

code[x_, y_] := N[(N[(N[(N[Exp[(-x)], $MachinePrecision] + N[Exp[x], $MachinePrecision]), $MachinePrecision] / 2.0), $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. Step-by-step derivation

   1. lift-cosh.f64 N/A

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

   2. cosh-neg-rev N/A

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

   3. cosh-def N/A

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

   4. lower-/.f64 N/A

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

   5. rec-exp N/A

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

   6. lower-+.f64 N/A

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

   7. rec-exp N/A

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

   8. lower-exp.f64 N/A

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

   9. lower-neg.f64 N/A

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

   10. lower-exp.f64 N/A

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

   11. lower-neg.f64 N/A

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

   12. lower-neg.f64 99.9

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

4. Applied rewrites 99.9%

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

5. Final simplification 99.9%

   \[\leadsto \frac{e^{-x} + e^{x}}{2} \cdot \frac{\sin y}{y} \]
6. Add Preprocessing

Alternative 2: 99.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ t_1 := \cosh x \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, y \cdot y, 0.008333333333333333\right) \cdot \left(y \cdot y\right) - 0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{elif}\;t\_1 \leq 0.05:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, 0.5, 1\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot \mathsf{fma}\left(0.008333333333333333 \cdot \left(y \cdot y\right) - 0.16666666666666666, y \cdot y, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)) (t_1 (* (cosh x) t_0)))
   (if (<= t_1 (- INFINITY))
     (*
      (fma (fma 0.041666666666666664 (* x x) 0.5) (* x x) 1.0)
      (fma
       (-
        (* (fma -0.0001984126984126984 (* y y) 0.008333333333333333) (* y y))
        0.16666666666666666)
       (* y y)
       1.0))
     (if (<= t_1 0.05)
       (* (fma (* x x) 0.5 1.0) t_0)
       (*
        (cosh x)
        (fma
         (- (* 0.008333333333333333 (* y y)) 0.16666666666666666)
         (* y y)
         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(0.041666666666666664, (x * x), 0.5), (x * x), 1.0) * fma(((fma(-0.0001984126984126984, (y * y), 0.008333333333333333) * (y * y)) - 0.16666666666666666), (y * y), 1.0);
	} else if (t_1 <= 0.05) {
		tmp = fma((x * x), 0.5, 1.0) * t_0;
	} else {
		tmp = cosh(x) * fma(((0.008333333333333333 * (y * y)) - 0.16666666666666666), (y * y), 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(0.041666666666666664, Float64(x * x), 0.5), Float64(x * x), 1.0) * fma(Float64(Float64(fma(-0.0001984126984126984, Float64(y * y), 0.008333333333333333) * Float64(y * y)) - 0.16666666666666666), Float64(y * y), 1.0));
	elseif (t_1 <= 0.05)
		tmp = Float64(fma(Float64(x * x), 0.5, 1.0) * t_0);
	else
		tmp = Float64(cosh(x) * fma(Float64(Float64(0.008333333333333333 * Float64(y * y)) - 0.16666666666666666), Float64(y * y), 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[(0.041666666666666664 * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(N[(N[(-0.0001984126984126984 * N[(y * y), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.05], N[(N[(N[(x * x), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[Cosh[x], $MachinePrecision] * N[(N[(N[(0.008333333333333333 * N[(y * y), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 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 54.8

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

   5. Applied rewrites 54.8%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot \left(\left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) - \frac{1}{6}\right) \cdot {y}^{2} + 1\right) \]

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot \mathsf{fma}\left(\left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) \cdot {y}^{2} - \frac{1}{6}, {y}^{2}, 1\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot \mathsf{fma}\left(\left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) \cdot {y}^{2} - \frac{1}{6}, {y}^{2}, 1\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot \mathsf{fma}\left(\left(\frac{-1}{5040} \cdot {y}^{2} + \frac{1}{120}\right) \cdot {y}^{2} - \frac{1}{6}, {y}^{2}, 1\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, {y}^{2}, \frac{1}{120}\right) \cdot {y}^{2} - \frac{1}{6}, {y}^{2}, 1\right) \]

      9. pow2 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot {y}^{2} - \frac{1}{6}, {y}^{2}, 1\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot {y}^{2} - \frac{1}{6}, {y}^{2}, 1\right) \]

      11. pow2 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot \left(y \cdot y\right) - \frac{1}{6}, {y}^{2}, 1\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot \left(y \cdot y\right) - \frac{1}{6}, {y}^{2}, 1\right) \]

      13. pow2 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot \left(y \cdot y\right) - \frac{1}{6}, y \cdot \color{blue}{y}, 1\right) \]

      14. lift-*.f64 97.2

          \[\leadsto \mathsf{fma}\left(x \cdot x, 0.5, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, y \cdot y, 0.008333333333333333\right) \cdot \left(y \cdot y\right) - 0.16666666666666666, y \cdot \color{blue}{y}, 1\right) \]

   8. Applied rewrites 97.2%

      \[\leadsto \mathsf{fma}\left(x \cdot x, 0.5, 1\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, y \cdot y, 0.008333333333333333\right) \cdot \left(y \cdot y\right) - 0.16666666666666666, y \cdot y, 1\right)} \]

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot \left(y \cdot y\right) - \frac{1}{6}, y \cdot y, 1\right) \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \left({x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) + \color{blue}{1}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot \left(y \cdot y\right) - \frac{1}{6}, y \cdot y, 1\right) \]

       2. *-commutative N/A

          \[\leadsto \left(\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) \cdot {x}^{2} + 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot \left(y \cdot y\right) - \frac{1}{6}, y \cdot y, 1\right) \]

       3. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, \color{blue}{{x}^{2}}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot \left(y \cdot y\right) - \frac{1}{6}, y \cdot y, 1\right) \]

       4. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{24} \cdot {x}^{2} + \frac{1}{2}, {\color{blue}{x}}^{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot \left(y \cdot y\right) - \frac{1}{6}, y \cdot y, 1\right) \]

       5. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, {x}^{2}, \frac{1}{2}\right), {\color{blue}{x}}^{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot \left(y \cdot y\right) - \frac{1}{6}, y \cdot y, 1\right) \]

       6. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot \left(y \cdot y\right) - \frac{1}{6}, y \cdot y, 1\right) \]

       7. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot \left(y \cdot y\right) - \frac{1}{6}, y \cdot y, 1\right) \]

       8. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot \color{blue}{x}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot \left(y \cdot y\right) - \frac{1}{6}, y \cdot y, 1\right) \]

       9. lift-*.f64 100.0

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot \color{blue}{x}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, y \cdot y, 0.008333333333333333\right) \cdot \left(y \cdot y\right) - 0.16666666666666666, y \cdot y, 1\right) \]

   11. Applied rewrites 100.0%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, y \cdot y, 0.008333333333333333\right) \cdot \left(y \cdot y\right) - 0.16666666666666666, y \cdot y, 1\right) \]

   if -inf.0 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < 0.050000000000000003

   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 99.6

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

   5. Applied rewrites 99.6%

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

   if 0.050000000000000003 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

Alternative 3: 99.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ t_1 := \cosh x \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, y \cdot y, 0.008333333333333333\right) \cdot \left(y \cdot y\right) - 0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{elif}\;t\_1 \leq 0.9999999999516638:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, 0.5, 1\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)) (t_1 (* (cosh x) t_0)))
   (if (<= t_1 (- INFINITY))
     (*
      (fma (fma 0.041666666666666664 (* x x) 0.5) (* x x) 1.0)
      (fma
       (-
        (* (fma -0.0001984126984126984 (* y y) 0.008333333333333333) (* y y))
        0.16666666666666666)
       (* y y)
       1.0))
     (if (<= t_1 0.9999999999516638)
       (* (fma (* x x) 0.5 1.0) t_0)
       (* (cosh x) 1.0)))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double t_1 = cosh(x) * t_0;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma(fma(0.041666666666666664, (x * x), 0.5), (x * x), 1.0) * fma(((fma(-0.0001984126984126984, (y * y), 0.008333333333333333) * (y * y)) - 0.16666666666666666), (y * y), 1.0);
	} else if (t_1 <= 0.9999999999516638) {
		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(fma(fma(0.041666666666666664, Float64(x * x), 0.5), Float64(x * x), 1.0) * fma(Float64(Float64(fma(-0.0001984126984126984, Float64(y * y), 0.008333333333333333) * Float64(y * y)) - 0.16666666666666666), Float64(y * y), 1.0));
	elseif (t_1 <= 0.9999999999516638)
		tmp = Float64(fma(Float64(x * x), 0.5, 1.0) * t_0);
	else
		tmp = Float64(cosh(x) * 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cosh[x], $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(0.041666666666666664 * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(N[(N[(-0.0001984126984126984 * N[(y * y), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.9999999999516638], N[(N[(N[(x * x), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[Cosh[x], $MachinePrecision] * 1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 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 54.8

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

   5. Applied rewrites 54.8%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot \left(\left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) - \frac{1}{6}\right) \cdot {y}^{2} + 1\right) \]

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot \mathsf{fma}\left(\left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) \cdot {y}^{2} - \frac{1}{6}, {y}^{2}, 1\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot \mathsf{fma}\left(\left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) \cdot {y}^{2} - \frac{1}{6}, {y}^{2}, 1\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot \mathsf{fma}\left(\left(\frac{-1}{5040} \cdot {y}^{2} + \frac{1}{120}\right) \cdot {y}^{2} - \frac{1}{6}, {y}^{2}, 1\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, {y}^{2}, \frac{1}{120}\right) \cdot {y}^{2} - \frac{1}{6}, {y}^{2}, 1\right) \]

      9. pow2 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot {y}^{2} - \frac{1}{6}, {y}^{2}, 1\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot {y}^{2} - \frac{1}{6}, {y}^{2}, 1\right) \]

      11. pow2 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot \left(y \cdot y\right) - \frac{1}{6}, {y}^{2}, 1\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot \left(y \cdot y\right) - \frac{1}{6}, {y}^{2}, 1\right) \]

      13. pow2 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot \left(y \cdot y\right) - \frac{1}{6}, y \cdot \color{blue}{y}, 1\right) \]

      14. lift-*.f64 97.2

          \[\leadsto \mathsf{fma}\left(x \cdot x, 0.5, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, y \cdot y, 0.008333333333333333\right) \cdot \left(y \cdot y\right) - 0.16666666666666666, y \cdot \color{blue}{y}, 1\right) \]

   8. Applied rewrites 97.2%

      \[\leadsto \mathsf{fma}\left(x \cdot x, 0.5, 1\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, y \cdot y, 0.008333333333333333\right) \cdot \left(y \cdot y\right) - 0.16666666666666666, y \cdot y, 1\right)} \]

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot \left(y \cdot y\right) - \frac{1}{6}, y \cdot y, 1\right) \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \left({x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) + \color{blue}{1}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot \left(y \cdot y\right) - \frac{1}{6}, y \cdot y, 1\right) \]

       2. *-commutative N/A

          \[\leadsto \left(\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) \cdot {x}^{2} + 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot \left(y \cdot y\right) - \frac{1}{6}, y \cdot y, 1\right) \]

       3. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, \color{blue}{{x}^{2}}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot \left(y \cdot y\right) - \frac{1}{6}, y \cdot y, 1\right) \]

       4. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{24} \cdot {x}^{2} + \frac{1}{2}, {\color{blue}{x}}^{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot \left(y \cdot y\right) - \frac{1}{6}, y \cdot y, 1\right) \]

       5. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, {x}^{2}, \frac{1}{2}\right), {\color{blue}{x}}^{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot \left(y \cdot y\right) - \frac{1}{6}, y \cdot y, 1\right) \]

       6. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot \left(y \cdot y\right) - \frac{1}{6}, y \cdot y, 1\right) \]

       7. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot \left(y \cdot y\right) - \frac{1}{6}, y \cdot y, 1\right) \]

       8. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot \color{blue}{x}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot \left(y \cdot y\right) - \frac{1}{6}, y \cdot y, 1\right) \]

       9. lift-*.f64 100.0

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot \color{blue}{x}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, y \cdot y, 0.008333333333333333\right) \cdot \left(y \cdot y\right) - 0.16666666666666666, y \cdot y, 1\right) \]

   11. Applied rewrites 100.0%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, y \cdot y, 0.008333333333333333\right) \cdot \left(y \cdot y\right) - 0.16666666666666666, y \cdot y, 1\right) \]

   if -inf.0 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < 0.99999999995166378

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 99.5

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

   5. Applied rewrites 99.5%

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

   if 0.99999999995166378 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 100.0%

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

Alternative 4: 99.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cosh x \cdot \frac{\sin y}{y}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, y \cdot y, 0.008333333333333333\right) \cdot \left(y \cdot y\right) - 0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{elif}\;t\_0 \leq 0.9999999999516638:\\ \;\;\;\;\left(\sin y \cdot \frac{\mathsf{fma}\left(x, x, 2\right)}{y}\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (cosh x) (/ (sin y) y))))
   (if (<= t_0 (- INFINITY))
     (*
      (fma (fma 0.041666666666666664 (* x x) 0.5) (* x x) 1.0)
      (fma
       (-
        (* (fma -0.0001984126984126984 (* y y) 0.008333333333333333) (* y y))
        0.16666666666666666)
       (* y y)
       1.0))
     (if (<= t_0 0.9999999999516638)
       (* (* (sin y) (/ (fma x x 2.0) y)) 0.5)
       (* (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 = fma(fma(0.041666666666666664, (x * x), 0.5), (x * x), 1.0) * fma(((fma(-0.0001984126984126984, (y * y), 0.008333333333333333) * (y * y)) - 0.16666666666666666), (y * y), 1.0);
	} else if (t_0 <= 0.9999999999516638) {
		tmp = (sin(y) * (fma(x, x, 2.0) / y)) * 0.5;
	} else {
		tmp = 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(fma(fma(0.041666666666666664, Float64(x * x), 0.5), Float64(x * x), 1.0) * fma(Float64(Float64(fma(-0.0001984126984126984, Float64(y * y), 0.008333333333333333) * Float64(y * y)) - 0.16666666666666666), Float64(y * y), 1.0));
	elseif (t_0 <= 0.9999999999516638)
		tmp = Float64(Float64(sin(y) * Float64(fma(x, x, 2.0) / y)) * 0.5);
	else
		tmp = Float64(cosh(x) * 1.0);
	end
	return 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[(N[(0.041666666666666664 * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(N[(N[(-0.0001984126984126984 * N[(y * y), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.9999999999516638], N[(N[(N[Sin[y], $MachinePrecision] * N[(N[(x * x + 2.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[Cosh[x], $MachinePrecision] * 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 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 54.8

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

   5. Applied rewrites 54.8%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot \left(\left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) - \frac{1}{6}\right) \cdot {y}^{2} + 1\right) \]

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot \mathsf{fma}\left(\left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) \cdot {y}^{2} - \frac{1}{6}, {y}^{2}, 1\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot \mathsf{fma}\left(\left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) \cdot {y}^{2} - \frac{1}{6}, {y}^{2}, 1\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot \mathsf{fma}\left(\left(\frac{-1}{5040} \cdot {y}^{2} + \frac{1}{120}\right) \cdot {y}^{2} - \frac{1}{6}, {y}^{2}, 1\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, {y}^{2}, \frac{1}{120}\right) \cdot {y}^{2} - \frac{1}{6}, {y}^{2}, 1\right) \]

      9. pow2 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot {y}^{2} - \frac{1}{6}, {y}^{2}, 1\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot {y}^{2} - \frac{1}{6}, {y}^{2}, 1\right) \]

      11. pow2 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot \left(y \cdot y\right) - \frac{1}{6}, {y}^{2}, 1\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot \left(y \cdot y\right) - \frac{1}{6}, {y}^{2}, 1\right) \]

      13. pow2 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot \left(y \cdot y\right) - \frac{1}{6}, y \cdot \color{blue}{y}, 1\right) \]

      14. lift-*.f64 97.2

          \[\leadsto \mathsf{fma}\left(x \cdot x, 0.5, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, y \cdot y, 0.008333333333333333\right) \cdot \left(y \cdot y\right) - 0.16666666666666666, y \cdot \color{blue}{y}, 1\right) \]

   8. Applied rewrites 97.2%

      \[\leadsto \mathsf{fma}\left(x \cdot x, 0.5, 1\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, y \cdot y, 0.008333333333333333\right) \cdot \left(y \cdot y\right) - 0.16666666666666666, y \cdot y, 1\right)} \]

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot \left(y \cdot y\right) - \frac{1}{6}, y \cdot y, 1\right) \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \left({x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) + \color{blue}{1}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot \left(y \cdot y\right) - \frac{1}{6}, y \cdot y, 1\right) \]

       2. *-commutative N/A

          \[\leadsto \left(\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) \cdot {x}^{2} + 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot \left(y \cdot y\right) - \frac{1}{6}, y \cdot y, 1\right) \]

       3. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, \color{blue}{{x}^{2}}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot \left(y \cdot y\right) - \frac{1}{6}, y \cdot y, 1\right) \]

       4. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{24} \cdot {x}^{2} + \frac{1}{2}, {\color{blue}{x}}^{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot \left(y \cdot y\right) - \frac{1}{6}, y \cdot y, 1\right) \]

       5. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, {x}^{2}, \frac{1}{2}\right), {\color{blue}{x}}^{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot \left(y \cdot y\right) - \frac{1}{6}, y \cdot y, 1\right) \]

       6. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot \left(y \cdot y\right) - \frac{1}{6}, y \cdot y, 1\right) \]

       7. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot \left(y \cdot y\right) - \frac{1}{6}, y \cdot y, 1\right) \]

       8. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot \color{blue}{x}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot \left(y \cdot y\right) - \frac{1}{6}, y \cdot y, 1\right) \]

       9. lift-*.f64 100.0

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot \color{blue}{x}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, y \cdot y, 0.008333333333333333\right) \cdot \left(y \cdot y\right) - 0.16666666666666666, y \cdot y, 1\right) \]

   11. Applied rewrites 100.0%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, y \cdot y, 0.008333333333333333\right) \cdot \left(y \cdot y\right) - 0.16666666666666666, y \cdot y, 1\right) \]

   if -inf.0 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < 0.99999999995166378

   1. Initial program 99.5%

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

      1. lift-cosh.f64 N/A

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

      2. cosh-neg-rev N/A

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

      3. cosh-def N/A

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

      4. lower-/.f64 N/A

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

      5. rec-exp N/A

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

      6. lower-+.f64 N/A

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

      7. rec-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-exp.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-neg.f64 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 99.4%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. pow2 N/A

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

      3. lower-fma.f64 99.4

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

   10. Applied rewrites 99.4%

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

   if 0.99999999995166378 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 100.0%

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

Alternative 5: 99.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ t_1 := \cosh x \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, y \cdot y, 0.008333333333333333\right) \cdot \left(y \cdot y\right) - 0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{elif}\;t\_1 \leq 0.9999999999516638:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)) (t_1 (* (cosh x) t_0)))
   (if (<= t_1 (- INFINITY))
     (*
      (fma (fma 0.041666666666666664 (* x x) 0.5) (* x x) 1.0)
      (fma
       (-
        (* (fma -0.0001984126984126984 (* y y) 0.008333333333333333) (* y y))
        0.16666666666666666)
       (* y y)
       1.0))
     (if (<= t_1 0.9999999999516638) 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(0.041666666666666664, (x * x), 0.5), (x * x), 1.0) * fma(((fma(-0.0001984126984126984, (y * y), 0.008333333333333333) * (y * y)) - 0.16666666666666666), (y * y), 1.0);
	} else if (t_1 <= 0.9999999999516638) {
		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(0.041666666666666664, Float64(x * x), 0.5), Float64(x * x), 1.0) * fma(Float64(Float64(fma(-0.0001984126984126984, Float64(y * y), 0.008333333333333333) * Float64(y * y)) - 0.16666666666666666), Float64(y * y), 1.0));
	elseif (t_1 <= 0.9999999999516638)
		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[(0.041666666666666664 * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(N[(N[(-0.0001984126984126984 * N[(y * y), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.9999999999516638], t$95$0, N[(N[Cosh[x], $MachinePrecision] * 1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 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 54.8

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

   5. Applied rewrites 54.8%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot \left(\left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) - \frac{1}{6}\right) \cdot {y}^{2} + 1\right) \]

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot \mathsf{fma}\left(\left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) \cdot {y}^{2} - \frac{1}{6}, {y}^{2}, 1\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot \mathsf{fma}\left(\left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) \cdot {y}^{2} - \frac{1}{6}, {y}^{2}, 1\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot \mathsf{fma}\left(\left(\frac{-1}{5040} \cdot {y}^{2} + \frac{1}{120}\right) \cdot {y}^{2} - \frac{1}{6}, {y}^{2}, 1\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, {y}^{2}, \frac{1}{120}\right) \cdot {y}^{2} - \frac{1}{6}, {y}^{2}, 1\right) \]

      9. pow2 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot {y}^{2} - \frac{1}{6}, {y}^{2}, 1\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot {y}^{2} - \frac{1}{6}, {y}^{2}, 1\right) \]

      11. pow2 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot \left(y \cdot y\right) - \frac{1}{6}, {y}^{2}, 1\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot \left(y \cdot y\right) - \frac{1}{6}, {y}^{2}, 1\right) \]

      13. pow2 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot \left(y \cdot y\right) - \frac{1}{6}, y \cdot \color{blue}{y}, 1\right) \]

      14. lift-*.f64 97.2

          \[\leadsto \mathsf{fma}\left(x \cdot x, 0.5, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, y \cdot y, 0.008333333333333333\right) \cdot \left(y \cdot y\right) - 0.16666666666666666, y \cdot \color{blue}{y}, 1\right) \]

   8. Applied rewrites 97.2%

      \[\leadsto \mathsf{fma}\left(x \cdot x, 0.5, 1\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, y \cdot y, 0.008333333333333333\right) \cdot \left(y \cdot y\right) - 0.16666666666666666, y \cdot y, 1\right)} \]

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot \left(y \cdot y\right) - \frac{1}{6}, y \cdot y, 1\right) \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \left({x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) + \color{blue}{1}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot \left(y \cdot y\right) - \frac{1}{6}, y \cdot y, 1\right) \]

       2. *-commutative N/A

          \[\leadsto \left(\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) \cdot {x}^{2} + 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot \left(y \cdot y\right) - \frac{1}{6}, y \cdot y, 1\right) \]

       3. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, \color{blue}{{x}^{2}}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot \left(y \cdot y\right) - \frac{1}{6}, y \cdot y, 1\right) \]

       4. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{24} \cdot {x}^{2} + \frac{1}{2}, {\color{blue}{x}}^{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot \left(y \cdot y\right) - \frac{1}{6}, y \cdot y, 1\right) \]

       5. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, {x}^{2}, \frac{1}{2}\right), {\color{blue}{x}}^{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot \left(y \cdot y\right) - \frac{1}{6}, y \cdot y, 1\right) \]

       6. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot \left(y \cdot y\right) - \frac{1}{6}, y \cdot y, 1\right) \]

       7. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot \left(y \cdot y\right) - \frac{1}{6}, y \cdot y, 1\right) \]

       8. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot \color{blue}{x}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot \left(y \cdot y\right) - \frac{1}{6}, y \cdot y, 1\right) \]

       9. lift-*.f64 100.0

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot \color{blue}{x}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, y \cdot y, 0.008333333333333333\right) \cdot \left(y \cdot y\right) - 0.16666666666666666, y \cdot y, 1\right) \]

   11. Applied rewrites 100.0%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, y \cdot y, 0.008333333333333333\right) \cdot \left(y \cdot y\right) - 0.16666666666666666, y \cdot y, 1\right) \]

   if -inf.0 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < 0.99999999995166378

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0

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

      1. lift-sin.f64 N/A

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

      2. lift-/.f64 99.3

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

   5. Applied rewrites 99.3%

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

   if 0.99999999995166378 < (*.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: 74.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -2 \cdot 10^{-151}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, y \cdot y, 0.008333333333333333\right) \cdot \left(y \cdot y\right) - 0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cosh x) (/ (sin y) y)) -2e-151)
   (*
    (fma (fma 0.041666666666666664 (* x x) 0.5) (* x x) 1.0)
    (fma
     (-
      (* (fma -0.0001984126984126984 (* y y) 0.008333333333333333) (* y y))
      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-151) {
		tmp = fma(fma(0.041666666666666664, (x * x), 0.5), (x * x), 1.0) * fma(((fma(-0.0001984126984126984, (y * y), 0.008333333333333333) * (y * y)) - 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-151)
		tmp = Float64(fma(fma(0.041666666666666664, Float64(x * x), 0.5), Float64(x * x), 1.0) * fma(Float64(Float64(fma(-0.0001984126984126984, Float64(y * y), 0.008333333333333333) * Float64(y * y)) - 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-151], N[(N[(N[(0.041666666666666664 * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(N[(N[(-0.0001984126984126984 * N[(y * y), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * 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)) < -1.9999999999999999e-151

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + \frac{1}{2} \cdot {x}^{2}\right)} \cdot \frac{\sin y}{y} \]
   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 73.5

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

   5. Applied rewrites 73.5%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot \left(\left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) - \frac{1}{6}\right) \cdot {y}^{2} + 1\right) \]

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot \mathsf{fma}\left(\left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) \cdot {y}^{2} - \frac{1}{6}, {y}^{2}, 1\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot \mathsf{fma}\left(\left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) \cdot {y}^{2} - \frac{1}{6}, {y}^{2}, 1\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot \mathsf{fma}\left(\left(\frac{-1}{5040} \cdot {y}^{2} + \frac{1}{120}\right) \cdot {y}^{2} - \frac{1}{6}, {y}^{2}, 1\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, {y}^{2}, \frac{1}{120}\right) \cdot {y}^{2} - \frac{1}{6}, {y}^{2}, 1\right) \]

      9. pow2 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot {y}^{2} - \frac{1}{6}, {y}^{2}, 1\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot {y}^{2} - \frac{1}{6}, {y}^{2}, 1\right) \]

      11. pow2 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot \left(y \cdot y\right) - \frac{1}{6}, {y}^{2}, 1\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot \left(y \cdot y\right) - \frac{1}{6}, {y}^{2}, 1\right) \]

      13. pow2 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot \left(y \cdot y\right) - \frac{1}{6}, y \cdot \color{blue}{y}, 1\right) \]

      14. lift-*.f64 58.2

          \[\leadsto \mathsf{fma}\left(x \cdot x, 0.5, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, y \cdot y, 0.008333333333333333\right) \cdot \left(y \cdot y\right) - 0.16666666666666666, y \cdot \color{blue}{y}, 1\right) \]

   8. Applied rewrites 58.2%

      \[\leadsto \mathsf{fma}\left(x \cdot x, 0.5, 1\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, y \cdot y, 0.008333333333333333\right) \cdot \left(y \cdot y\right) - 0.16666666666666666, y \cdot y, 1\right)} \]

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot \left(y \cdot y\right) - \frac{1}{6}, y \cdot y, 1\right) \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \left({x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) + \color{blue}{1}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot \left(y \cdot y\right) - \frac{1}{6}, y \cdot y, 1\right) \]

       2. *-commutative N/A

          \[\leadsto \left(\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) \cdot {x}^{2} + 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot \left(y \cdot y\right) - \frac{1}{6}, y \cdot y, 1\right) \]

       3. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, \color{blue}{{x}^{2}}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot \left(y \cdot y\right) - \frac{1}{6}, y \cdot y, 1\right) \]

       4. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{24} \cdot {x}^{2} + \frac{1}{2}, {\color{blue}{x}}^{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot \left(y \cdot y\right) - \frac{1}{6}, y \cdot y, 1\right) \]

       5. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, {x}^{2}, \frac{1}{2}\right), {\color{blue}{x}}^{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot \left(y \cdot y\right) - \frac{1}{6}, y \cdot y, 1\right) \]

       6. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot \left(y \cdot y\right) - \frac{1}{6}, y \cdot y, 1\right) \]

       7. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot \left(y \cdot y\right) - \frac{1}{6}, y \cdot y, 1\right) \]

       8. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot \color{blue}{x}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot \left(y \cdot y\right) - \frac{1}{6}, y \cdot y, 1\right) \]

       9. lift-*.f64 59.8

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot \color{blue}{x}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, y \cdot y, 0.008333333333333333\right) \cdot \left(y \cdot y\right) - 0.16666666666666666, y \cdot y, 1\right) \]

   11. Applied rewrites 59.8%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, y \cdot y, 0.008333333333333333\right) \cdot \left(y \cdot y\right) - 0.16666666666666666, y \cdot y, 1\right) \]

   if -1.9999999999999999e-151 < (*.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 75.4%

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

Alternative 7: 69.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -2 \cdot 10^{-285}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, y \cdot y, 0.008333333333333333\right) \cdot \left(y \cdot y\right) - 0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot \mathsf{fma}\left(0.008333333333333333 \cdot \left(y \cdot y\right) - 0.16666666666666666, y \cdot y, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cosh x) (/ (sin y) y)) -2e-285)
   (*
    (fma (fma 0.041666666666666664 (* x x) 0.5) (* x x) 1.0)
    (fma
     (-
      (* (fma -0.0001984126984126984 (* y y) 0.008333333333333333) (* y y))
      0.16666666666666666)
     (* y y)
     1.0))
   (*
    (fma
     (fma (fma 0.001388888888888889 (* x x) 0.041666666666666664) (* x x) 0.5)
     (* x x)
     1.0)
    (fma
     (- (* 0.008333333333333333 (* y y)) 0.16666666666666666)
     (* y y)
     1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((cosh(x) * (sin(y) / y)) <= -2e-285) {
		tmp = fma(fma(0.041666666666666664, (x * x), 0.5), (x * x), 1.0) * fma(((fma(-0.0001984126984126984, (y * y), 0.008333333333333333) * (y * y)) - 0.16666666666666666), (y * y), 1.0);
	} else {
		tmp = fma(fma(fma(0.001388888888888889, (x * x), 0.041666666666666664), (x * x), 0.5), (x * x), 1.0) * fma(((0.008333333333333333 * (y * y)) - 0.16666666666666666), (y * y), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(cosh(x) * Float64(sin(y) / y)) <= -2e-285)
		tmp = Float64(fma(fma(0.041666666666666664, Float64(x * x), 0.5), Float64(x * x), 1.0) * fma(Float64(Float64(fma(-0.0001984126984126984, Float64(y * y), 0.008333333333333333) * Float64(y * y)) - 0.16666666666666666), Float64(y * y), 1.0));
	else
		tmp = Float64(fma(fma(fma(0.001388888888888889, Float64(x * x), 0.041666666666666664), Float64(x * x), 0.5), Float64(x * x), 1.0) * fma(Float64(Float64(0.008333333333333333 * Float64(y * y)) - 0.16666666666666666), Float64(y * y), 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-285], N[(N[(N[(0.041666666666666664 * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(N[(N[(-0.0001984126984126984 * N[(y * y), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], 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[(N[(N[(0.008333333333333333 * N[(y * y), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < -2.00000000000000015e-285

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + \frac{1}{2} \cdot {x}^{2}\right)} \cdot \frac{\sin y}{y} \]
   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 79.7

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

   5. Applied rewrites 79.7%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot \left(\left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) - \frac{1}{6}\right) \cdot {y}^{2} + 1\right) \]

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot \mathsf{fma}\left(\left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) \cdot {y}^{2} - \frac{1}{6}, {y}^{2}, 1\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot \mathsf{fma}\left(\left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) \cdot {y}^{2} - \frac{1}{6}, {y}^{2}, 1\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot \mathsf{fma}\left(\left(\frac{-1}{5040} \cdot {y}^{2} + \frac{1}{120}\right) \cdot {y}^{2} - \frac{1}{6}, {y}^{2}, 1\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, {y}^{2}, \frac{1}{120}\right) \cdot {y}^{2} - \frac{1}{6}, {y}^{2}, 1\right) \]

      9. pow2 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot {y}^{2} - \frac{1}{6}, {y}^{2}, 1\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot {y}^{2} - \frac{1}{6}, {y}^{2}, 1\right) \]

      11. pow2 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot \left(y \cdot y\right) - \frac{1}{6}, {y}^{2}, 1\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot \left(y \cdot y\right) - \frac{1}{6}, {y}^{2}, 1\right) \]

      13. pow2 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot \left(y \cdot y\right) - \frac{1}{6}, y \cdot \color{blue}{y}, 1\right) \]

      14. lift-*.f64 44.9

          \[\leadsto \mathsf{fma}\left(x \cdot x, 0.5, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, y \cdot y, 0.008333333333333333\right) \cdot \left(y \cdot y\right) - 0.16666666666666666, y \cdot \color{blue}{y}, 1\right) \]

   8. Applied rewrites 44.9%

      \[\leadsto \mathsf{fma}\left(x \cdot x, 0.5, 1\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, y \cdot y, 0.008333333333333333\right) \cdot \left(y \cdot y\right) - 0.16666666666666666, y \cdot y, 1\right)} \]

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot \left(y \cdot y\right) - \frac{1}{6}, y \cdot y, 1\right) \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \left({x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) + \color{blue}{1}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot \left(y \cdot y\right) - \frac{1}{6}, y \cdot y, 1\right) \]

       2. *-commutative N/A

          \[\leadsto \left(\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) \cdot {x}^{2} + 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot \left(y \cdot y\right) - \frac{1}{6}, y \cdot y, 1\right) \]

       3. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, \color{blue}{{x}^{2}}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot \left(y \cdot y\right) - \frac{1}{6}, y \cdot y, 1\right) \]

       4. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{24} \cdot {x}^{2} + \frac{1}{2}, {\color{blue}{x}}^{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot \left(y \cdot y\right) - \frac{1}{6}, y \cdot y, 1\right) \]

       5. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, {x}^{2}, \frac{1}{2}\right), {\color{blue}{x}}^{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot \left(y \cdot y\right) - \frac{1}{6}, y \cdot y, 1\right) \]

       6. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot \left(y \cdot y\right) - \frac{1}{6}, y \cdot y, 1\right) \]

       7. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot \left(y \cdot y\right) - \frac{1}{6}, y \cdot y, 1\right) \]

       8. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot \color{blue}{x}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot \left(y \cdot y\right) - \frac{1}{6}, y \cdot y, 1\right) \]

       9. lift-*.f64 46.2

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot \color{blue}{x}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, y \cdot y, 0.008333333333333333\right) \cdot \left(y \cdot y\right) - 0.16666666666666666, y \cdot y, 1\right) \]

   11. Applied rewrites 46.2%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, y \cdot y, 0.008333333333333333\right) \cdot \left(y \cdot y\right) - 0.16666666666666666, y \cdot y, 1\right) \]

   if -2.00000000000000015e-285 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y))

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + \frac{1}{2} \cdot {x}^{2}\right)} \cdot \frac{\sin y}{y} \]
   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 75.6

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

   5. Applied rewrites 75.6%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 66.8

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

   8. Applied rewrites 66.8%

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

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)} \cdot \mathsf{fma}\left(\frac{1}{120} \cdot \left(y \cdot y\right) - \frac{1}{6}, y \cdot y, 1\right) \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \left({x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) + \color{blue}{1}\right) \cdot \mathsf{fma}\left(\frac{1}{120} \cdot \left(y \cdot y\right) - \frac{1}{6}, y \cdot y, 1\right) \]

       2. *-commutative N/A

          \[\leadsto \left(\left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) \cdot {x}^{2} + 1\right) \cdot \mathsf{fma}\left(\frac{1}{120} \cdot \left(y \cdot y\right) - \frac{1}{6}, y \cdot y, 1\right) \]

       3. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right), \color{blue}{{x}^{2}}, 1\right) \cdot \mathsf{fma}\left(\frac{1}{120} \cdot \left(y \cdot y\right) - \frac{1}{6}, y \cdot y, 1\right) \]

       4. +-commutative N/A

          \[\leadsto \mathsf{fma}\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right) + \frac{1}{2}, {\color{blue}{x}}^{2}, 1\right) \cdot \mathsf{fma}\left(\frac{1}{120} \cdot \left(y \cdot y\right) - \frac{1}{6}, y \cdot y, 1\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right) \cdot {x}^{2} + \frac{1}{2}, {x}^{2}, 1\right) \cdot \mathsf{fma}\left(\frac{1}{120} \cdot \left(y \cdot y\right) - \frac{1}{6}, y \cdot y, 1\right) \]

       6. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}, {x}^{2}, \frac{1}{2}\right), {\color{blue}{x}}^{2}, 1\right) \cdot \mathsf{fma}\left(\frac{1}{120} \cdot \left(y \cdot y\right) - \frac{1}{6}, y \cdot y, 1\right) \]

       7. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720} \cdot {x}^{2} + \frac{1}{24}, {x}^{2}, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot \mathsf{fma}\left(\frac{1}{120} \cdot \left(y \cdot y\right) - \frac{1}{6}, y \cdot y, 1\right) \]

       8. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, {x}^{2}, \frac{1}{24}\right), {x}^{2}, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot \mathsf{fma}\left(\frac{1}{120} \cdot \left(y \cdot y\right) - \frac{1}{6}, y \cdot y, 1\right) \]

       9. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, x \cdot x, \frac{1}{24}\right), {x}^{2}, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot \mathsf{fma}\left(\frac{1}{120} \cdot \left(y \cdot y\right) - \frac{1}{6}, y \cdot y, 1\right) \]

       10. lift-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, x \cdot x, \frac{1}{24}\right), {x}^{2}, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot \mathsf{fma}\left(\frac{1}{120} \cdot \left(y \cdot y\right) - \frac{1}{6}, y \cdot y, 1\right) \]

       11. pow2 N/A

           \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, x \cdot x, \frac{1}{24}\right), x \cdot x, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot \mathsf{fma}\left(\frac{1}{120} \cdot \left(y \cdot y\right) - \frac{1}{6}, y \cdot y, 1\right) \]

       12. lift-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, x \cdot x, \frac{1}{24}\right), x \cdot x, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot \mathsf{fma}\left(\frac{1}{120} \cdot \left(y \cdot y\right) - \frac{1}{6}, y \cdot y, 1\right) \]

       13. pow2 N/A

           \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, x \cdot x, \frac{1}{24}\right), x \cdot x, \frac{1}{2}\right), x \cdot \color{blue}{x}, 1\right) \cdot \mathsf{fma}\left(\frac{1}{120} \cdot \left(y \cdot y\right) - \frac{1}{6}, y \cdot y, 1\right) \]

       14. lift-*.f64 77.3

           \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot \color{blue}{x}, 1\right) \cdot \mathsf{fma}\left(0.008333333333333333 \cdot \left(y \cdot y\right) - 0.16666666666666666, y \cdot y, 1\right) \]

   11. Applied rewrites 77.3%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right)} \cdot \mathsf{fma}\left(0.008333333333333333 \cdot \left(y \cdot y\right) - 0.16666666666666666, y \cdot y, 1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 68.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right)\\ \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -2 \cdot 10^{-285}:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, y \cdot y, 0.008333333333333333\right) \cdot \left(y \cdot y\right) - 0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(0.008333333333333333 \cdot \left(y \cdot y\right) - 0.16666666666666666, y \cdot y, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma (fma 0.041666666666666664 (* x x) 0.5) (* x x) 1.0)))
   (if (<= (* (cosh x) (/ (sin y) y)) -2e-285)
     (*
      t_0
      (fma
       (-
        (* (fma -0.0001984126984126984 (* y y) 0.008333333333333333) (* y y))
        0.16666666666666666)
       (* y y)
       1.0))
     (*
      t_0
      (fma
       (- (* 0.008333333333333333 (* y y)) 0.16666666666666666)
       (* y y)
       1.0)))))

C

double code(double x, double y) {
	double t_0 = fma(fma(0.041666666666666664, (x * x), 0.5), (x * x), 1.0);
	double tmp;
	if ((cosh(x) * (sin(y) / y)) <= -2e-285) {
		tmp = t_0 * fma(((fma(-0.0001984126984126984, (y * y), 0.008333333333333333) * (y * y)) - 0.16666666666666666), (y * y), 1.0);
	} else {
		tmp = t_0 * fma(((0.008333333333333333 * (y * y)) - 0.16666666666666666), (y * y), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(fma(0.041666666666666664, Float64(x * x), 0.5), Float64(x * x), 1.0)
	tmp = 0.0
	if (Float64(cosh(x) * Float64(sin(y) / y)) <= -2e-285)
		tmp = Float64(t_0 * fma(Float64(Float64(fma(-0.0001984126984126984, Float64(y * y), 0.008333333333333333) * Float64(y * y)) - 0.16666666666666666), Float64(y * y), 1.0));
	else
		tmp = Float64(t_0 * fma(Float64(Float64(0.008333333333333333 * Float64(y * y)) - 0.16666666666666666), Float64(y * y), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(0.041666666666666664 * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[N[(N[Cosh[x], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], -2e-285], N[(t$95$0 * N[(N[(N[(N[(-0.0001984126984126984 * N[(y * y), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(N[(N[(0.008333333333333333 * N[(y * y), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < -2.00000000000000015e-285

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + \frac{1}{2} \cdot {x}^{2}\right)} \cdot \frac{\sin y}{y} \]
   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 79.7

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

   5. Applied rewrites 79.7%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot \left(\left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) - \frac{1}{6}\right) \cdot {y}^{2} + 1\right) \]

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot \mathsf{fma}\left(\left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) \cdot {y}^{2} - \frac{1}{6}, {y}^{2}, 1\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot \mathsf{fma}\left(\left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) \cdot {y}^{2} - \frac{1}{6}, {y}^{2}, 1\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot \mathsf{fma}\left(\left(\frac{-1}{5040} \cdot {y}^{2} + \frac{1}{120}\right) \cdot {y}^{2} - \frac{1}{6}, {y}^{2}, 1\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, {y}^{2}, \frac{1}{120}\right) \cdot {y}^{2} - \frac{1}{6}, {y}^{2}, 1\right) \]

      9. pow2 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot {y}^{2} - \frac{1}{6}, {y}^{2}, 1\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot {y}^{2} - \frac{1}{6}, {y}^{2}, 1\right) \]

      11. pow2 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot \left(y \cdot y\right) - \frac{1}{6}, {y}^{2}, 1\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot \left(y \cdot y\right) - \frac{1}{6}, {y}^{2}, 1\right) \]

      13. pow2 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot \left(y \cdot y\right) - \frac{1}{6}, y \cdot \color{blue}{y}, 1\right) \]

      14. lift-*.f64 44.9

          \[\leadsto \mathsf{fma}\left(x \cdot x, 0.5, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, y \cdot y, 0.008333333333333333\right) \cdot \left(y \cdot y\right) - 0.16666666666666666, y \cdot \color{blue}{y}, 1\right) \]

   8. Applied rewrites 44.9%

      \[\leadsto \mathsf{fma}\left(x \cdot x, 0.5, 1\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, y \cdot y, 0.008333333333333333\right) \cdot \left(y \cdot y\right) - 0.16666666666666666, y \cdot y, 1\right)} \]

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot \left(y \cdot y\right) - \frac{1}{6}, y \cdot y, 1\right) \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \left({x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) + \color{blue}{1}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot \left(y \cdot y\right) - \frac{1}{6}, y \cdot y, 1\right) \]

       2. *-commutative N/A

          \[\leadsto \left(\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) \cdot {x}^{2} + 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot \left(y \cdot y\right) - \frac{1}{6}, y \cdot y, 1\right) \]

       3. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, \color{blue}{{x}^{2}}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot \left(y \cdot y\right) - \frac{1}{6}, y \cdot y, 1\right) \]

       4. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{24} \cdot {x}^{2} + \frac{1}{2}, {\color{blue}{x}}^{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot \left(y \cdot y\right) - \frac{1}{6}, y \cdot y, 1\right) \]

       5. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, {x}^{2}, \frac{1}{2}\right), {\color{blue}{x}}^{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot \left(y \cdot y\right) - \frac{1}{6}, y \cdot y, 1\right) \]

       6. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot \left(y \cdot y\right) - \frac{1}{6}, y \cdot y, 1\right) \]

       7. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot \left(y \cdot y\right) - \frac{1}{6}, y \cdot y, 1\right) \]

       8. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot \color{blue}{x}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot \left(y \cdot y\right) - \frac{1}{6}, y \cdot y, 1\right) \]

       9. lift-*.f64 46.2

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot \color{blue}{x}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, y \cdot y, 0.008333333333333333\right) \cdot \left(y \cdot y\right) - 0.16666666666666666, y \cdot y, 1\right) \]

   11. Applied rewrites 46.2%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, y \cdot y, 0.008333333333333333\right) \cdot \left(y \cdot y\right) - 0.16666666666666666, y \cdot y, 1\right) \]

   if -2.00000000000000015e-285 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y))

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + \frac{1}{2} \cdot {x}^{2}\right)} \cdot \frac{\sin y}{y} \]
   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 75.6

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

   5. Applied rewrites 75.6%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 66.8

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

   8. Applied rewrites 66.8%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. +-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. pow2 N/A

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

       7. lift-*.f64 N/A

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

       8. pow2 N/A

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

       9. lift-*.f64 74.2

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

   11. Applied rewrites 74.2%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right)} \cdot \mathsf{fma}\left(0.008333333333333333 \cdot \left(y \cdot y\right) - 0.16666666666666666, y \cdot y, 1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 66.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ t_1 := \mathsf{fma}\left(x \cdot x, 0.5, 1\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-291}:\\ \;\;\;\;\frac{t\_1 \cdot \left(\mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right) \cdot y\right)}{y}\\ \mathbf{elif}\;t\_0 \leq 0.99999999996:\\ \;\;\;\;t\_1 \cdot \mathsf{fma}\left(0.008333333333333333 \cdot \left(y \cdot y\right) - 0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot \frac{\mathsf{fma}\left(x, x, 2\right)}{y}\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)) (t_1 (fma (* x x) 0.5 1.0)))
   (if (<= t_0 -2e-291)
     (/ (* t_1 (* (fma (* y y) -0.16666666666666666 1.0) y)) y)
     (if (<= t_0 0.99999999996)
       (*
        t_1
        (fma
         (- (* 0.008333333333333333 (* y y)) 0.16666666666666666)
         (* y y)
         1.0))
       (* (* y (/ (fma x x 2.0) y)) 0.5)))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double t_1 = fma((x * x), 0.5, 1.0);
	double tmp;
	if (t_0 <= -2e-291) {
		tmp = (t_1 * (fma((y * y), -0.16666666666666666, 1.0) * y)) / y;
	} else if (t_0 <= 0.99999999996) {
		tmp = t_1 * fma(((0.008333333333333333 * (y * y)) - 0.16666666666666666), (y * y), 1.0);
	} else {
		tmp = (y * (fma(x, x, 2.0) / y)) * 0.5;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(y) / y)
	t_1 = fma(Float64(x * x), 0.5, 1.0)
	tmp = 0.0
	if (t_0 <= -2e-291)
		tmp = Float64(Float64(t_1 * Float64(fma(Float64(y * y), -0.16666666666666666, 1.0) * y)) / y);
	elseif (t_0 <= 0.99999999996)
		tmp = Float64(t_1 * fma(Float64(Float64(0.008333333333333333 * Float64(y * y)) - 0.16666666666666666), Float64(y * y), 1.0));
	else
		tmp = Float64(Float64(y * Float64(fma(x, x, 2.0) / y)) * 0.5);
	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[(x * x), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-291], N[(N[(t$95$1 * N[(N[(N[(y * y), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[t$95$0, 0.99999999996], N[(t$95$1 * N[(N[(N[(0.008333333333333333 * N[(y * y), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(y * N[(N[(x * x + 2.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 y) y) < -1.99999999999999992e-291

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + \frac{1}{2} \cdot {x}^{2}\right)} \cdot \frac{\sin y}{y} \]
   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 79.7

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

   5. Applied rewrites 79.7%

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

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. pow2 N/A

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

      3. +-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 42.7

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

   8. Applied rewrites 42.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 45.3

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

   10. Applied rewrites 45.3%

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

   if -1.99999999999999992e-291 < (/.f64 (sin.f64 y) y) < 0.99999999996

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + \frac{1}{2} \cdot {x}^{2}\right)} \cdot \frac{\sin y}{y} \]
   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 72.5

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

   5. Applied rewrites 72.5%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 49.6

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

   8. Applied rewrites 49.6%

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

   if 0.99999999996 < (/.f64 (sin.f64 y) y)

   1. Initial program 100.0%

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

      1. lift-cosh.f64 N/A

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

      2. cosh-neg-rev N/A

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

      3. cosh-def N/A

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

      4. lower-/.f64 N/A

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

      5. rec-exp N/A

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

      6. lower-+.f64 N/A

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

      7. rec-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-exp.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-neg.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 99.0%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. pow2 N/A

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

      3. lower-fma.f64 90.0

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

   10. Applied rewrites 90.0%

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

   11. Taylor expanded in y around 0

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

   13. Applied rewrites 90.0%

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

Alternative 10: 65.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ t_1 := \mathsf{fma}\left(x \cdot x, 0.5, 1\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-291}:\\ \;\;\;\;t\_1 \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot y, y, 1\right)\\ \mathbf{elif}\;t\_0 \leq 0.99999999996:\\ \;\;\;\;t\_1 \cdot \mathsf{fma}\left(0.008333333333333333 \cdot \left(y \cdot y\right) - 0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot \frac{\mathsf{fma}\left(x, x, 2\right)}{y}\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)) (t_1 (fma (* x x) 0.5 1.0)))
   (if (<= t_0 -2e-291)
     (* t_1 (fma (* -0.16666666666666666 y) y 1.0))
     (if (<= t_0 0.99999999996)
       (*
        t_1
        (fma
         (- (* 0.008333333333333333 (* y y)) 0.16666666666666666)
         (* y y)
         1.0))
       (* (* y (/ (fma x x 2.0) y)) 0.5)))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double t_1 = fma((x * x), 0.5, 1.0);
	double tmp;
	if (t_0 <= -2e-291) {
		tmp = t_1 * fma((-0.16666666666666666 * y), y, 1.0);
	} else if (t_0 <= 0.99999999996) {
		tmp = t_1 * fma(((0.008333333333333333 * (y * y)) - 0.16666666666666666), (y * y), 1.0);
	} else {
		tmp = (y * (fma(x, x, 2.0) / y)) * 0.5;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(y) / y)
	t_1 = fma(Float64(x * x), 0.5, 1.0)
	tmp = 0.0
	if (t_0 <= -2e-291)
		tmp = Float64(t_1 * fma(Float64(-0.16666666666666666 * y), y, 1.0));
	elseif (t_0 <= 0.99999999996)
		tmp = Float64(t_1 * fma(Float64(Float64(0.008333333333333333 * Float64(y * y)) - 0.16666666666666666), Float64(y * y), 1.0));
	else
		tmp = Float64(Float64(y * Float64(fma(x, x, 2.0) / y)) * 0.5);
	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[(x * x), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-291], N[(t$95$1 * N[(N[(-0.16666666666666666 * y), $MachinePrecision] * y + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.99999999996], N[(t$95$1 * N[(N[(N[(0.008333333333333333 * N[(y * y), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(y * N[(N[(x * x + 2.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 y) y) < -1.99999999999999992e-291

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + \frac{1}{2} \cdot {x}^{2}\right)} \cdot \frac{\sin y}{y} \]
   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 79.7

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

   5. Applied rewrites 79.7%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 0.7

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

   8. Applied rewrites 0.7%

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

   9. Taylor expanded in y around 0

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

       1. pow2 N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. +-commutative N/A

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

       5. *-commutative N/A

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

       6. lower-fma.f64 N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 42.8

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

   11. Applied rewrites 42.8%

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

   if -1.99999999999999992e-291 < (/.f64 (sin.f64 y) y) < 0.99999999996

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + \frac{1}{2} \cdot {x}^{2}\right)} \cdot \frac{\sin y}{y} \]
   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 72.5

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

   5. Applied rewrites 72.5%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 49.6

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

   8. Applied rewrites 49.6%

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

   if 0.99999999996 < (/.f64 (sin.f64 y) y)

   1. Initial program 100.0%

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

      1. lift-cosh.f64 N/A

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

      2. cosh-neg-rev N/A

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

      3. cosh-def N/A

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

      4. lower-/.f64 N/A

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

      5. rec-exp N/A

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

      6. lower-+.f64 N/A

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

      7. rec-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-exp.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-neg.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 99.0%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. pow2 N/A

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

      3. lower-fma.f64 90.0

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

   10. Applied rewrites 90.0%

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

   11. Taylor expanded in y around 0

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

   13. Applied rewrites 90.0%

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

Alternative 11: 67.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -2 \cdot 10^{-285}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot x, 0.5, 1\right) \cdot \left(\mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right) \cdot y\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot \mathsf{fma}\left(0.008333333333333333 \cdot \left(y \cdot y\right) - 0.16666666666666666, y \cdot y, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cosh x) (/ (sin y) y)) -2e-285)
   (/ (* (fma (* x x) 0.5 1.0) (* (fma (* y y) -0.16666666666666666 1.0) y)) y)
   (*
    (fma (fma 0.041666666666666664 (* x x) 0.5) (* x x) 1.0)
    (fma
     (- (* 0.008333333333333333 (* y y)) 0.16666666666666666)
     (* y y)
     1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((cosh(x) * (sin(y) / y)) <= -2e-285) {
		tmp = (fma((x * x), 0.5, 1.0) * (fma((y * y), -0.16666666666666666, 1.0) * y)) / y;
	} else {
		tmp = fma(fma(0.041666666666666664, (x * x), 0.5), (x * x), 1.0) * fma(((0.008333333333333333 * (y * y)) - 0.16666666666666666), (y * y), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(cosh(x) * Float64(sin(y) / y)) <= -2e-285)
		tmp = Float64(Float64(fma(Float64(x * x), 0.5, 1.0) * Float64(fma(Float64(y * y), -0.16666666666666666, 1.0) * y)) / y);
	else
		tmp = Float64(fma(fma(0.041666666666666664, Float64(x * x), 0.5), Float64(x * x), 1.0) * fma(Float64(Float64(0.008333333333333333 * Float64(y * y)) - 0.16666666666666666), Float64(y * y), 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-285], N[(N[(N[(N[(x * x), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] * N[(N[(N[(y * y), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], N[(N[(N[(0.041666666666666664 * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(N[(0.008333333333333333 * N[(y * y), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < -2.00000000000000015e-285

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + \frac{1}{2} \cdot {x}^{2}\right)} \cdot \frac{\sin y}{y} \]
   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 79.7

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

   5. Applied rewrites 79.7%

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

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. pow2 N/A

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

      3. +-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 42.7

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

   8. Applied rewrites 42.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 45.3

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

   10. Applied rewrites 45.3%

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

   if -2.00000000000000015e-285 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y))

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + \frac{1}{2} \cdot {x}^{2}\right)} \cdot \frac{\sin y}{y} \]
   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 75.6

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

   5. Applied rewrites 75.6%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 66.8

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

   8. Applied rewrites 66.8%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. +-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. pow2 N/A

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

       7. lift-*.f64 N/A

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

       8. pow2 N/A

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

       9. lift-*.f64 74.2

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

   11. Applied rewrites 74.2%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right)} \cdot \mathsf{fma}\left(0.008333333333333333 \cdot \left(y \cdot y\right) - 0.16666666666666666, y \cdot y, 1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 65.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ t_1 := \mathsf{fma}\left(x \cdot x, 0.5, 1\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-291}:\\ \;\;\;\;t\_1 \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot y, y, 1\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-71}:\\ \;\;\;\;t\_1 \cdot \mathsf{fma}\left(0.008333333333333333 \cdot \left(y \cdot y\right), y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot \frac{\mathsf{fma}\left(x, x, 2\right)}{y}\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)) (t_1 (fma (* x x) 0.5 1.0)))
   (if (<= t_0 -2e-291)
     (* t_1 (fma (* -0.16666666666666666 y) y 1.0))
     (if (<= t_0 2e-71)
       (* t_1 (fma (* 0.008333333333333333 (* y y)) (* y y) 1.0))
       (* (* y (/ (fma x x 2.0) y)) 0.5)))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double t_1 = fma((x * x), 0.5, 1.0);
	double tmp;
	if (t_0 <= -2e-291) {
		tmp = t_1 * fma((-0.16666666666666666 * y), y, 1.0);
	} else if (t_0 <= 2e-71) {
		tmp = t_1 * fma((0.008333333333333333 * (y * y)), (y * y), 1.0);
	} else {
		tmp = (y * (fma(x, x, 2.0) / y)) * 0.5;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(y) / y)
	t_1 = fma(Float64(x * x), 0.5, 1.0)
	tmp = 0.0
	if (t_0 <= -2e-291)
		tmp = Float64(t_1 * fma(Float64(-0.16666666666666666 * y), y, 1.0));
	elseif (t_0 <= 2e-71)
		tmp = Float64(t_1 * fma(Float64(0.008333333333333333 * Float64(y * y)), Float64(y * y), 1.0));
	else
		tmp = Float64(Float64(y * Float64(fma(x, x, 2.0) / y)) * 0.5);
	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[(x * x), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-291], N[(t$95$1 * N[(N[(-0.16666666666666666 * y), $MachinePrecision] * y + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e-71], N[(t$95$1 * N[(N[(0.008333333333333333 * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(y * N[(N[(x * x + 2.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 y) y) < -1.99999999999999992e-291

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + \frac{1}{2} \cdot {x}^{2}\right)} \cdot \frac{\sin y}{y} \]
   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 79.7

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

   5. Applied rewrites 79.7%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 0.7

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

   8. Applied rewrites 0.7%

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

   9. Taylor expanded in y around 0

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

       1. pow2 N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. +-commutative N/A

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

       5. *-commutative N/A

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

       6. lower-fma.f64 N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 42.8

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

   11. Applied rewrites 42.8%

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

   if -1.99999999999999992e-291 < (/.f64 (sin.f64 y) y) < 1.9999999999999998e-71

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + \frac{1}{2} \cdot {x}^{2}\right)} \cdot \frac{\sin y}{y} \]
   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 69.2

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

   5. Applied rewrites 69.2%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 52.8

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

   8. Applied rewrites 52.8%

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

   9. Taylor expanded in y around inf

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

       1. pow2 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 52.8

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

   11. Applied rewrites 52.8%

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

   if 1.9999999999999998e-71 < (/.f64 (sin.f64 y) y)

   1. Initial program 100.0%

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

      1. lift-cosh.f64 N/A

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

      2. cosh-neg-rev N/A

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

      3. cosh-def N/A

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

      4. lower-/.f64 N/A

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

      5. rec-exp N/A

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

      6. lower-+.f64 N/A

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

      7. rec-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-exp.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-neg.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 99.1%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. pow2 N/A

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

      3. lower-fma.f64 89.0

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

   10. Applied rewrites 89.0%

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

   11. Taylor expanded in y around 0

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

   13. Applied rewrites 83.0%

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

Alternative 13: 64.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-291}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, 0.5, 1\right) \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot y, y, 1\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-90}:\\ \;\;\;\;\mathsf{fma}\left(0.008333333333333333 \cdot \left(y \cdot y\right) - 0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot \frac{\mathsf{fma}\left(x, x, 2\right)}{y}\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)))
   (if (<= t_0 -2e-291)
     (* (fma (* x x) 0.5 1.0) (fma (* -0.16666666666666666 y) y 1.0))
     (if (<= t_0 2e-90)
       (fma
        (- (* 0.008333333333333333 (* y y)) 0.16666666666666666)
        (* y y)
        1.0)
       (* (* y (/ (fma x x 2.0) y)) 0.5)))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double tmp;
	if (t_0 <= -2e-291) {
		tmp = fma((x * x), 0.5, 1.0) * fma((-0.16666666666666666 * y), y, 1.0);
	} else if (t_0 <= 2e-90) {
		tmp = fma(((0.008333333333333333 * (y * y)) - 0.16666666666666666), (y * y), 1.0);
	} else {
		tmp = (y * (fma(x, x, 2.0) / y)) * 0.5;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(y) / y)
	tmp = 0.0
	if (t_0 <= -2e-291)
		tmp = Float64(fma(Float64(x * x), 0.5, 1.0) * fma(Float64(-0.16666666666666666 * y), y, 1.0));
	elseif (t_0 <= 2e-90)
		tmp = fma(Float64(Float64(0.008333333333333333 * Float64(y * y)) - 0.16666666666666666), Float64(y * y), 1.0);
	else
		tmp = Float64(Float64(y * Float64(fma(x, x, 2.0) / y)) * 0.5);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-291], N[(N[(N[(x * x), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] * N[(N[(-0.16666666666666666 * y), $MachinePrecision] * y + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e-90], N[(N[(N[(0.008333333333333333 * N[(y * y), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(y * N[(N[(x * x + 2.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 y) y) < -1.99999999999999992e-291

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + \frac{1}{2} \cdot {x}^{2}\right)} \cdot \frac{\sin y}{y} \]
   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 79.7

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

   5. Applied rewrites 79.7%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 0.7

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

   8. Applied rewrites 0.7%

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

   9. Taylor expanded in y around 0

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

       1. pow2 N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. +-commutative N/A

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

       5. *-commutative N/A

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

       6. lower-fma.f64 N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 42.8

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

   11. Applied rewrites 42.8%

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

   if -1.99999999999999992e-291 < (/.f64 (sin.f64 y) y) < 1.99999999999999999e-90

   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 50.8

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

   5. Applied rewrites 50.8%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 51.0

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

   8. Applied rewrites 51.0%

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

   if 1.99999999999999999e-90 < (/.f64 (sin.f64 y) y)

   1. Initial program 100.0%

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

      1. lift-cosh.f64 N/A

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

      2. cosh-neg-rev N/A

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

      3. cosh-def N/A

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

      4. lower-/.f64 N/A

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

      5. rec-exp N/A

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

      6. lower-+.f64 N/A

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

      7. rec-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-exp.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-neg.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 99.1%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. pow2 N/A

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

      3. lower-fma.f64 88.6

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

   10. Applied rewrites 88.6%

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

   11. Taylor expanded in y around 0

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

   13. Applied rewrites 82.1%

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

Alternative 14: 62.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-291}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, y \cdot -0.16666666666666666, 1\right) \cdot y}{y}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-90}:\\ \;\;\;\;\mathsf{fma}\left(0.008333333333333333 \cdot \left(y \cdot y\right) - 0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot \frac{\mathsf{fma}\left(x, x, 2\right)}{y}\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)))
   (if (<= t_0 -2e-291)
     (/ (* (fma y (* y -0.16666666666666666) 1.0) y) y)
     (if (<= t_0 2e-90)
       (fma
        (- (* 0.008333333333333333 (* y y)) 0.16666666666666666)
        (* y y)
        1.0)
       (* (* y (/ (fma x x 2.0) y)) 0.5)))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double tmp;
	if (t_0 <= -2e-291) {
		tmp = (fma(y, (y * -0.16666666666666666), 1.0) * y) / y;
	} else if (t_0 <= 2e-90) {
		tmp = fma(((0.008333333333333333 * (y * y)) - 0.16666666666666666), (y * y), 1.0);
	} else {
		tmp = (y * (fma(x, x, 2.0) / y)) * 0.5;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(y) / y)
	tmp = 0.0
	if (t_0 <= -2e-291)
		tmp = Float64(Float64(fma(y, Float64(y * -0.16666666666666666), 1.0) * y) / y);
	elseif (t_0 <= 2e-90)
		tmp = fma(Float64(Float64(0.008333333333333333 * Float64(y * y)) - 0.16666666666666666), Float64(y * y), 1.0);
	else
		tmp = Float64(Float64(y * Float64(fma(x, x, 2.0) / y)) * 0.5);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-291], N[(N[(N[(y * N[(y * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[t$95$0, 2e-90], N[(N[(N[(0.008333333333333333 * N[(y * y), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(y * N[(N[(x * x + 2.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 y) y) < -1.99999999999999992e-291

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. lift-sin.f64 N/A

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

      2. lift-/.f64 56.1

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

   5. Applied rewrites 56.1%

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

   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 33.2

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

   8. Applied rewrites 33.2%

      \[\leadsto \frac{\mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right) \cdot y}{y} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 33.2

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

   10. Applied rewrites 33.2%

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

   if -1.99999999999999992e-291 < (/.f64 (sin.f64 y) y) < 1.99999999999999999e-90

   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 50.8

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

   5. Applied rewrites 50.8%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 51.0

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

   8. Applied rewrites 51.0%

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

   if 1.99999999999999999e-90 < (/.f64 (sin.f64 y) y)

   1. Initial program 100.0%

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

      1. lift-cosh.f64 N/A

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

      2. cosh-neg-rev N/A

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

      3. cosh-def N/A

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

      4. lower-/.f64 N/A

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

      5. rec-exp N/A

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

      6. lower-+.f64 N/A

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

      7. rec-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-exp.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-neg.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 99.1%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. pow2 N/A

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

      3. lower-fma.f64 88.6

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

   10. Applied rewrites 88.6%

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

   11. Taylor expanded in y around 0

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

   13. Applied rewrites 82.1%

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

Alternative 15: 62.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-291}:\\ \;\;\;\;\frac{\left(\left(y \cdot y\right) \cdot -0.16666666666666666\right) \cdot y}{y}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-90}:\\ \;\;\;\;\mathsf{fma}\left(0.008333333333333333 \cdot \left(y \cdot y\right) - 0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot \frac{\mathsf{fma}\left(x, x, 2\right)}{y}\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)))
   (if (<= t_0 -2e-291)
     (/ (* (* (* y y) -0.16666666666666666) y) y)
     (if (<= t_0 2e-90)
       (fma
        (- (* 0.008333333333333333 (* y y)) 0.16666666666666666)
        (* y y)
        1.0)
       (* (* y (/ (fma x x 2.0) y)) 0.5)))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double tmp;
	if (t_0 <= -2e-291) {
		tmp = (((y * y) * -0.16666666666666666) * y) / y;
	} else if (t_0 <= 2e-90) {
		tmp = fma(((0.008333333333333333 * (y * y)) - 0.16666666666666666), (y * y), 1.0);
	} else {
		tmp = (y * (fma(x, x, 2.0) / y)) * 0.5;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(y) / y)
	tmp = 0.0
	if (t_0 <= -2e-291)
		tmp = Float64(Float64(Float64(Float64(y * y) * -0.16666666666666666) * y) / y);
	elseif (t_0 <= 2e-90)
		tmp = fma(Float64(Float64(0.008333333333333333 * Float64(y * y)) - 0.16666666666666666), Float64(y * y), 1.0);
	else
		tmp = Float64(Float64(y * Float64(fma(x, x, 2.0) / y)) * 0.5);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-291], N[(N[(N[(N[(y * y), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * y), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[t$95$0, 2e-90], N[(N[(N[(0.008333333333333333 * N[(y * y), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(y * N[(N[(x * x + 2.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 y) y) < -1.99999999999999992e-291

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. lift-sin.f64 N/A

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

      2. lift-/.f64 56.1

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

   5. Applied rewrites 56.1%

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

   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 33.2

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

   8. Applied rewrites 33.2%

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

   9. Taylor expanded in y around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 33.2

          \[\leadsto \frac{\left(\left(y \cdot y\right) \cdot -0.16666666666666666\right) \cdot y}{y} \]

   11. Applied rewrites 33.2%

       \[\leadsto \frac{\left(\left(y \cdot y\right) \cdot -0.16666666666666666\right) \cdot y}{y} \]

   if -1.99999999999999992e-291 < (/.f64 (sin.f64 y) y) < 1.99999999999999999e-90

   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 50.8

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

   5. Applied rewrites 50.8%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 51.0

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

   8. Applied rewrites 51.0%

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

   if 1.99999999999999999e-90 < (/.f64 (sin.f64 y) y)

   1. Initial program 100.0%

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

      1. lift-cosh.f64 N/A

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

      2. cosh-neg-rev N/A

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

      3. cosh-def N/A

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

      4. lower-/.f64 N/A

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

      5. rec-exp N/A

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

      6. lower-+.f64 N/A

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

      7. rec-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-exp.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-neg.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 99.1%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. pow2 N/A

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

      3. lower-fma.f64 88.6

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

   10. Applied rewrites 88.6%

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

   11. Taylor expanded in y around 0

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

   13. Applied rewrites 82.1%

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

Alternative 16: 57.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-291}:\\ \;\;\;\;\frac{\left(\left(y \cdot y\right) \cdot -0.16666666666666666\right) \cdot y}{y}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-90}:\\ \;\;\;\;\mathsf{fma}\left(0.008333333333333333 \cdot \left(y \cdot y\right) - 0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, 0.5, 1\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)))
   (if (<= t_0 -2e-291)
     (/ (* (* (* y y) -0.16666666666666666) y) y)
     (if (<= t_0 2e-90)
       (fma
        (- (* 0.008333333333333333 (* y y)) 0.16666666666666666)
        (* y y)
        1.0)
       (* (fma (* x x) 0.5 1.0) 1.0)))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double tmp;
	if (t_0 <= -2e-291) {
		tmp = (((y * y) * -0.16666666666666666) * y) / y;
	} else if (t_0 <= 2e-90) {
		tmp = fma(((0.008333333333333333 * (y * y)) - 0.16666666666666666), (y * y), 1.0);
	} else {
		tmp = fma((x * x), 0.5, 1.0) * 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(y) / y)
	tmp = 0.0
	if (t_0 <= -2e-291)
		tmp = Float64(Float64(Float64(Float64(y * y) * -0.16666666666666666) * y) / y);
	elseif (t_0 <= 2e-90)
		tmp = fma(Float64(Float64(0.008333333333333333 * Float64(y * y)) - 0.16666666666666666), Float64(y * y), 1.0);
	else
		tmp = Float64(fma(Float64(x * x), 0.5, 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, -2e-291], N[(N[(N[(N[(y * y), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * y), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[t$95$0, 2e-90], N[(N[(N[(0.008333333333333333 * N[(y * y), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(N[(x * x), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] * 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 y) y) < -1.99999999999999992e-291

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. lift-sin.f64 N/A

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

      2. lift-/.f64 56.1

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

   5. Applied rewrites 56.1%

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

   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 33.2

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

   8. Applied rewrites 33.2%

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

   9. Taylor expanded in y around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 33.2

          \[\leadsto \frac{\left(\left(y \cdot y\right) \cdot -0.16666666666666666\right) \cdot y}{y} \]

   11. Applied rewrites 33.2%

       \[\leadsto \frac{\left(\left(y \cdot y\right) \cdot -0.16666666666666666\right) \cdot y}{y} \]

   if -1.99999999999999992e-291 < (/.f64 (sin.f64 y) y) < 1.99999999999999999e-90

   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 50.8

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

   5. Applied rewrites 50.8%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 51.0

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

   8. Applied rewrites 51.0%

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

   if 1.99999999999999999e-90 < (/.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

      \[\leadsto \color{blue}{\left(1 + \frac{1}{2} \cdot {x}^{2}\right)} \cdot \frac{\sin y}{y} \]
   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 78.2

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

   5. Applied rewrites 78.2%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 72.7

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

   8. Applied rewrites 72.7%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 71.6%

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

Alternative 17: 55.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-291}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666 \cdot y, y, 1\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-90}:\\ \;\;\;\;\mathsf{fma}\left(0.008333333333333333 \cdot \left(y \cdot y\right) - 0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, 0.5, 1\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)))
   (if (<= t_0 -2e-291)
     (fma (* -0.16666666666666666 y) y 1.0)
     (if (<= t_0 2e-90)
       (fma
        (- (* 0.008333333333333333 (* y y)) 0.16666666666666666)
        (* y y)
        1.0)
       (* (fma (* x x) 0.5 1.0) 1.0)))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double tmp;
	if (t_0 <= -2e-291) {
		tmp = fma((-0.16666666666666666 * y), y, 1.0);
	} else if (t_0 <= 2e-90) {
		tmp = fma(((0.008333333333333333 * (y * y)) - 0.16666666666666666), (y * y), 1.0);
	} else {
		tmp = fma((x * x), 0.5, 1.0) * 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(y) / y)
	tmp = 0.0
	if (t_0 <= -2e-291)
		tmp = fma(Float64(-0.16666666666666666 * y), y, 1.0);
	elseif (t_0 <= 2e-90)
		tmp = fma(Float64(Float64(0.008333333333333333 * Float64(y * y)) - 0.16666666666666666), Float64(y * y), 1.0);
	else
		tmp = Float64(fma(Float64(x * x), 0.5, 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, -2e-291], N[(N[(-0.16666666666666666 * y), $MachinePrecision] * y + 1.0), $MachinePrecision], If[LessEqual[t$95$0, 2e-90], N[(N[(N[(0.008333333333333333 * N[(y * y), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(N[(x * x), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] * 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 y) y) < -1.99999999999999992e-291

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. lift-sin.f64 N/A

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

      2. lift-/.f64 56.1

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

   5. Applied rewrites 56.1%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

         \[\leadsto {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) - \frac{1}{6}\right) + 1 \]

      2. *-commutative N/A

         \[\leadsto \left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) - \frac{1}{6}\right) \cdot {y}^{2} + 1 \]

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) - \frac{1}{6}, {y}^{2}, 1\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) \cdot {y}^{2} - \frac{1}{6}, {y}^{2}, 1\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) \cdot {y}^{2} - \frac{1}{6}, {y}^{2}, 1\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{-1}{5040} \cdot {y}^{2} + \frac{1}{120}\right) \cdot {y}^{2} - \frac{1}{6}, {y}^{2}, 1\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, {y}^{2}, \frac{1}{120}\right) \cdot {y}^{2} - \frac{1}{6}, {y}^{2}, 1\right) \]

      9. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot {y}^{2} - \frac{1}{6}, {y}^{2}, 1\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot {y}^{2} - \frac{1}{6}, {y}^{2}, 1\right) \]

      11. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot \left(y \cdot y\right) - \frac{1}{6}, {y}^{2}, 1\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot \left(y \cdot y\right) - \frac{1}{6}, {y}^{2}, 1\right) \]

      13. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot \left(y \cdot y\right) - \frac{1}{6}, y \cdot y, 1\right) \]

      14. lift-*.f64 40.9

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, y \cdot y, 0.008333333333333333\right) \cdot \left(y \cdot y\right) - 0.16666666666666666, y \cdot y, 1\right) \]

   8. Applied rewrites 40.9%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, y \cdot y, 0.008333333333333333\right) \cdot \left(y \cdot y\right) - 0.16666666666666666, \color{blue}{y \cdot y}, 1\right) \]

   9. Taylor expanded in y around 0

      \[\leadsto 1 \]
   10. Step-by-step derivation

   11. Applied rewrites 1.5%

       \[\leadsto 1 \]

   12. Taylor expanded in y around 0

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

       1. pow2 N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. +-commutative N/A

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

       5. *-commutative N/A

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

       6. lower-fma.f64 N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 28.1

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

   14. Applied rewrites 28.1%

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

   if -1.99999999999999992e-291 < (/.f64 (sin.f64 y) y) < 1.99999999999999999e-90

   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 50.8

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

   5. Applied rewrites 50.8%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 51.0

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

   8. Applied rewrites 51.0%

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

   if 1.99999999999999999e-90 < (/.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

      \[\leadsto \color{blue}{\left(1 + \frac{1}{2} \cdot {x}^{2}\right)} \cdot \frac{\sin y}{y} \]
   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 78.2

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

   5. Applied rewrites 78.2%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 72.7

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

   8. Applied rewrites 72.7%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 71.6%

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

Alternative 18: 51.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -2 \cdot 10^{-151}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666 \cdot y, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, 0.5, 1\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cosh x) (/ (sin y) y)) -2e-151)
   (fma (* -0.16666666666666666 y) y 1.0)
   (* (fma (* x x) 0.5 1.0) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((cosh(x) * (sin(y) / y)) <= -2e-151) {
		tmp = fma((-0.16666666666666666 * y), y, 1.0);
	} else {
		tmp = fma((x * x), 0.5, 1.0) * 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(cosh(x) * Float64(sin(y) / y)) <= -2e-151)
		tmp = fma(Float64(-0.16666666666666666 * y), y, 1.0);
	else
		tmp = Float64(fma(Float64(x * x), 0.5, 1.0) * 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Cosh[x], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], -2e-151], N[(N[(-0.16666666666666666 * y), $MachinePrecision] * y + 1.0), $MachinePrecision], N[(N[(N[(x * x), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < -1.9999999999999999e-151

   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 42.9

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

   5. Applied rewrites 42.9%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

         \[\leadsto {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) - \frac{1}{6}\right) + 1 \]

      2. *-commutative N/A

         \[\leadsto \left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) - \frac{1}{6}\right) \cdot {y}^{2} + 1 \]

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) - \frac{1}{6}, {y}^{2}, 1\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) \cdot {y}^{2} - \frac{1}{6}, {y}^{2}, 1\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) \cdot {y}^{2} - \frac{1}{6}, {y}^{2}, 1\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{-1}{5040} \cdot {y}^{2} + \frac{1}{120}\right) \cdot {y}^{2} - \frac{1}{6}, {y}^{2}, 1\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, {y}^{2}, \frac{1}{120}\right) \cdot {y}^{2} - \frac{1}{6}, {y}^{2}, 1\right) \]

      9. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot {y}^{2} - \frac{1}{6}, {y}^{2}, 1\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot {y}^{2} - \frac{1}{6}, {y}^{2}, 1\right) \]

      11. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot \left(y \cdot y\right) - \frac{1}{6}, {y}^{2}, 1\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot \left(y \cdot y\right) - \frac{1}{6}, {y}^{2}, 1\right) \]

      13. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot \left(y \cdot y\right) - \frac{1}{6}, y \cdot y, 1\right) \]

      14. lift-*.f64 52.9

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, y \cdot y, 0.008333333333333333\right) \cdot \left(y \cdot y\right) - 0.16666666666666666, y \cdot y, 1\right) \]

   8. Applied rewrites 52.9%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, y \cdot y, 0.008333333333333333\right) \cdot \left(y \cdot y\right) - 0.16666666666666666, \color{blue}{y \cdot y}, 1\right) \]

   9. Taylor expanded in y around 0

      \[\leadsto 1 \]
   10. Step-by-step derivation

   11. Applied rewrites 1.1%

       \[\leadsto 1 \]

   12. Taylor expanded in y around 0

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

       1. pow2 N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. +-commutative N/A

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

       5. *-commutative N/A

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

       6. lower-fma.f64 N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 36.2

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

   14. Applied rewrites 36.2%

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

   if -1.9999999999999999e-151 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y))

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + \frac{1}{2} \cdot {x}^{2}\right)} \cdot \frac{\sin y}{y} \]
   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 77.6

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

   5. Applied rewrites 77.6%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 61.2

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

   8. Applied rewrites 61.2%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 53.4%

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

Alternative 19: 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 20: 32.4% accurate, 18.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in x around 0

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

   1. lift-sin.f64 N/A

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

   2. lift-/.f64 52.0

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

5. Applied rewrites 52.0%

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

6. Taylor expanded in y around 0

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

   1. +-commutative N/A

      \[\leadsto {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) - \frac{1}{6}\right) + 1 \]

   2. *-commutative N/A

      \[\leadsto \left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) - \frac{1}{6}\right) \cdot {y}^{2} + 1 \]

   3. lower-fma.f64 N/A

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

   4. lower--.f64 N/A

      \[\leadsto \mathsf{fma}\left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) - \frac{1}{6}, {y}^{2}, 1\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) \cdot {y}^{2} - \frac{1}{6}, {y}^{2}, 1\right) \]

   6. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) \cdot {y}^{2} - \frac{1}{6}, {y}^{2}, 1\right) \]

   7. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(\left(\frac{-1}{5040} \cdot {y}^{2} + \frac{1}{120}\right) \cdot {y}^{2} - \frac{1}{6}, {y}^{2}, 1\right) \]

   8. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, {y}^{2}, \frac{1}{120}\right) \cdot {y}^{2} - \frac{1}{6}, {y}^{2}, 1\right) \]

   9. pow2 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot {y}^{2} - \frac{1}{6}, {y}^{2}, 1\right) \]

   10. lift-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot {y}^{2} - \frac{1}{6}, {y}^{2}, 1\right) \]

   11. pow2 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot \left(y \cdot y\right) - \frac{1}{6}, {y}^{2}, 1\right) \]

   12. lift-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot \left(y \cdot y\right) - \frac{1}{6}, {y}^{2}, 1\right) \]

   13. pow2 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot \left(y \cdot y\right) - \frac{1}{6}, y \cdot y, 1\right) \]

   14. lift-*.f64 34.4

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, y \cdot y, 0.008333333333333333\right) \cdot \left(y \cdot y\right) - 0.16666666666666666, y \cdot y, 1\right) \]

8. Applied rewrites 34.4%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, y \cdot y, 0.008333333333333333\right) \cdot \left(y \cdot y\right) - 0.16666666666666666, \color{blue}{y \cdot y}, 1\right) \]

9. Taylor expanded in y around 0

   \[\leadsto 1 \]
10. Step-by-step derivation

11. Applied rewrites 23.9%

    \[\leadsto 1 \]

12. Taylor expanded in y around 0

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

    1. pow2 N/A

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

    2. *-commutative N/A

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

    3. associate-*r* N/A

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

    4. +-commutative N/A

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

    5. *-commutative N/A

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

    6. lower-fma.f64 N/A

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

    7. *-commutative N/A

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

    8. lower-*.f64 30.7

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

14. Applied rewrites 30.7%

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

Alternative 21: 26.8% accurate, 217.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 1.0)

C

double code(double x, double y) {
	return 1.0;
}

Fortran

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
end function

Java

public static double code(double x, double y) {
	return 1.0;
}

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0;
end

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in 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 52.0

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

5. Applied rewrites 52.0%

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

6. Taylor expanded in y around 0

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

   1. +-commutative N/A

      \[\leadsto {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) - \frac{1}{6}\right) + 1 \]

   2. *-commutative N/A

      \[\leadsto \left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) - \frac{1}{6}\right) \cdot {y}^{2} + 1 \]

   3. lower-fma.f64 N/A

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

   4. lower--.f64 N/A

      \[\leadsto \mathsf{fma}\left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) - \frac{1}{6}, {y}^{2}, 1\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) \cdot {y}^{2} - \frac{1}{6}, {y}^{2}, 1\right) \]

   6. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) \cdot {y}^{2} - \frac{1}{6}, {y}^{2}, 1\right) \]

   7. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(\left(\frac{-1}{5040} \cdot {y}^{2} + \frac{1}{120}\right) \cdot {y}^{2} - \frac{1}{6}, {y}^{2}, 1\right) \]

   8. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, {y}^{2}, \frac{1}{120}\right) \cdot {y}^{2} - \frac{1}{6}, {y}^{2}, 1\right) \]

   9. pow2 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot {y}^{2} - \frac{1}{6}, {y}^{2}, 1\right) \]

   10. lift-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot {y}^{2} - \frac{1}{6}, {y}^{2}, 1\right) \]

   11. pow2 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot \left(y \cdot y\right) - \frac{1}{6}, {y}^{2}, 1\right) \]

   12. lift-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot \left(y \cdot y\right) - \frac{1}{6}, {y}^{2}, 1\right) \]

   13. pow2 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot \left(y \cdot y\right) - \frac{1}{6}, y \cdot y, 1\right) \]

   14. lift-*.f64 34.4

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, y \cdot y, 0.008333333333333333\right) \cdot \left(y \cdot y\right) - 0.16666666666666666, y \cdot y, 1\right) \]

8. Applied rewrites 34.4%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, y \cdot y, 0.008333333333333333\right) \cdot \left(y \cdot y\right) - 0.16666666666666666, \color{blue}{y \cdot y}, 1\right) \]

9. Taylor expanded in y around 0

   \[\leadsto 1 \]
10. Step-by-step derivation

11. Applied rewrites 23.9%

    \[\leadsto 1 \]
12. 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 2025038 
(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)))