Linear.Quaternion:$csinh from linear-1.19.1.3

Percentage Accurate: 99.9% → 99.9%

Time: 4.8s

Alternatives: 23

Speedup: 1.0×

• 50.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

Alternative 2: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ t_1 := \cosh x \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\cosh x \cdot \left(\left(y \cdot y\right) \cdot -0.16666666666666666\right)\\ \mathbf{elif}\;t\_1 \leq 0.9999999999999869:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)) (t_1 (* (cosh x) t_0)))
   (if (<= t_1 (- INFINITY))
     (* (cosh x) (* (* y y) -0.16666666666666666))
     (if (<= t_1 0.9999999999999869)
       (* (fma (fma 0.041666666666666664 (* x x) 0.5) (* x x) 1.0) t_0)
       (* (cosh x) 1.0)))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double t_1 = cosh(x) * t_0;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = cosh(x) * ((y * y) * -0.16666666666666666);
	} else if (t_1 <= 0.9999999999999869) {
		tmp = fma(fma(0.041666666666666664, (x * x), 0.5), (x * x), 1.0) * t_0;
	} else {
		tmp = cosh(x) * 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(y) / y)
	t_1 = Float64(cosh(x) * t_0)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(cosh(x) * Float64(Float64(y * y) * -0.16666666666666666));
	elseif (t_1 <= 0.9999999999999869)
		tmp = Float64(fma(fma(0.041666666666666664, Float64(x * x), 0.5), Float64(x * x), 1.0) * t_0);
	else
		tmp = Float64(cosh(x) * 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cosh[x], $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[Cosh[x], $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.9999999999999869], N[(N[(N[(0.041666666666666664 * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[Cosh[x], $MachinePrecision] * 1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 100.0

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

   8. Applied rewrites 100.0%

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

   if -inf.0 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < 0.9999999999999869

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 99.6

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

   5. Applied rewrites 99.6%

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

   if 0.9999999999999869 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 100.0%

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

Alternative 3: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ t_1 := \cosh x \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\cosh x \cdot \left(\left(y \cdot y\right) \cdot -0.16666666666666666\right)\\ \mathbf{elif}\;t\_1 \leq 0.9999999999999869:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, 0.5, 1\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)) (t_1 (* (cosh x) t_0)))
   (if (<= t_1 (- INFINITY))
     (* (cosh x) (* (* y y) -0.16666666666666666))
     (if (<= t_1 0.9999999999999869)
       (* (fma (* x x) 0.5 1.0) t_0)
       (* (cosh x) 1.0)))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double t_1 = cosh(x) * t_0;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = cosh(x) * ((y * y) * -0.16666666666666666);
	} else if (t_1 <= 0.9999999999999869) {
		tmp = fma((x * x), 0.5, 1.0) * t_0;
	} else {
		tmp = cosh(x) * 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(y) / y)
	t_1 = Float64(cosh(x) * t_0)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(cosh(x) * Float64(Float64(y * y) * -0.16666666666666666));
	elseif (t_1 <= 0.9999999999999869)
		tmp = Float64(fma(Float64(x * x), 0.5, 1.0) * t_0);
	else
		tmp = Float64(cosh(x) * 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cosh[x], $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[Cosh[x], $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.9999999999999869], N[(N[(N[(x * x), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[Cosh[x], $MachinePrecision] * 1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 100.0

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

   8. Applied rewrites 100.0%

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

   if -inf.0 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < 0.9999999999999869

   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 99.1

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

   5. Applied rewrites 99.1%

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

   if 0.9999999999999869 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 100.0%

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

Alternative 4: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ t_1 := \cosh x \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\cosh x \cdot \left(\left(y \cdot y\right) \cdot -0.16666666666666666\right)\\ \mathbf{elif}\;t\_1 \leq 0.9999999999999869:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)) (t_1 (* (cosh x) t_0)))
   (if (<= t_1 (- INFINITY))
     (* (cosh x) (* (* y y) -0.16666666666666666))
     (if (<= t_1 0.9999999999999869) t_0 (* (cosh x) 1.0)))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double t_1 = cosh(x) * t_0;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = cosh(x) * ((y * y) * -0.16666666666666666);
	} else if (t_1 <= 0.9999999999999869) {
		tmp = t_0;
	} else {
		tmp = cosh(x) * 1.0;
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double t_0 = Math.sin(y) / y;
	double t_1 = Math.cosh(x) * t_0;
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = Math.cosh(x) * ((y * y) * -0.16666666666666666);
	} else if (t_1 <= 0.9999999999999869) {
		tmp = t_0;
	} else {
		tmp = Math.cosh(x) * 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sin(y) / y
	t_1 = math.cosh(x) * t_0
	tmp = 0
	if t_1 <= -math.inf:
		tmp = math.cosh(x) * ((y * y) * -0.16666666666666666)
	elif t_1 <= 0.9999999999999869:
		tmp = t_0
	else:
		tmp = math.cosh(x) * 1.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sin(y) / y)
	t_1 = Float64(cosh(x) * t_0)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(cosh(x) * Float64(Float64(y * y) * -0.16666666666666666));
	elseif (t_1 <= 0.9999999999999869)
		tmp = t_0;
	else
		tmp = Float64(cosh(x) * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sin(y) / y;
	t_1 = cosh(x) * t_0;
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = cosh(x) * ((y * y) * -0.16666666666666666);
	elseif (t_1 <= 0.9999999999999869)
		tmp = t_0;
	else
		tmp = cosh(x) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cosh[x], $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[Cosh[x], $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.9999999999999869], t$95$0, N[(N[Cosh[x], $MachinePrecision] * 1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 100.0

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

   8. Applied rewrites 100.0%

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

   if -inf.0 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < 0.9999999999999869

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. lift-sin.f64 N/A

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

      2. lift-/.f64 98.6

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

   5. Applied rewrites 98.6%

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

   if 0.9999999999999869 < (*.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: 98.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ t_1 := \cosh x \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{elif}\;t\_1 \leq 0.9999999999999869:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)) (t_1 (* (cosh x) t_0)))
   (if (<= t_1 (- INFINITY))
     (*
      (fma
       (fma
        (fma 0.001388888888888889 (* x x) 0.041666666666666664)
        (* x x)
        0.5)
       (* x x)
       1.0)
      (fma -0.16666666666666666 (* y y) 1.0))
     (if (<= t_1 0.9999999999999869) t_0 (* (cosh x) 1.0)))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double t_1 = cosh(x) * t_0;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma(fma(fma(0.001388888888888889, (x * x), 0.041666666666666664), (x * x), 0.5), (x * x), 1.0) * fma(-0.16666666666666666, (y * y), 1.0);
	} else if (t_1 <= 0.9999999999999869) {
		tmp = t_0;
	} else {
		tmp = cosh(x) * 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(y) / y)
	t_1 = Float64(cosh(x) * t_0)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(fma(fma(fma(0.001388888888888889, Float64(x * x), 0.041666666666666664), Float64(x * x), 0.5), Float64(x * x), 1.0) * fma(-0.16666666666666666, Float64(y * y), 1.0));
	elseif (t_1 <= 0.9999999999999869)
		tmp = t_0;
	else
		tmp = Float64(cosh(x) * 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cosh[x], $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(N[(0.001388888888888889 * N[(x * x), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.9999999999999869], t$95$0, N[(N[Cosh[x], $MachinePrecision] * 1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 93.2

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

   8. Applied rewrites 93.2%

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

   if -inf.0 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < 0.9999999999999869

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. lift-sin.f64 N/A

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

      2. lift-/.f64 98.6

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

   5. Applied rewrites 98.6%

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

   if 0.9999999999999869 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 100.0%

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

Alternative 6: 75.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -1 \cdot 10^{-128}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cosh x) (/ (sin y) y)) -1e-128)
   (*
    (fma
     (fma (fma 0.001388888888888889 (* x x) 0.041666666666666664) (* x x) 0.5)
     (* x x)
     1.0)
    (fma -0.16666666666666666 (* y y) 1.0))
   (* (cosh x) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((cosh(x) * (sin(y) / y)) <= -1e-128) {
		tmp = fma(fma(fma(0.001388888888888889, (x * x), 0.041666666666666664), (x * x), 0.5), (x * x), 1.0) * fma(-0.16666666666666666, (y * y), 1.0);
	} else {
		tmp = cosh(x) * 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(cosh(x) * Float64(sin(y) / y)) <= -1e-128)
		tmp = Float64(fma(fma(fma(0.001388888888888889, Float64(x * x), 0.041666666666666664), Float64(x * x), 0.5), Float64(x * x), 1.0) * fma(-0.16666666666666666, Float64(y * y), 1.0));
	else
		tmp = Float64(cosh(x) * 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Cosh[x], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], -1e-128], N[(N[(N[(N[(0.001388888888888889 * N[(x * x), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Cosh[x], $MachinePrecision] * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < -1.00000000000000005e-128

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 68.1

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

   5. Applied rewrites 68.1%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 63.5

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

   8. Applied rewrites 63.5%

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

   if -1.00000000000000005e-128 < (*.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 82.6%

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

Alternative 7: 66.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-304}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, 0.5, 1\right) \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-64}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(y \cdot y\right) \cdot 0.008333333333333333, y \cdot y, 1\right) \cdot y}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)))
   (if (<= t_0 -5e-304)
     (* (fma (* x x) 0.5 1.0) (fma -0.16666666666666666 (* y y) 1.0))
     (if (<= t_0 2e-64)
       (/ (* (fma (* (* y y) 0.008333333333333333) (* y y) 1.0) y) y)
       (* (fma (fma 0.041666666666666664 (* x x) 0.5) (* x x) 1.0) 1.0)))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double tmp;
	if (t_0 <= -5e-304) {
		tmp = fma((x * x), 0.5, 1.0) * fma(-0.16666666666666666, (y * y), 1.0);
	} else if (t_0 <= 2e-64) {
		tmp = (fma(((y * y) * 0.008333333333333333), (y * y), 1.0) * y) / y;
	} else {
		tmp = fma(fma(0.041666666666666664, (x * x), 0.5), (x * x), 1.0) * 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(y) / y)
	tmp = 0.0
	if (t_0 <= -5e-304)
		tmp = Float64(fma(Float64(x * x), 0.5, 1.0) * fma(-0.16666666666666666, Float64(y * y), 1.0));
	elseif (t_0 <= 2e-64)
		tmp = Float64(Float64(fma(Float64(Float64(y * y) * 0.008333333333333333), Float64(y * y), 1.0) * y) / y);
	else
		tmp = Float64(fma(fma(0.041666666666666664, Float64(x * x), 0.5), Float64(x * x), 1.0) * 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-304], N[(N[(N[(x * x), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] * N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e-64], N[(N[(N[(N[(N[(y * y), $MachinePrecision] * 0.008333333333333333), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision] / y), $MachinePrecision], N[(N[(N[(0.041666666666666664 * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 y) y) < -4.99999999999999965e-304

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 55.3

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

   5. Applied rewrites 55.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 29.8%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. pow2 N/A

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

       5. lift-*.f64 44.6

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

   11. Applied rewrites 44.6%

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

   if -4.99999999999999965e-304 < (/.f64 (sin.f64 y) y) < 1.99999999999999993e-64

   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 42.4

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

   5. Applied rewrites 42.4%

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

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 59.6

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

   8. Applied rewrites 59.6%

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

   9. Taylor expanded in y around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 59.6

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

   11. Applied rewrites 59.6%

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

   if 1.99999999999999993e-64 < (/.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 98.0%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 87.2

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

   8. Applied rewrites 87.2%

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

Alternative 8: 68.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -5 \cdot 10^{-304}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, -0.16666666666666666\right), y \cdot y, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cosh x) (/ (sin y) y)) -5e-304)
   (*
    (fma
     (fma (fma 0.001388888888888889 (* x x) 0.041666666666666664) (* x x) 0.5)
     (* x x)
     1.0)
    (fma -0.16666666666666666 (* y y) 1.0))
   (*
    (fma (fma 0.041666666666666664 (* x x) 0.5) (* x x) 1.0)
    (fma
     (fma (* y y) 0.008333333333333333 -0.16666666666666666)
     (* y y)
     1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((cosh(x) * (sin(y) / y)) <= -5e-304) {
		tmp = fma(fma(fma(0.001388888888888889, (x * x), 0.041666666666666664), (x * x), 0.5), (x * x), 1.0) * fma(-0.16666666666666666, (y * y), 1.0);
	} else {
		tmp = fma(fma(0.041666666666666664, (x * x), 0.5), (x * x), 1.0) * fma(fma((y * y), 0.008333333333333333, -0.16666666666666666), (y * y), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(cosh(x) * Float64(sin(y) / y)) <= -5e-304)
		tmp = Float64(fma(fma(fma(0.001388888888888889, Float64(x * x), 0.041666666666666664), Float64(x * x), 0.5), Float64(x * x), 1.0) * fma(-0.16666666666666666, Float64(y * y), 1.0));
	else
		tmp = Float64(fma(fma(0.041666666666666664, Float64(x * x), 0.5), Float64(x * x), 1.0) * fma(fma(Float64(y * y), 0.008333333333333333, -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], -5e-304], N[(N[(N[(N[(0.001388888888888889 * N[(x * x), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.041666666666666664 * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + -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)) < -4.99999999999999965e-304

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 55.3

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

   5. Applied rewrites 55.3%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 51.7

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

   8. Applied rewrites 51.7%

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

   if -4.99999999999999965e-304 < (*.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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 88.2

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

   5. Applied rewrites 88.2%

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

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. pow2 N/A

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

      3. pow2 N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. lift-*.f64 78.7

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

   8. Applied rewrites 78.7%

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

Alternative 9: 66.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -5 \cdot 10^{-304}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, y \cdot y, 0.008333333333333333\right) \cdot \left(y \cdot y\right) - 0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, -0.16666666666666666\right), y \cdot y, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cosh x) (/ (sin y) y)) -5e-304)
   (*
    1.0
    (fma
     (-
      (* (fma -0.0001984126984126984 (* y y) 0.008333333333333333) (* y y))
      0.16666666666666666)
     (* y y)
     1.0))
   (*
    (fma (fma 0.041666666666666664 (* x x) 0.5) (* x x) 1.0)
    (fma
     (fma (* y y) 0.008333333333333333 -0.16666666666666666)
     (* y y)
     1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((cosh(x) * (sin(y) / y)) <= -5e-304) {
		tmp = 1.0 * fma(((fma(-0.0001984126984126984, (y * y), 0.008333333333333333) * (y * y)) - 0.16666666666666666), (y * y), 1.0);
	} else {
		tmp = fma(fma(0.041666666666666664, (x * x), 0.5), (x * x), 1.0) * fma(fma((y * y), 0.008333333333333333, -0.16666666666666666), (y * y), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(cosh(x) * Float64(sin(y) / y)) <= -5e-304)
		tmp = Float64(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(0.041666666666666664, Float64(x * x), 0.5), Float64(x * x), 1.0) * fma(fma(Float64(y * y), 0.008333333333333333, -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], -5e-304], N[(1.0 * 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[(N[(N[(0.041666666666666664 * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + -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)) < -4.99999999999999965e-304

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 55.3

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

   5. Applied rewrites 55.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 29.8%

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

   9. Taylor expanded in y around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. lower--.f64 N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. +-commutative N/A

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

       8. lower-fma.f64 N/A

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

       9. pow2 N/A

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

       10. lift-*.f64 N/A

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

       11. pow2 N/A

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

       12. lift-*.f64 N/A

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

       13. pow2 N/A

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

       14. lift-*.f64 47.8

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

   11. Applied rewrites 47.8%

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

   if -4.99999999999999965e-304 < (*.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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 88.2

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

   5. Applied rewrites 88.2%

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

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. pow2 N/A

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

      3. pow2 N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. lift-*.f64 78.7

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

   8. Applied rewrites 78.7%

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

Alternative 10: 66.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-304}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, 0.5, 1\right) \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-87}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(\left(y \cdot y\right) \cdot 0.008333333333333333 - 0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)))
   (if (<= t_0 -5e-304)
     (* (fma (* x x) 0.5 1.0) (fma -0.16666666666666666 (* y y) 1.0))
     (if (<= t_0 2e-87)
       (*
        1.0
        (fma
         (- (* (* y y) 0.008333333333333333) 0.16666666666666666)
         (* y y)
         1.0))
       (* (fma (fma 0.041666666666666664 (* x x) 0.5) (* x x) 1.0) 1.0)))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double tmp;
	if (t_0 <= -5e-304) {
		tmp = fma((x * x), 0.5, 1.0) * fma(-0.16666666666666666, (y * y), 1.0);
	} else if (t_0 <= 2e-87) {
		tmp = 1.0 * fma((((y * y) * 0.008333333333333333) - 0.16666666666666666), (y * y), 1.0);
	} else {
		tmp = fma(fma(0.041666666666666664, (x * x), 0.5), (x * x), 1.0) * 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(y) / y)
	tmp = 0.0
	if (t_0 <= -5e-304)
		tmp = Float64(fma(Float64(x * x), 0.5, 1.0) * fma(-0.16666666666666666, Float64(y * y), 1.0));
	elseif (t_0 <= 2e-87)
		tmp = Float64(1.0 * fma(Float64(Float64(Float64(y * y) * 0.008333333333333333) - 0.16666666666666666), Float64(y * y), 1.0));
	else
		tmp = Float64(fma(fma(0.041666666666666664, Float64(x * x), 0.5), Float64(x * x), 1.0) * 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-304], N[(N[(N[(x * x), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] * N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e-87], N[(1.0 * N[(N[(N[(N[(y * y), $MachinePrecision] * 0.008333333333333333), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.041666666666666664 * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 y) y) < -4.99999999999999965e-304

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 55.3

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

   5. Applied rewrites 55.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 29.8%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. pow2 N/A

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

       5. lift-*.f64 44.6

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

   11. Applied rewrites 44.6%

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

   if -4.99999999999999965e-304 < (/.f64 (sin.f64 y) y) < 2.00000000000000004e-87

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 0.5

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

   5. Applied rewrites 0.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 0.5%

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

   9. Taylor expanded in y around 0

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

       1. *-commutative N/A

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

       2. pow2 N/A

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

       3. pow2 N/A

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

       4. +-commutative N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. lift--.f64 N/A

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

       8. lift-fma.f64 N/A

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

       9. lift-*.f64 60.4

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

       10. lift-*.f64 N/A

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

       11. lift-*.f64 N/A

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

       12. pow2 N/A

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

       13. *-commutative N/A

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

       14. lower-*.f64 N/A

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

       15. pow2 N/A

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

       16. lift-*.f64 60.4

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

   11. Applied rewrites 60.4%

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

   if 2.00000000000000004e-87 < (/.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 95.1%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 83.8

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

   8. Applied rewrites 83.8%

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

Alternative 11: 67.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -1 \cdot 10^{-128}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, y \cdot y, 0.008333333333333333\right) \cdot \left(y \cdot y\right) - 0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cosh x) (/ (sin y) y)) -1e-128)
   (*
    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)
    1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((cosh(x) * (sin(y) / y)) <= -1e-128) {
		tmp = 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) * 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(cosh(x) * Float64(sin(y) / y)) <= -1e-128)
		tmp = Float64(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) * 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], -1e-128], N[(1.0 * 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[(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] * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < -1.00000000000000005e-128

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 68.1

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

   5. Applied rewrites 68.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 36.5%

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

   9. Taylor expanded in y around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. lower--.f64 N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. +-commutative N/A

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

       8. lower-fma.f64 N/A

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

       9. pow2 N/A

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

       10. lift-*.f64 N/A

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

       11. pow2 N/A

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

       12. lift-*.f64 N/A

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

       13. pow2 N/A

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

       14. lift-*.f64 58.7

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

   11. Applied rewrites 58.7%

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

   if -1.00000000000000005e-128 < (*.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 82.6%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 74.6

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

   8. Applied rewrites 74.6%

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

Alternative 12: 68.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -1 \cdot 10^{-128}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cosh x) (/ (sin y) y)) -1e-128)
   (*
    (fma (fma 0.041666666666666664 (* x x) 0.5) (* x x) 1.0)
    (fma -0.16666666666666666 (* y y) 1.0))
   (*
    (fma
     (fma (fma 0.001388888888888889 (* x x) 0.041666666666666664) (* x x) 0.5)
     (* x x)
     1.0)
    1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((cosh(x) * (sin(y) / y)) <= -1e-128) {
		tmp = fma(fma(0.041666666666666664, (x * x), 0.5), (x * x), 1.0) * fma(-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) * 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(cosh(x) * Float64(sin(y) / y)) <= -1e-128)
		tmp = Float64(fma(fma(0.041666666666666664, Float64(x * x), 0.5), Float64(x * x), 1.0) * fma(-0.16666666666666666, Float64(y * y), 1.0));
	else
		tmp = Float64(fma(fma(fma(0.001388888888888889, Float64(x * x), 0.041666666666666664), Float64(x * x), 0.5), Float64(x * x), 1.0) * 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Cosh[x], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], -1e-128], N[(N[(N[(0.041666666666666664 * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], 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] * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < -1.00000000000000005e-128

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 68.1

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

   5. Applied rewrites 68.1%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 57.0

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

   8. Applied rewrites 57.0%

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

   if -1.00000000000000005e-128 < (*.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 82.6%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 74.6

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

   8. Applied rewrites 74.6%

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

Alternative 13: 68.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -1 \cdot 10^{-128}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left(x \cdot x\right) \cdot 0.001388888888888889, x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cosh x) (/ (sin y) y)) -1e-128)
   (*
    (fma (fma 0.041666666666666664 (* x x) 0.5) (* x x) 1.0)
    (fma -0.16666666666666666 (* y y) 1.0))
   (*
    (fma (fma (* (* x x) 0.001388888888888889) (* x x) 0.5) (* x x) 1.0)
    1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((cosh(x) * (sin(y) / y)) <= -1e-128) {
		tmp = fma(fma(0.041666666666666664, (x * x), 0.5), (x * x), 1.0) * fma(-0.16666666666666666, (y * y), 1.0);
	} else {
		tmp = fma(fma(((x * x) * 0.001388888888888889), (x * x), 0.5), (x * x), 1.0) * 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(cosh(x) * Float64(sin(y) / y)) <= -1e-128)
		tmp = Float64(fma(fma(0.041666666666666664, Float64(x * x), 0.5), Float64(x * x), 1.0) * fma(-0.16666666666666666, Float64(y * y), 1.0));
	else
		tmp = Float64(fma(fma(Float64(Float64(x * x) * 0.001388888888888889), Float64(x * x), 0.5), Float64(x * x), 1.0) * 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Cosh[x], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], -1e-128], N[(N[(N[(0.041666666666666664 * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(x * x), $MachinePrecision] * 0.001388888888888889), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < -1.00000000000000005e-128

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 68.1

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

   5. Applied rewrites 68.1%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 57.0

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

   8. Applied rewrites 57.0%

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

   if -1.00000000000000005e-128 < (*.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 82.6%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 74.6

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

   8. Applied rewrites 74.6%

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

   9. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 74.4

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

   11. Applied rewrites 74.4%

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

Alternative 14: 68.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -1 \cdot 10^{-128}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, 0.5, 1\right) \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left(x \cdot x\right) \cdot 0.001388888888888889, x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cosh x) (/ (sin y) y)) -1e-128)
   (* (fma (* x x) 0.5 1.0) (fma -0.16666666666666666 (* y y) 1.0))
   (*
    (fma (fma (* (* x x) 0.001388888888888889) (* x x) 0.5) (* x x) 1.0)
    1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((cosh(x) * (sin(y) / y)) <= -1e-128) {
		tmp = fma((x * x), 0.5, 1.0) * fma(-0.16666666666666666, (y * y), 1.0);
	} else {
		tmp = fma(fma(((x * x) * 0.001388888888888889), (x * x), 0.5), (x * x), 1.0) * 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(cosh(x) * Float64(sin(y) / y)) <= -1e-128)
		tmp = Float64(fma(Float64(x * x), 0.5, 1.0) * fma(-0.16666666666666666, Float64(y * y), 1.0));
	else
		tmp = Float64(fma(fma(Float64(Float64(x * x) * 0.001388888888888889), Float64(x * x), 0.5), Float64(x * x), 1.0) * 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Cosh[x], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], -1e-128], N[(N[(N[(x * x), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] * N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(x * x), $MachinePrecision] * 0.001388888888888889), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < -1.00000000000000005e-128

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 68.1

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

   5. Applied rewrites 68.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 36.5%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. pow2 N/A

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

       5. lift-*.f64 54.8

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

   11. Applied rewrites 54.8%

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

   if -1.00000000000000005e-128 < (*.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 82.6%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 74.6

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

   8. Applied rewrites 74.6%

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

   9. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 74.4

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

   11. Applied rewrites 74.4%

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

Alternative 15: 64.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -1 \cdot 10^{-128}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, 0.5, 1\right) \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cosh x) (/ (sin y) y)) -1e-128)
   (* (fma (* x x) 0.5 1.0) (fma -0.16666666666666666 (* y y) 1.0))
   (* (fma (fma 0.041666666666666664 (* x x) 0.5) (* x x) 1.0) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((cosh(x) * (sin(y) / y)) <= -1e-128) {
		tmp = fma((x * x), 0.5, 1.0) * fma(-0.16666666666666666, (y * y), 1.0);
	} else {
		tmp = fma(fma(0.041666666666666664, (x * x), 0.5), (x * x), 1.0) * 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(cosh(x) * Float64(sin(y) / y)) <= -1e-128)
		tmp = Float64(fma(Float64(x * x), 0.5, 1.0) * fma(-0.16666666666666666, Float64(y * y), 1.0));
	else
		tmp = Float64(fma(fma(0.041666666666666664, Float64(x * x), 0.5), Float64(x * x), 1.0) * 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Cosh[x], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], -1e-128], N[(N[(N[(x * x), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] * N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.041666666666666664 * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < -1.00000000000000005e-128

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 68.1

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

   5. Applied rewrites 68.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 36.5%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. pow2 N/A

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

       5. lift-*.f64 54.8

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

   11. Applied rewrites 54.8%

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

   if -1.00000000000000005e-128 < (*.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 82.6%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 71.5

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

   8. Applied rewrites 71.5%

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

Alternative 16: 64.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -1 \cdot 10^{-128}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, 0.5, 1\right) \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(x \cdot x\right) \cdot 0.041666666666666664, x \cdot x, 1\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cosh x) (/ (sin y) y)) -1e-128)
   (* (fma (* x x) 0.5 1.0) (fma -0.16666666666666666 (* y y) 1.0))
   (* (fma (* (* x x) 0.041666666666666664) (* x x) 1.0) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((cosh(x) * (sin(y) / y)) <= -1e-128) {
		tmp = fma((x * x), 0.5, 1.0) * fma(-0.16666666666666666, (y * y), 1.0);
	} else {
		tmp = fma(((x * x) * 0.041666666666666664), (x * x), 1.0) * 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(cosh(x) * Float64(sin(y) / y)) <= -1e-128)
		tmp = Float64(fma(Float64(x * x), 0.5, 1.0) * fma(-0.16666666666666666, Float64(y * y), 1.0));
	else
		tmp = Float64(fma(Float64(Float64(x * x) * 0.041666666666666664), Float64(x * x), 1.0) * 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Cosh[x], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], -1e-128], N[(N[(N[(x * x), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] * N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(x * x), $MachinePrecision] * 0.041666666666666664), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < -1.00000000000000005e-128

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 68.1

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

   5. Applied rewrites 68.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 36.5%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. pow2 N/A

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

       5. lift-*.f64 54.8

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

   11. Applied rewrites 54.8%

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

   if -1.00000000000000005e-128 < (*.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 82.6%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 71.5

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

   8. Applied rewrites 71.5%

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

   9. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 71.0

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

   11. Applied rewrites 71.0%

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

Alternative 17: 62.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -1 \cdot 10^{-128}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right) \cdot y}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(x \cdot x\right) \cdot 0.041666666666666664, x \cdot x, 1\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cosh x) (/ (sin y) y)) -1e-128)
   (/ (* (fma (* y y) -0.16666666666666666 1.0) y) y)
   (* (fma (* (* x x) 0.041666666666666664) (* x x) 1.0) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((cosh(x) * (sin(y) / y)) <= -1e-128) {
		tmp = (fma((y * y), -0.16666666666666666, 1.0) * y) / y;
	} else {
		tmp = fma(((x * x) * 0.041666666666666664), (x * x), 1.0) * 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(cosh(x) * Float64(sin(y) / y)) <= -1e-128)
		tmp = Float64(Float64(fma(Float64(y * y), -0.16666666666666666, 1.0) * y) / y);
	else
		tmp = Float64(fma(Float64(Float64(x * x) * 0.041666666666666664), Float64(x * x), 1.0) * 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Cosh[x], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], -1e-128], N[(N[(N[(N[(y * y), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * y), $MachinePrecision] / y), $MachinePrecision], N[(N[(N[(N[(x * x), $MachinePrecision] * 0.041666666666666664), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < -1.00000000000000005e-128

   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 33.4

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

   5. Applied rewrites 33.4%

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

   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 49.8

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

   8. Applied rewrites 49.8%

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

   if -1.00000000000000005e-128 < (*.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 82.6%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 71.5

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

   8. Applied rewrites 71.5%

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

   9. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 71.0

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

   11. Applied rewrites 71.0%

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

Alternative 18: 60.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -1 \cdot 10^{-128}:\\ \;\;\;\;1 \cdot \left(\left(-0.16666666666666666 \cdot y\right) \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(x \cdot x\right) \cdot 0.041666666666666664, x \cdot x, 1\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cosh x) (/ (sin y) y)) -1e-128)
   (* 1.0 (* (* -0.16666666666666666 y) y))
   (* (fma (* (* x x) 0.041666666666666664) (* x x) 1.0) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((cosh(x) * (sin(y) / y)) <= -1e-128) {
		tmp = 1.0 * ((-0.16666666666666666 * y) * y);
	} else {
		tmp = fma(((x * x) * 0.041666666666666664), (x * x), 1.0) * 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(cosh(x) * Float64(sin(y) / y)) <= -1e-128)
		tmp = Float64(1.0 * Float64(Float64(-0.16666666666666666 * y) * y));
	else
		tmp = Float64(fma(Float64(Float64(x * x) * 0.041666666666666664), Float64(x * x), 1.0) * 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Cosh[x], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], -1e-128], N[(1.0 * N[(N[(-0.16666666666666666 * y), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(x * x), $MachinePrecision] * 0.041666666666666664), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < -1.00000000000000005e-128

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 68.1

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

   5. Applied rewrites 68.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 36.5%

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

   9. Taylor expanded in y around inf

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

       1. pow2 N/A

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

       2. *-commutative N/A

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

       3. pow2 N/A

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

       4. pow2 N/A

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

       5. associate-*r* N/A

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

       6. lower-*.f64 N/A

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

       7. lower-*.f64 36.5

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

   11. Applied rewrites 36.5%

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

   if -1.00000000000000005e-128 < (*.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 82.6%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 71.5

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

   8. Applied rewrites 71.5%

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

   9. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 71.0

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

   11. Applied rewrites 71.0%

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

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -1 \cdot 10^{-128}:\\ \;\;\;\;1 \cdot \left(\left(-0.16666666666666666 \cdot y\right) \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(x \cdot x\right) \cdot 0.041666666666666664, x \cdot x, 1\right) \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 59.8% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cosh x) (/ (sin y) y)) 2.0)
   (* 1.0 (fma -0.16666666666666666 (* y y) 1.0))
   (* (* (* (fma (* x x) 0.041666666666666664 0.5) x) x) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((cosh(x) * (sin(y) / y)) <= 2.0) {
		tmp = 1.0 * fma(-0.16666666666666666, (y * y), 1.0);
	} else {
		tmp = ((fma((x * x), 0.041666666666666664, 0.5) * x) * x) * 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(cosh(x) * Float64(sin(y) / y)) <= 2.0)
		tmp = Float64(1.0 * fma(-0.16666666666666666, Float64(y * y), 1.0));
	else
		tmp = Float64(Float64(Float64(fma(Float64(x * x), 0.041666666666666664, 0.5) * x) * x) * 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Cosh[x], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], 2.0], N[(1.0 * N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(x * x), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 60.4

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

   5. Applied rewrites 60.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 49.5%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 81.0

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

   8. Applied rewrites 81.0%

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

   9. Taylor expanded in x around inf

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

       1. metadata-eval N/A

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

       2. pow-sqr N/A

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

       3. associate-*l* N/A

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

       4. distribute-lft-in N/A

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

       5. *-commutative N/A

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

       6. *-commutative N/A

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

       7. associate-*r* N/A

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

       8. rgt-mult-inverse N/A

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

       9. metadata-eval N/A

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

       10. pow2 N/A

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

       11. *-commutative N/A

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

       12. pow2 N/A

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

       13. associate-*r* N/A

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

       14. lower-*.f64 N/A

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

   11. Applied rewrites 81.0%

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

Alternative 20: 51.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -1 \cdot 10^{-128}:\\ \;\;\;\;1 \cdot \left(\left(-0.16666666666666666 \cdot y\right) \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, x \cdot x, 1\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cosh x) (/ (sin y) y)) -1e-128)
   (* 1.0 (* (* -0.16666666666666666 y) y))
   (* (fma 0.5 (* x x) 1.0) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((cosh(x) * (sin(y) / y)) <= -1e-128) {
		tmp = 1.0 * ((-0.16666666666666666 * y) * y);
	} else {
		tmp = fma(0.5, (x * x), 1.0) * 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(cosh(x) * Float64(sin(y) / y)) <= -1e-128)
		tmp = Float64(1.0 * Float64(Float64(-0.16666666666666666 * y) * y));
	else
		tmp = Float64(fma(0.5, Float64(x * x), 1.0) * 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Cosh[x], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], -1e-128], N[(1.0 * N[(N[(-0.16666666666666666 * y), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < -1.00000000000000005e-128

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 68.1

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

   5. Applied rewrites 68.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 36.5%

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

   9. Taylor expanded in y around inf

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

       1. pow2 N/A

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

       2. *-commutative N/A

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

       3. pow2 N/A

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

       4. pow2 N/A

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

       5. associate-*r* N/A

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

       6. lower-*.f64 N/A

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

       7. lower-*.f64 36.5

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

   11. Applied rewrites 36.5%

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

   if -1.00000000000000005e-128 < (*.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 82.6%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 71.5

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

   8. Applied rewrites 71.5%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 57.3%

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

4. Final simplification 53.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -1 \cdot 10^{-128}:\\ \;\;\;\;1 \cdot \left(\left(-0.16666666666666666 \cdot y\right) \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, x \cdot x, 1\right) \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 32.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -1 \cdot 10^{-128}:\\ \;\;\;\;1 \cdot \left(\left(-0.16666666666666666 \cdot y\right) \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cosh x) (/ (sin y) y)) -1e-128)
   (* 1.0 (* (* -0.16666666666666666 y) y))
   (* 1.0 1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((cosh(x) * (sin(y) / y)) <= -1e-128) {
		tmp = 1.0 * ((-0.16666666666666666 * y) * y);
	} else {
		tmp = 1.0 * 1.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((cosh(x) * (sin(y) / y)) <= (-1d-128)) then
        tmp = 1.0d0 * (((-0.16666666666666666d0) * y) * y)
    else
        tmp = 1.0d0 * 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((Math.cosh(x) * (Math.sin(y) / y)) <= -1e-128) {
		tmp = 1.0 * ((-0.16666666666666666 * y) * y);
	} else {
		tmp = 1.0 * 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (math.cosh(x) * (math.sin(y) / y)) <= -1e-128:
		tmp = 1.0 * ((-0.16666666666666666 * y) * y)
	else:
		tmp = 1.0 * 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(cosh(x) * Float64(sin(y) / y)) <= -1e-128)
		tmp = Float64(1.0 * Float64(Float64(-0.16666666666666666 * y) * y));
	else
		tmp = Float64(1.0 * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((cosh(x) * (sin(y) / y)) <= -1e-128)
		tmp = 1.0 * ((-0.16666666666666666 * y) * y);
	else
		tmp = 1.0 * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Cosh[x], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], -1e-128], N[(1.0 * N[(N[(-0.16666666666666666 * y), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], N[(1.0 * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < -1.00000000000000005e-128

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 68.1

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

   5. Applied rewrites 68.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 36.5%

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

   9. Taylor expanded in y around inf

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

       1. pow2 N/A

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

       2. *-commutative N/A

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

       3. pow2 N/A

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

       4. pow2 N/A

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

       5. associate-*r* N/A

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

       6. lower-*.f64 N/A

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

       7. lower-*.f64 36.5

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

   11. Applied rewrites 36.5%

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

   if -1.00000000000000005e-128 < (*.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 82.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 25.5%

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

4. Final simplification 27.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -1 \cdot 10^{-128}:\\ \;\;\;\;1 \cdot \left(\left(-0.16666666666666666 \cdot y\right) \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 31.6% accurate, 12.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 62.6

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

5. Applied rewrites 62.6%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 26.3%

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

Alternative 23: 26.3% accurate, 36.2× speedup

Math

\[\begin{array}{l} \\ 1 \cdot 1 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* 1.0 1.0))

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 * 1.0d0
end function

Java

public static double code(double x, double y) {
	return 1.0 * 1.0;
}

Python

def code(x, y):
	return 1.0 * 1.0

Julia

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

MATLAB

function tmp = code(x, y)
	tmp = 1.0 * 1.0;
end

Wolfram

code[x_, y_] := N[(1.0 * 1.0), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in y around 0

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

5. Applied rewrites 69.1%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 21.5%

   \[\leadsto \color{blue}{1} \cdot 1 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025085 
(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)))