Linear.Quaternion:$csinh from linear-1.19.1.3

Percentage Accurate: 99.9% → 99.9%

Time: 4.2s

Alternatives: 16

Speedup: 1.0×

• 49.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

Alternative 2: 98.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ t_1 := \cosh x \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{elif}\;t\_1 \leq 10^{-14}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)) (t_1 (* (cosh x) t_0)))
   (if (<= t_1 (- INFINITY))
     (*
      (fma
       (fma
        (fma (* x x) 0.001388888888888889 0.041666666666666664)
        (* x x)
        0.5)
       (* x x)
       1.0)
      (fma -0.16666666666666666 (* y y) 1.0))
     (if (<= t_1 1e-14)
       (* (fma (fma 0.041666666666666664 (* x x) 0.5) (* x x) 1.0) t_0)
       (* (cosh x) 1.0)))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double t_1 = cosh(x) * t_0;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma(fma(fma((x * x), 0.001388888888888889, 0.041666666666666664), (x * x), 0.5), (x * x), 1.0) * fma(-0.16666666666666666, (y * y), 1.0);
	} else if (t_1 <= 1e-14) {
		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(fma(fma(fma(Float64(x * x), 0.001388888888888889, 0.041666666666666664), Float64(x * x), 0.5), Float64(x * x), 1.0) * fma(-0.16666666666666666, Float64(y * y), 1.0));
	elseif (t_1 <= 1e-14)
		tmp = Float64(fma(fma(0.041666666666666664, Float64(x * x), 0.5), Float64(x * x), 1.0) * t_0);
	else
		tmp = Float64(cosh(x) * 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cosh[x], $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(N[(N[(x * x), $MachinePrecision] * 0.001388888888888889 + 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, 1e-14], 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 x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 100.0

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

   8. Applied rewrites 100.0%

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

   if -inf.0 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < 9.99999999999999999e-15

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 98.4

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

   5. Applied rewrites 98.4%

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

   if 9.99999999999999999e-15 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 100.0%

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

Alternative 3: 98.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ t_1 := \cosh x \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{elif}\;t\_1 \leq 10^{-14}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, 0.5, 1\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)) (t_1 (* (cosh x) t_0)))
   (if (<= t_1 (- INFINITY))
     (*
      (fma
       (fma
        (fma (* x x) 0.001388888888888889 0.041666666666666664)
        (* x x)
        0.5)
       (* x x)
       1.0)
      (fma -0.16666666666666666 (* y y) 1.0))
     (if (<= t_1 1e-14) (* (fma (* x x) 0.5 1.0) t_0) (* (cosh x) 1.0)))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double t_1 = cosh(x) * t_0;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma(fma(fma((x * x), 0.001388888888888889, 0.041666666666666664), (x * x), 0.5), (x * x), 1.0) * fma(-0.16666666666666666, (y * y), 1.0);
	} else if (t_1 <= 1e-14) {
		tmp = fma((x * x), 0.5, 1.0) * t_0;
	} else {
		tmp = cosh(x) * 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(y) / y)
	t_1 = Float64(cosh(x) * t_0)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(fma(fma(fma(Float64(x * x), 0.001388888888888889, 0.041666666666666664), Float64(x * x), 0.5), Float64(x * x), 1.0) * fma(-0.16666666666666666, Float64(y * y), 1.0));
	elseif (t_1 <= 1e-14)
		tmp = Float64(fma(Float64(x * x), 0.5, 1.0) * t_0);
	else
		tmp = Float64(cosh(x) * 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cosh[x], $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(N[(N[(x * x), $MachinePrecision] * 0.001388888888888889 + 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, 1e-14], 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 x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 100.0

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

   8. Applied rewrites 100.0%

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

   if -inf.0 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < 9.99999999999999999e-15

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + \frac{1}{2} \cdot {x}^{2}\right)} \cdot \frac{\sin y}{y} \]
   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 97.7

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

   5. Applied rewrites 97.7%

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

   if 9.99999999999999999e-15 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 100.0%

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

Alternative 4: 98.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cosh x \cdot \frac{\sin y}{y}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{elif}\;t\_0 \leq 10^{-14}:\\ \;\;\;\;\left(0.5 \cdot \sin y\right) \cdot \frac{\mathsf{fma}\left(x, x, 2\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (cosh x) (/ (sin y) y))))
   (if (<= t_0 (- INFINITY))
     (*
      (fma
       (fma
        (fma (* x x) 0.001388888888888889 0.041666666666666664)
        (* x x)
        0.5)
       (* x x)
       1.0)
      (fma -0.16666666666666666 (* y y) 1.0))
     (if (<= t_0 1e-14)
       (* (* 0.5 (sin y)) (/ (fma x x 2.0) y))
       (* (cosh x) 1.0)))))

C

double code(double x, double y) {
	double t_0 = cosh(x) * (sin(y) / y);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma(fma(fma((x * x), 0.001388888888888889, 0.041666666666666664), (x * x), 0.5), (x * x), 1.0) * fma(-0.16666666666666666, (y * y), 1.0);
	} else if (t_0 <= 1e-14) {
		tmp = (0.5 * sin(y)) * (fma(x, x, 2.0) / y);
	} else {
		tmp = cosh(x) * 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cosh(x) * Float64(sin(y) / y))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(fma(fma(fma(Float64(x * x), 0.001388888888888889, 0.041666666666666664), Float64(x * x), 0.5), Float64(x * x), 1.0) * fma(-0.16666666666666666, Float64(y * y), 1.0));
	elseif (t_0 <= 1e-14)
		tmp = Float64(Float64(0.5 * sin(y)) * Float64(fma(x, x, 2.0) / y));
	else
		tmp = Float64(cosh(x) * 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cosh[x], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[(N[(x * x), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e-14], N[(N[(0.5 * N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(N[(x * x + 2.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(N[Cosh[x], $MachinePrecision] * 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 100.0

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

   8. Applied rewrites 100.0%

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

   if -inf.0 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < 9.99999999999999999e-15

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. rec-exp N/A

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

      8. cosh-undef N/A

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

      9. lower-*.f64 N/A

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

      10. lift-cosh.f64 99.4

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

   5. Applied rewrites 99.4%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. pow2 N/A

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

      3. lower-fma.f64 97.5

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

   8. Applied rewrites 97.5%

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

   if 9.99999999999999999e-15 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 100.0%

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

Alternative 5: 98.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ t_1 := \cosh x \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{elif}\;t\_1 \leq 10^{-14}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot 1\\ \end{array} \end{array} \]

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 (* x x) 0.001388888888888889 0.041666666666666664)
        (* x x)
        0.5)
       (* x x)
       1.0)
      (fma -0.16666666666666666 (* y y) 1.0))
     (if (<= t_1 1e-14) 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((x * x), 0.001388888888888889, 0.041666666666666664), (x * x), 0.5), (x * x), 1.0) * fma(-0.16666666666666666, (y * y), 1.0);
	} else if (t_1 <= 1e-14) {
		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(Float64(x * x), 0.001388888888888889, 0.041666666666666664), Float64(x * x), 0.5), Float64(x * x), 1.0) * fma(-0.16666666666666666, Float64(y * y), 1.0));
	elseif (t_1 <= 1e-14)
		tmp = t_0;
	else
		tmp = Float64(cosh(x) * 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cosh[x], $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(N[(N[(x * x), $MachinePrecision] * 0.001388888888888889 + 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, 1e-14], 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. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 100.0

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

   8. Applied rewrites 100.0%

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

   if -inf.0 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < 9.99999999999999999e-15

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. lift-sin.f64 N/A

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

      2. lift-/.f64 96.4

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

   5. Applied rewrites 96.4%

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

   if 9.99999999999999999e-15 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 100.0%

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

Alternative 6: 96.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ \mathbf{if}\;\cosh x \cdot t\_0 \leq 10^{-14}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)))
   (if (<= (* (cosh x) t_0) 1e-14)
     (*
      (fma
       (fma
        (fma 0.001388888888888889 (* x x) 0.041666666666666664)
        (* x x)
        0.5)
       (* x x)
       1.0)
      t_0)
     (* (cosh x) 1.0))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double tmp;
	if ((cosh(x) * t_0) <= 1e-14) {
		tmp = fma(fma(fma(0.001388888888888889, (x * x), 0.041666666666666664), (x * x), 0.5), (x * x), 1.0) * t_0;
	} else {
		tmp = cosh(x) * 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(y) / y)
	tmp = 0.0
	if (Float64(cosh(x) * t_0) <= 1e-14)
		tmp = Float64(fma(fma(fma(0.001388888888888889, Float64(x * x), 0.041666666666666664), Float64(x * x), 0.5), Float64(x * x), 1.0) * t_0);
	else
		tmp = Float64(cosh(x) * 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[N[(N[Cosh[x], $MachinePrecision] * t$95$0), $MachinePrecision], 1e-14], N[(N[(N[(N[(0.001388888888888889 * N[(x * x), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[Cosh[x], $MachinePrecision] * 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < 9.99999999999999999e-15

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 96.1

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

   5. Applied rewrites 96.1%

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

   if 9.99999999999999999e-15 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 100.0%

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

Alternative 7: 74.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -1 \cdot 10^{-153}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cosh x) (/ (sin y) y)) -1e-153)
   (*
    (fma
     (fma (fma (* x x) 0.001388888888888889 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-153) {
		tmp = fma(fma(fma((x * x), 0.001388888888888889, 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-153)
		tmp = Float64(fma(fma(fma(Float64(x * x), 0.001388888888888889, 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-153], N[(N[(N[(N[(N[(x * x), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(-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)) < -1.00000000000000004e-153

   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 56.9

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

   5. Applied rewrites 56.9%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 56.9

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

   8. Applied rewrites 56.9%

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

   if -1.00000000000000004e-153 < (*.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 77.2%

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

Alternative 8: 70.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -1 \cdot 10^{-153}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot y\right) \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, x \cdot x, 0.08333333333333333\right), x \cdot x, 1\right), x \cdot x, 2\right)}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cosh x) (/ (sin y) y)) -1e-153)
   (*
    (fma
     (fma (fma (* x x) 0.001388888888888889 0.041666666666666664) (* x x) 0.5)
     (* x x)
     1.0)
    (fma -0.16666666666666666 (* y y) 1.0))
   (*
    (* 0.5 y)
    (/
     (fma
      (fma (fma 0.002777777777777778 (* x x) 0.08333333333333333) (* x x) 1.0)
      (* x x)
      2.0)
     y))))

C

double code(double x, double y) {
	double tmp;
	if ((cosh(x) * (sin(y) / y)) <= -1e-153) {
		tmp = fma(fma(fma((x * x), 0.001388888888888889, 0.041666666666666664), (x * x), 0.5), (x * x), 1.0) * fma(-0.16666666666666666, (y * y), 1.0);
	} else {
		tmp = (0.5 * y) * (fma(fma(fma(0.002777777777777778, (x * x), 0.08333333333333333), (x * x), 1.0), (x * x), 2.0) / y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(cosh(x) * Float64(sin(y) / y)) <= -1e-153)
		tmp = Float64(fma(fma(fma(Float64(x * x), 0.001388888888888889, 0.041666666666666664), Float64(x * x), 0.5), Float64(x * x), 1.0) * fma(-0.16666666666666666, Float64(y * y), 1.0));
	else
		tmp = Float64(Float64(0.5 * y) * Float64(fma(fma(fma(0.002777777777777778, Float64(x * x), 0.08333333333333333), Float64(x * x), 1.0), Float64(x * x), 2.0) / y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Cosh[x], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], -1e-153], N[(N[(N[(N[(N[(x * x), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * y), $MachinePrecision] * N[(N[(N[(N[(0.002777777777777778 * N[(x * x), $MachinePrecision] + 0.08333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < -1.00000000000000004e-153

   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 56.9

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

   5. Applied rewrites 56.9%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 56.9

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

   8. Applied rewrites 56.9%

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

   if -1.00000000000000004e-153 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y))

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. rec-exp N/A

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

      8. cosh-undef N/A

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

      9. lower-*.f64 N/A

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

      10. lift-cosh.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 95.8

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

   8. Applied rewrites 95.8%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 73.5%

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

Alternative 9: 68.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), x \cdot x, 0.5\right)\\ \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -1 \cdot 10^{-153}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, x \cdot x, 1\right) \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0 \cdot x, x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (fma
          (fma (* x x) 0.001388888888888889 0.041666666666666664)
          (* x x)
          0.5)))
   (if (<= (* (cosh x) (/ (sin y) y)) -1e-153)
     (* (fma t_0 (* x x) 1.0) (fma -0.16666666666666666 (* y y) 1.0))
     (fma (* t_0 x) x 1.0))))

C

double code(double x, double y) {
	double t_0 = fma(fma((x * x), 0.001388888888888889, 0.041666666666666664), (x * x), 0.5);
	double tmp;
	if ((cosh(x) * (sin(y) / y)) <= -1e-153) {
		tmp = fma(t_0, (x * x), 1.0) * fma(-0.16666666666666666, (y * y), 1.0);
	} else {
		tmp = fma((t_0 * x), x, 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(fma(Float64(x * x), 0.001388888888888889, 0.041666666666666664), Float64(x * x), 0.5)
	tmp = 0.0
	if (Float64(cosh(x) * Float64(sin(y) / y)) <= -1e-153)
		tmp = Float64(fma(t_0, Float64(x * x), 1.0) * fma(-0.16666666666666666, Float64(y * y), 1.0));
	else
		tmp = fma(Float64(t_0 * x), x, 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(x * x), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision]}, If[LessEqual[N[(N[Cosh[x], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], -1e-153], N[(N[(t$95$0 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * x), $MachinePrecision] * x + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < -1.00000000000000004e-153

   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 56.9

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

   5. Applied rewrites 56.9%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 56.9

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

   8. Applied rewrites 56.9%

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

   if -1.00000000000000004e-153 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y))

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 93.1%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 70.8

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

   8. Applied rewrites 70.8%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-fma.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-fma.f64 N/A

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

      13. lift-*.f64 70.8

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

   10. Applied rewrites 70.8%

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

Alternative 10: 68.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -1 \cdot 10^{-153}:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot 0.5\right) \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), x \cdot x, 0.5\right) \cdot x, x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cosh x) (/ (sin y) y)) -1e-153)
   (* (* (* x x) 0.5) (fma -0.16666666666666666 (* y y) 1.0))
   (fma
    (*
     (fma (fma (* x x) 0.001388888888888889 0.041666666666666664) (* x x) 0.5)
     x)
    x
    1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((cosh(x) * (sin(y) / y)) <= -1e-153) {
		tmp = ((x * x) * 0.5) * fma(-0.16666666666666666, (y * y), 1.0);
	} else {
		tmp = fma((fma(fma((x * x), 0.001388888888888889, 0.041666666666666664), (x * x), 0.5) * x), x, 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(cosh(x) * Float64(sin(y) / y)) <= -1e-153)
		tmp = Float64(Float64(Float64(x * x) * 0.5) * fma(-0.16666666666666666, Float64(y * y), 1.0));
	else
		tmp = fma(Float64(fma(fma(Float64(x * x), 0.001388888888888889, 0.041666666666666664), Float64(x * x), 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], -1e-153], N[(N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision] * N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(x * x), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision] * x), $MachinePrecision] * x + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < -1.00000000000000004e-153

   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 56.9

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

   5. Applied rewrites 56.9%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 49.5

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

   8. Applied rewrites 49.5%

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

   9. Taylor expanded in x around inf

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

       1. pow2 N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 N/A

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

       4. lift-*.f64 49.9

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

   11. Applied rewrites 49.9%

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

   if -1.00000000000000004e-153 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y))

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 93.1%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 70.8

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

   8. Applied rewrites 70.8%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-fma.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-fma.f64 N/A

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

      13. lift-*.f64 70.8

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

   10. Applied rewrites 70.8%

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

Alternative 11: 67.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -1 \cdot 10^{-153}:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot 0.5\right) \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left(x \cdot x\right) \cdot 0.001388888888888889, x \cdot x, 0.5\right), x \cdot x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cosh x) (/ (sin y) y)) -1e-153)
   (* (* (* x x) 0.5) (fma -0.16666666666666666 (* y y) 1.0))
   (fma (fma (* (* x x) 0.001388888888888889) (* x x) 0.5) (* x x) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((cosh(x) * (sin(y) / y)) <= -1e-153) {
		tmp = ((x * x) * 0.5) * fma(-0.16666666666666666, (y * y), 1.0);
	} else {
		tmp = fma(fma(((x * x) * 0.001388888888888889), (x * x), 0.5), (x * x), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(cosh(x) * Float64(sin(y) / y)) <= -1e-153)
		tmp = Float64(Float64(Float64(x * x) * 0.5) * fma(-0.16666666666666666, Float64(y * y), 1.0));
	else
		tmp = fma(fma(Float64(Float64(x * x) * 0.001388888888888889), Float64(x * x), 0.5), Float64(x * x), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Cosh[x], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], -1e-153], N[(N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision] * N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(x * x), $MachinePrecision] * 0.001388888888888889), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < -1.00000000000000004e-153

   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 56.9

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

   5. Applied rewrites 56.9%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 49.5

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

   8. Applied rewrites 49.5%

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

   9. Taylor expanded in x around inf

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

       1. pow2 N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 N/A

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

       4. lift-*.f64 49.9

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

   11. Applied rewrites 49.9%

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

   if -1.00000000000000004e-153 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y))

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 93.1%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 70.8

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

   8. Applied rewrites 70.8%

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

   9. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 70.6

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

   11. Applied rewrites 70.6%

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

Alternative 12: 64.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -1 \cdot 10^{-153}:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot 0.5\right) \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cosh x) (/ (sin y) y)) -1e-153)
   (* (* (* x x) 0.5) (fma -0.16666666666666666 (* y y) 1.0))
   (fma (fma 0.041666666666666664 (* x x) 0.5) (* x x) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((cosh(x) * (sin(y) / y)) <= -1e-153) {
		tmp = ((x * x) * 0.5) * fma(-0.16666666666666666, (y * y), 1.0);
	} else {
		tmp = fma(fma(0.041666666666666664, (x * x), 0.5), (x * x), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(cosh(x) * Float64(sin(y) / y)) <= -1e-153)
		tmp = Float64(Float64(Float64(x * x) * 0.5) * fma(-0.16666666666666666, Float64(y * y), 1.0));
	else
		tmp = fma(fma(0.041666666666666664, Float64(x * x), 0.5), Float64(x * x), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Cosh[x], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], -1e-153], N[(N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision] * N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(0.041666666666666664 * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < -1.00000000000000004e-153

   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 56.9

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

   5. Applied rewrites 56.9%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 49.5

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

   8. Applied rewrites 49.5%

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

   9. Taylor expanded in x around inf

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

       1. pow2 N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 N/A

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

       4. lift-*.f64 49.9

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

   11. Applied rewrites 49.9%

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

   if -1.00000000000000004e-153 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y))

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 93.1%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 70.8

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

   8. Applied rewrites 70.8%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 67.9%

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

Alternative 13: 60.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -1 \cdot 10^{-153}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cosh x) (/ (sin y) y)) -1e-153)
   (* 1.0 (fma -0.16666666666666666 (* y y) 1.0))
   (fma (fma 0.041666666666666664 (* x x) 0.5) (* x x) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((cosh(x) * (sin(y) / y)) <= -1e-153) {
		tmp = 1.0 * fma(-0.16666666666666666, (y * y), 1.0);
	} else {
		tmp = fma(fma(0.041666666666666664, (x * x), 0.5), (x * x), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(cosh(x) * Float64(sin(y) / y)) <= -1e-153)
		tmp = Float64(1.0 * fma(-0.16666666666666666, Float64(y * y), 1.0));
	else
		tmp = fma(fma(0.041666666666666664, Float64(x * x), 0.5), Float64(x * x), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Cosh[x], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], -1e-153], N[(1.0 * N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(0.041666666666666664 * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < -1.00000000000000004e-153

   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 56.9

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

   5. Applied rewrites 56.9%

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

   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 34.5%

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

   if -1.00000000000000004e-153 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y))

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 93.1%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 70.8

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

   8. Applied rewrites 70.8%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 67.9%

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

Alternative 14: 51.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -1 \cdot 10^{-153}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, x \cdot x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cosh x) (/ (sin y) y)) -1e-153)
   (* 1.0 (fma -0.16666666666666666 (* y y) 1.0))
   (fma 0.5 (* x x) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((cosh(x) * (sin(y) / y)) <= -1e-153) {
		tmp = 1.0 * fma(-0.16666666666666666, (y * y), 1.0);
	} else {
		tmp = fma(0.5, (x * x), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(cosh(x) * Float64(sin(y) / y)) <= -1e-153)
		tmp = Float64(1.0 * fma(-0.16666666666666666, Float64(y * y), 1.0));
	else
		tmp = fma(0.5, Float64(x * x), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Cosh[x], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], -1e-153], N[(1.0 * N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < -1.00000000000000004e-153

   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 56.9

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

   5. Applied rewrites 56.9%

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

   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 34.5%

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

   if -1.00000000000000004e-153 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y))

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 93.1%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 70.8

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

   8. Applied rewrites 70.8%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 53.9%

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

Alternative 15: 45.2% accurate, 18.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(0.5, x \cdot x, 1\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (fma 0.5 (* x x) 1.0))

C

double code(double x, double y) {
	return fma(0.5, (x * x), 1.0);
}

Julia

function code(x, y)
	return fma(0.5, Float64(x * x), 1.0)
end

Wolfram

code[x_, y_] := N[(0.5 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-fma.f64 N/A

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

5. Applied rewrites 92.3%

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

6. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. +-commutative N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. pow2 N/A

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

   11. lift-*.f64 N/A

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

   12. pow2 N/A

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

   13. lift-*.f64 N/A

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

   14. pow2 N/A

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

   15. lift-*.f64 56.9

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

8. Applied rewrites 56.9%

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

9. Taylor expanded in x around 0

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

11. Applied rewrites 43.4%

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

Alternative 16: 26.0% accurate, 217.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 1.0)

C

double code(double x, double y) {
	return 1.0;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0
end function

Java

public static double code(double x, double y) {
	return 1.0;
}

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0;
end

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-fma.f64 N/A

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

5. Applied rewrites 92.3%

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

6. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. +-commutative N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. pow2 N/A

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

   11. lift-*.f64 N/A

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

   12. pow2 N/A

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

   13. lift-*.f64 N/A

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

   14. pow2 N/A

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

   15. lift-*.f64 56.9

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

8. Applied rewrites 56.9%

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

9. Taylor expanded in x around 0

   \[\leadsto 1 \]
10. Step-by-step derivation

11. Applied rewrites 23.2%

    \[\leadsto 1 \]
12. Add Preprocessing

Developer Target 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2025060 
(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)))