Linear.Quaternion:$csinh from linear-1.19.1.3

Percentage Accurate: 99.9% → 99.9%

Time: 10.5s

Alternatives: 22

Speedup: 1.0×

• 48.9% 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

real(8) function code(x, y)
    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

real(8) function code(x, y)
    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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Final simplification 99.9%

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

Alternative 2: 99.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ t_1 := t\_0 \cdot \cosh x\\ t_2 := \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, y \cdot y, 0.008333333333333333\right), y \cdot y, -0.16666666666666666\right), y \cdot y, 1\right) \cdot t\_2\\ \mathbf{elif}\;t\_1 \leq 1.0000002:\\ \;\;\;\;t\_2 \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-1}{y} \cdot \cosh x\right) \cdot \left(-y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y))
        (t_1 (* t_0 (cosh x)))
        (t_2
         (fma
          (fma
           (fma 0.001388888888888889 (* x x) 0.041666666666666664)
           (* x x)
           0.5)
          (* x x)
          1.0)))
   (if (<= t_1 (- INFINITY))
     (*
      (fma
       (fma
        (fma -0.0001984126984126984 (* y y) 0.008333333333333333)
        (* y y)
        -0.16666666666666666)
       (* y y)
       1.0)
      t_2)
     (if (<= t_1 1.0000002) (* t_2 t_0) (* (* (/ -1.0 y) (cosh x)) (- y))))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double t_1 = t_0 * cosh(x);
	double t_2 = fma(fma(fma(0.001388888888888889, (x * x), 0.041666666666666664), (x * x), 0.5), (x * x), 1.0);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma(fma(fma(-0.0001984126984126984, (y * y), 0.008333333333333333), (y * y), -0.16666666666666666), (y * y), 1.0) * t_2;
	} else if (t_1 <= 1.0000002) {
		tmp = t_2 * t_0;
	} else {
		tmp = ((-1.0 / y) * cosh(x)) * -y;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(y) / y)
	t_1 = Float64(t_0 * cosh(x))
	t_2 = fma(fma(fma(0.001388888888888889, Float64(x * x), 0.041666666666666664), Float64(x * x), 0.5), Float64(x * x), 1.0)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(fma(fma(fma(-0.0001984126984126984, Float64(y * y), 0.008333333333333333), Float64(y * y), -0.16666666666666666), Float64(y * y), 1.0) * t_2);
	elseif (t_1 <= 1.0000002)
		tmp = Float64(t_2 * t_0);
	else
		tmp = Float64(Float64(Float64(-1.0 / y) * cosh(x)) * Float64(-y));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[Cosh[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = 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]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(N[(-0.0001984126984126984 * N[(y * y), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] * N[(y * y), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[t$95$1, 1.0000002], N[(t$95$2 * t$95$0), $MachinePrecision], N[(N[(N[(-1.0 / y), $MachinePrecision] * N[Cosh[x], $MachinePrecision]), $MachinePrecision] * (-y)), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 78.2

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

   5. Applied rewrites 78.2%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 94.1

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

   8. Applied rewrites 94.1%

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

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right), y \cdot y, \frac{-1}{6}\right), y \cdot y, 1\right) \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) + 1\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right), y \cdot y, \frac{-1}{6}\right), y \cdot y, 1\right) \]

       2. *-commutative N/A

          \[\leadsto \left(\color{blue}{\left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) \cdot {x}^{2}} + 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right), y \cdot y, \frac{-1}{6}\right), y \cdot y, 1\right) \]

       3. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right), {x}^{2}, 1\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right), y \cdot y, \frac{-1}{6}\right), y \cdot y, 1\right) \]

       4. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right) + \frac{1}{2}}, {x}^{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right), y \cdot y, \frac{-1}{6}\right), y \cdot y, 1\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right) \cdot {x}^{2}} + \frac{1}{2}, {x}^{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right), y \cdot y, \frac{-1}{6}\right), y \cdot y, 1\right) \]

       6. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}, {x}^{2}, \frac{1}{2}\right)}, {x}^{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right), y \cdot y, \frac{-1}{6}\right), y \cdot y, 1\right) \]

       7. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{720} \cdot {x}^{2} + \frac{1}{24}}, {x}^{2}, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right), y \cdot y, \frac{-1}{6}\right), y \cdot y, 1\right) \]

       8. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{720}, {x}^{2}, \frac{1}{24}\right)}, {x}^{2}, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right), y \cdot y, \frac{-1}{6}\right), y \cdot y, 1\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, \color{blue}{x \cdot x}, \frac{1}{24}\right), {x}^{2}, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right), y \cdot y, \frac{-1}{6}\right), y \cdot y, 1\right) \]

       10. lower-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, \color{blue}{x \cdot x}, \frac{1}{24}\right), {x}^{2}, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right), y \cdot y, \frac{-1}{6}\right), y \cdot y, 1\right) \]

       11. unpow2 N/A

           \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, x \cdot x, \frac{1}{24}\right), \color{blue}{x \cdot x}, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right), y \cdot y, \frac{-1}{6}\right), y \cdot y, 1\right) \]

       12. lower-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, x \cdot x, \frac{1}{24}\right), \color{blue}{x \cdot x}, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right), y \cdot y, \frac{-1}{6}\right), y \cdot y, 1\right) \]

       13. unpow2 N/A

           \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, x \cdot x, \frac{1}{24}\right), x \cdot x, \frac{1}{2}\right), \color{blue}{x \cdot x}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right), y \cdot y, \frac{-1}{6}\right), y \cdot y, 1\right) \]

       14. lower-*.f64 100.0

           \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right), \color{blue}{x \cdot x}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, y \cdot y, 0.008333333333333333\right), y \cdot y, -0.16666666666666666\right), y \cdot y, 1\right) \]

   11. Applied rewrites 100.0%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, y \cdot y, 0.008333333333333333\right), y \cdot y, -0.16666666666666666\right), y \cdot y, 1\right) \]

   if -inf.0 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < 1.00000019999999989

   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

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 98.4

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

   5. Applied rewrites 98.4%

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

   if 1.00000019999999989 < (*.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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\frac{y}{\sin y}}{\cosh x}}} \]

      3. associate-/r/ N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. frac-2neg N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. div-inv N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. frac-2neg N/A

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

      14. metadata-eval N/A

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

      15. lift-neg.f64 N/A

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

      16. remove-double-neg N/A

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

      17. lower-/.f64 100.0

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 100.0

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

   9. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\left(-y\right)} \cdot \left(\frac{-1}{y} \cdot \cosh x\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \cdot \cosh x \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, y \cdot y, 0.008333333333333333\right), y \cdot y, -0.16666666666666666\right), y \cdot y, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right)\\ \mathbf{elif}\;\frac{\sin y}{y} \cdot \cosh x \leq 1.0000002:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot \frac{\sin y}{y}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-1}{y} \cdot \cosh x\right) \cdot \left(-y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ t_1 := t\_0 \cdot \cosh x\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, y \cdot y, 0.008333333333333333\right), y \cdot y, -0.16666666666666666\right), y \cdot y, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right)\\ \mathbf{elif}\;t\_1 \leq 1.0000002:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-1}{y} \cdot \cosh x\right) \cdot \left(-y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)) (t_1 (* t_0 (cosh x))))
   (if (<= t_1 (- INFINITY))
     (*
      (fma
       (fma
        (fma -0.0001984126984126984 (* y y) 0.008333333333333333)
        (* y y)
        -0.16666666666666666)
       (* y y)
       1.0)
      (fma
       (fma
        (fma 0.001388888888888889 (* x x) 0.041666666666666664)
        (* x x)
        0.5)
       (* x x)
       1.0))
     (if (<= t_1 1.0000002)
       (* (fma (fma 0.041666666666666664 (* x x) 0.5) (* x x) 1.0) t_0)
       (* (* (/ -1.0 y) (cosh x)) (- y))))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double t_1 = t_0 * cosh(x);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma(fma(fma(-0.0001984126984126984, (y * y), 0.008333333333333333), (y * y), -0.16666666666666666), (y * y), 1.0) * fma(fma(fma(0.001388888888888889, (x * x), 0.041666666666666664), (x * x), 0.5), (x * x), 1.0);
	} else if (t_1 <= 1.0000002) {
		tmp = fma(fma(0.041666666666666664, (x * x), 0.5), (x * x), 1.0) * t_0;
	} else {
		tmp = ((-1.0 / y) * cosh(x)) * -y;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(y) / y)
	t_1 = Float64(t_0 * cosh(x))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(fma(fma(fma(-0.0001984126984126984, Float64(y * y), 0.008333333333333333), Float64(y * y), -0.16666666666666666), Float64(y * y), 1.0) * fma(fma(fma(0.001388888888888889, Float64(x * x), 0.041666666666666664), Float64(x * x), 0.5), Float64(x * x), 1.0));
	elseif (t_1 <= 1.0000002)
		tmp = Float64(fma(fma(0.041666666666666664, Float64(x * x), 0.5), Float64(x * x), 1.0) * t_0);
	else
		tmp = Float64(Float64(Float64(-1.0 / y) * cosh(x)) * Float64(-y));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[Cosh[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(N[(-0.0001984126984126984 * N[(y * y), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] * N[(y * y), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * 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]), $MachinePrecision], If[LessEqual[t$95$1, 1.0000002], 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[(N[(-1.0 / y), $MachinePrecision] * N[Cosh[x], $MachinePrecision]), $MachinePrecision] * (-y)), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 78.2

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

   5. Applied rewrites 78.2%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 94.1

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

   8. Applied rewrites 94.1%

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

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right), y \cdot y, \frac{-1}{6}\right), y \cdot y, 1\right) \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) + 1\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right), y \cdot y, \frac{-1}{6}\right), y \cdot y, 1\right) \]

       2. *-commutative N/A

          \[\leadsto \left(\color{blue}{\left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) \cdot {x}^{2}} + 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right), y \cdot y, \frac{-1}{6}\right), y \cdot y, 1\right) \]

       3. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right), {x}^{2}, 1\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right), y \cdot y, \frac{-1}{6}\right), y \cdot y, 1\right) \]

       4. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right) + \frac{1}{2}}, {x}^{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right), y \cdot y, \frac{-1}{6}\right), y \cdot y, 1\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right) \cdot {x}^{2}} + \frac{1}{2}, {x}^{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right), y \cdot y, \frac{-1}{6}\right), y \cdot y, 1\right) \]

       6. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}, {x}^{2}, \frac{1}{2}\right)}, {x}^{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right), y \cdot y, \frac{-1}{6}\right), y \cdot y, 1\right) \]

       7. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{720} \cdot {x}^{2} + \frac{1}{24}}, {x}^{2}, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right), y \cdot y, \frac{-1}{6}\right), y \cdot y, 1\right) \]

       8. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{720}, {x}^{2}, \frac{1}{24}\right)}, {x}^{2}, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right), y \cdot y, \frac{-1}{6}\right), y \cdot y, 1\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, \color{blue}{x \cdot x}, \frac{1}{24}\right), {x}^{2}, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right), y \cdot y, \frac{-1}{6}\right), y \cdot y, 1\right) \]

       10. lower-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, \color{blue}{x \cdot x}, \frac{1}{24}\right), {x}^{2}, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right), y \cdot y, \frac{-1}{6}\right), y \cdot y, 1\right) \]

       11. unpow2 N/A

           \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, x \cdot x, \frac{1}{24}\right), \color{blue}{x \cdot x}, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right), y \cdot y, \frac{-1}{6}\right), y \cdot y, 1\right) \]

       12. lower-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, x \cdot x, \frac{1}{24}\right), \color{blue}{x \cdot x}, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right), y \cdot y, \frac{-1}{6}\right), y \cdot y, 1\right) \]

       13. unpow2 N/A

           \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, x \cdot x, \frac{1}{24}\right), x \cdot x, \frac{1}{2}\right), \color{blue}{x \cdot x}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right), y \cdot y, \frac{-1}{6}\right), y \cdot y, 1\right) \]

       14. lower-*.f64 100.0

           \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right), \color{blue}{x \cdot x}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, y \cdot y, 0.008333333333333333\right), y \cdot y, -0.16666666666666666\right), y \cdot y, 1\right) \]

   11. Applied rewrites 100.0%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, y \cdot y, 0.008333333333333333\right), y \cdot y, -0.16666666666666666\right), y \cdot y, 1\right) \]

   if -inf.0 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < 1.00000019999999989

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 98.3

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

   5. Applied rewrites 98.3%

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

   if 1.00000019999999989 < (*.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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\frac{y}{\sin y}}{\cosh x}}} \]

      3. associate-/r/ N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. frac-2neg N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. div-inv N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. frac-2neg N/A

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

      14. metadata-eval N/A

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

      15. lift-neg.f64 N/A

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

      16. remove-double-neg N/A

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

      17. lower-/.f64 100.0

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 100.0

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

   9. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\left(-y\right)} \cdot \left(\frac{-1}{y} \cdot \cosh x\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \cdot \cosh x \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, y \cdot y, 0.008333333333333333\right), y \cdot y, -0.16666666666666666\right), y \cdot y, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right)\\ \mathbf{elif}\;\frac{\sin y}{y} \cdot \cosh x \leq 1.0000002:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot \frac{\sin y}{y}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-1}{y} \cdot \cosh x\right) \cdot \left(-y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ t_1 := t\_0 \cdot \cosh x\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, y \cdot y, 0.008333333333333333\right), y \cdot y, -0.16666666666666666\right), y \cdot y, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right)\\ \mathbf{elif}\;t\_1 \leq 1.0000002:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, 0.5, 1\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-1}{y} \cdot \cosh x\right) \cdot \left(-y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)) (t_1 (* t_0 (cosh x))))
   (if (<= t_1 (- INFINITY))
     (*
      (fma
       (fma
        (fma -0.0001984126984126984 (* y y) 0.008333333333333333)
        (* y y)
        -0.16666666666666666)
       (* y y)
       1.0)
      (fma
       (fma
        (fma 0.001388888888888889 (* x x) 0.041666666666666664)
        (* x x)
        0.5)
       (* x x)
       1.0))
     (if (<= t_1 1.0000002)
       (* (fma (* x x) 0.5 1.0) t_0)
       (* (* (/ -1.0 y) (cosh x)) (- y))))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double t_1 = t_0 * cosh(x);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma(fma(fma(-0.0001984126984126984, (y * y), 0.008333333333333333), (y * y), -0.16666666666666666), (y * y), 1.0) * fma(fma(fma(0.001388888888888889, (x * x), 0.041666666666666664), (x * x), 0.5), (x * x), 1.0);
	} else if (t_1 <= 1.0000002) {
		tmp = fma((x * x), 0.5, 1.0) * t_0;
	} else {
		tmp = ((-1.0 / y) * cosh(x)) * -y;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(y) / y)
	t_1 = Float64(t_0 * cosh(x))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(fma(fma(fma(-0.0001984126984126984, Float64(y * y), 0.008333333333333333), Float64(y * y), -0.16666666666666666), Float64(y * y), 1.0) * fma(fma(fma(0.001388888888888889, Float64(x * x), 0.041666666666666664), Float64(x * x), 0.5), Float64(x * x), 1.0));
	elseif (t_1 <= 1.0000002)
		tmp = Float64(fma(Float64(x * x), 0.5, 1.0) * t_0);
	else
		tmp = Float64(Float64(Float64(-1.0 / y) * cosh(x)) * Float64(-y));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[Cosh[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(N[(-0.0001984126984126984 * N[(y * y), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] * N[(y * y), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * 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]), $MachinePrecision], If[LessEqual[t$95$1, 1.0000002], N[(N[(N[(x * x), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(N[(-1.0 / y), $MachinePrecision] * N[Cosh[x], $MachinePrecision]), $MachinePrecision] * (-y)), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 78.2

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

   5. Applied rewrites 78.2%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 94.1

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

   8. Applied rewrites 94.1%

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

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right), y \cdot y, \frac{-1}{6}\right), y \cdot y, 1\right) \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) + 1\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right), y \cdot y, \frac{-1}{6}\right), y \cdot y, 1\right) \]

       2. *-commutative N/A

          \[\leadsto \left(\color{blue}{\left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) \cdot {x}^{2}} + 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right), y \cdot y, \frac{-1}{6}\right), y \cdot y, 1\right) \]

       3. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right), {x}^{2}, 1\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right), y \cdot y, \frac{-1}{6}\right), y \cdot y, 1\right) \]

       4. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right) + \frac{1}{2}}, {x}^{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right), y \cdot y, \frac{-1}{6}\right), y \cdot y, 1\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right) \cdot {x}^{2}} + \frac{1}{2}, {x}^{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right), y \cdot y, \frac{-1}{6}\right), y \cdot y, 1\right) \]

       6. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}, {x}^{2}, \frac{1}{2}\right)}, {x}^{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right), y \cdot y, \frac{-1}{6}\right), y \cdot y, 1\right) \]

       7. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{720} \cdot {x}^{2} + \frac{1}{24}}, {x}^{2}, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right), y \cdot y, \frac{-1}{6}\right), y \cdot y, 1\right) \]

       8. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{720}, {x}^{2}, \frac{1}{24}\right)}, {x}^{2}, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right), y \cdot y, \frac{-1}{6}\right), y \cdot y, 1\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, \color{blue}{x \cdot x}, \frac{1}{24}\right), {x}^{2}, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right), y \cdot y, \frac{-1}{6}\right), y \cdot y, 1\right) \]

       10. lower-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, \color{blue}{x \cdot x}, \frac{1}{24}\right), {x}^{2}, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right), y \cdot y, \frac{-1}{6}\right), y \cdot y, 1\right) \]

       11. unpow2 N/A

           \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, x \cdot x, \frac{1}{24}\right), \color{blue}{x \cdot x}, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right), y \cdot y, \frac{-1}{6}\right), y \cdot y, 1\right) \]

       12. lower-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, x \cdot x, \frac{1}{24}\right), \color{blue}{x \cdot x}, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right), y \cdot y, \frac{-1}{6}\right), y \cdot y, 1\right) \]

       13. unpow2 N/A

           \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, x \cdot x, \frac{1}{24}\right), x \cdot x, \frac{1}{2}\right), \color{blue}{x \cdot x}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right), y \cdot y, \frac{-1}{6}\right), y \cdot y, 1\right) \]

       14. lower-*.f64 100.0

           \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right), \color{blue}{x \cdot x}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, y \cdot y, 0.008333333333333333\right), y \cdot y, -0.16666666666666666\right), y \cdot y, 1\right) \]

   11. Applied rewrites 100.0%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, y \cdot y, 0.008333333333333333\right), y \cdot y, -0.16666666666666666\right), y \cdot y, 1\right) \]

   if -inf.0 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < 1.00000019999999989

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 98.2

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

   5. Applied rewrites 98.2%

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

   if 1.00000019999999989 < (*.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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\frac{y}{\sin y}}{\cosh x}}} \]

      3. associate-/r/ N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. frac-2neg N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. div-inv N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. frac-2neg N/A

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

      14. metadata-eval N/A

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

      15. lift-neg.f64 N/A

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

      16. remove-double-neg N/A

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

      17. lower-/.f64 100.0

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 100.0

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

   9. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\left(-y\right)} \cdot \left(\frac{-1}{y} \cdot \cosh x\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \cdot \cosh x \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, y \cdot y, 0.008333333333333333\right), y \cdot y, -0.16666666666666666\right), y \cdot y, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right)\\ \mathbf{elif}\;\frac{\sin y}{y} \cdot \cosh x \leq 1.0000002:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, 0.5, 1\right) \cdot \frac{\sin y}{y}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-1}{y} \cdot \cosh x\right) \cdot \left(-y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ t_1 := t\_0 \cdot \cosh x\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, y \cdot y, 0.008333333333333333\right), y \cdot y, -0.16666666666666666\right), y \cdot y, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right)\\ \mathbf{elif}\;t\_1 \leq 0.9999999999408081:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \cosh x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)) (t_1 (* t_0 (cosh x))))
   (if (<= t_1 (- INFINITY))
     (*
      (fma
       (fma
        (fma -0.0001984126984126984 (* y y) 0.008333333333333333)
        (* y y)
        -0.16666666666666666)
       (* y y)
       1.0)
      (fma
       (fma
        (fma 0.001388888888888889 (* x x) 0.041666666666666664)
        (* x x)
        0.5)
       (* x x)
       1.0))
     (if (<= t_1 0.9999999999408081) t_0 (* 1.0 (cosh x))))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double t_1 = t_0 * cosh(x);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma(fma(fma(-0.0001984126984126984, (y * y), 0.008333333333333333), (y * y), -0.16666666666666666), (y * y), 1.0) * fma(fma(fma(0.001388888888888889, (x * x), 0.041666666666666664), (x * x), 0.5), (x * x), 1.0);
	} else if (t_1 <= 0.9999999999408081) {
		tmp = t_0;
	} else {
		tmp = 1.0 * cosh(x);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(y) / y)
	t_1 = Float64(t_0 * cosh(x))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(fma(fma(fma(-0.0001984126984126984, Float64(y * y), 0.008333333333333333), Float64(y * y), -0.16666666666666666), Float64(y * y), 1.0) * fma(fma(fma(0.001388888888888889, Float64(x * x), 0.041666666666666664), Float64(x * x), 0.5), Float64(x * x), 1.0));
	elseif (t_1 <= 0.9999999999408081)
		tmp = t_0;
	else
		tmp = Float64(1.0 * cosh(x));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[Cosh[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(N[(-0.0001984126984126984 * N[(y * y), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] * N[(y * y), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * 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]), $MachinePrecision], If[LessEqual[t$95$1, 0.9999999999408081], t$95$0, N[(1.0 * N[Cosh[x], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 78.2

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

   5. Applied rewrites 78.2%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 94.1

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

   8. Applied rewrites 94.1%

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

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right), y \cdot y, \frac{-1}{6}\right), y \cdot y, 1\right) \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) + 1\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right), y \cdot y, \frac{-1}{6}\right), y \cdot y, 1\right) \]

       2. *-commutative N/A

          \[\leadsto \left(\color{blue}{\left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) \cdot {x}^{2}} + 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right), y \cdot y, \frac{-1}{6}\right), y \cdot y, 1\right) \]

       3. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right), {x}^{2}, 1\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right), y \cdot y, \frac{-1}{6}\right), y \cdot y, 1\right) \]

       4. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right) + \frac{1}{2}}, {x}^{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right), y \cdot y, \frac{-1}{6}\right), y \cdot y, 1\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right) \cdot {x}^{2}} + \frac{1}{2}, {x}^{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right), y \cdot y, \frac{-1}{6}\right), y \cdot y, 1\right) \]

       6. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}, {x}^{2}, \frac{1}{2}\right)}, {x}^{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right), y \cdot y, \frac{-1}{6}\right), y \cdot y, 1\right) \]

       7. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{720} \cdot {x}^{2} + \frac{1}{24}}, {x}^{2}, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right), y \cdot y, \frac{-1}{6}\right), y \cdot y, 1\right) \]

       8. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{720}, {x}^{2}, \frac{1}{24}\right)}, {x}^{2}, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right), y \cdot y, \frac{-1}{6}\right), y \cdot y, 1\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, \color{blue}{x \cdot x}, \frac{1}{24}\right), {x}^{2}, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right), y \cdot y, \frac{-1}{6}\right), y \cdot y, 1\right) \]

       10. lower-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, \color{blue}{x \cdot x}, \frac{1}{24}\right), {x}^{2}, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right), y \cdot y, \frac{-1}{6}\right), y \cdot y, 1\right) \]

       11. unpow2 N/A

           \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, x \cdot x, \frac{1}{24}\right), \color{blue}{x \cdot x}, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right), y \cdot y, \frac{-1}{6}\right), y \cdot y, 1\right) \]

       12. lower-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, x \cdot x, \frac{1}{24}\right), \color{blue}{x \cdot x}, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right), y \cdot y, \frac{-1}{6}\right), y \cdot y, 1\right) \]

       13. unpow2 N/A

           \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, x \cdot x, \frac{1}{24}\right), x \cdot x, \frac{1}{2}\right), \color{blue}{x \cdot x}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right), y \cdot y, \frac{-1}{6}\right), y \cdot y, 1\right) \]

       14. lower-*.f64 100.0

           \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right), \color{blue}{x \cdot x}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, y \cdot y, 0.008333333333333333\right), y \cdot y, -0.16666666666666666\right), y \cdot y, 1\right) \]

   11. Applied rewrites 100.0%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, y \cdot y, 0.008333333333333333\right), y \cdot y, -0.16666666666666666\right), y \cdot y, 1\right) \]

   if -inf.0 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < 0.999999999940808126

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 96.4

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

   5. Applied rewrites 96.4%

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

   if 0.999999999940808126 < (*.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. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \cdot \cosh x \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, y \cdot y, 0.008333333333333333\right), y \cdot y, -0.16666666666666666\right), y \cdot y, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right)\\ \mathbf{elif}\;\frac{\sin y}{y} \cdot \cosh x \leq 0.9999999999408081:\\ \;\;\;\;\frac{\sin y}{y}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \cosh x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \cdot \cosh x \leq -2 \cdot 10^{-147}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, y \cdot y, 0.008333333333333333\right), y \cdot y, -0.16666666666666666\right), y \cdot y, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \cosh x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (/ (sin y) y) (cosh x)) -2e-147)
   (*
    (fma
     (fma
      (fma -0.0001984126984126984 (* y y) 0.008333333333333333)
      (* y y)
      -0.16666666666666666)
     (* y y)
     1.0)
    (fma
     (fma (fma 0.001388888888888889 (* x x) 0.041666666666666664) (* x x) 0.5)
     (* x x)
     1.0))
   (* 1.0 (cosh x))))

C

double code(double x, double y) {
	double tmp;
	if (((sin(y) / y) * cosh(x)) <= -2e-147) {
		tmp = fma(fma(fma(-0.0001984126984126984, (y * y), 0.008333333333333333), (y * y), -0.16666666666666666), (y * y), 1.0) * fma(fma(fma(0.001388888888888889, (x * x), 0.041666666666666664), (x * x), 0.5), (x * x), 1.0);
	} else {
		tmp = 1.0 * cosh(x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(sin(y) / y) * cosh(x)) <= -2e-147)
		tmp = Float64(fma(fma(fma(-0.0001984126984126984, Float64(y * y), 0.008333333333333333), Float64(y * y), -0.16666666666666666), Float64(y * y), 1.0) * fma(fma(fma(0.001388888888888889, Float64(x * x), 0.041666666666666664), Float64(x * x), 0.5), Float64(x * x), 1.0));
	else
		tmp = Float64(1.0 * cosh(x));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision] * N[Cosh[x], $MachinePrecision]), $MachinePrecision], -2e-147], N[(N[(N[(N[(-0.0001984126984126984 * N[(y * y), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] * N[(y * y), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * 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]), $MachinePrecision], N[(1.0 * N[Cosh[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < -1.9999999999999999e-147

   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

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 86.0

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

   5. Applied rewrites 86.0%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 54.3

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

   8. Applied rewrites 54.3%

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

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right), y \cdot y, \frac{-1}{6}\right), y \cdot y, 1\right) \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) + 1\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right), y \cdot y, \frac{-1}{6}\right), y \cdot y, 1\right) \]

       2. *-commutative N/A

          \[\leadsto \left(\color{blue}{\left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) \cdot {x}^{2}} + 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right), y \cdot y, \frac{-1}{6}\right), y \cdot y, 1\right) \]

       3. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right), {x}^{2}, 1\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right), y \cdot y, \frac{-1}{6}\right), y \cdot y, 1\right) \]

       4. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right) + \frac{1}{2}}, {x}^{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right), y \cdot y, \frac{-1}{6}\right), y \cdot y, 1\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right) \cdot {x}^{2}} + \frac{1}{2}, {x}^{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right), y \cdot y, \frac{-1}{6}\right), y \cdot y, 1\right) \]

       6. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}, {x}^{2}, \frac{1}{2}\right)}, {x}^{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right), y \cdot y, \frac{-1}{6}\right), y \cdot y, 1\right) \]

       7. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{720} \cdot {x}^{2} + \frac{1}{24}}, {x}^{2}, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right), y \cdot y, \frac{-1}{6}\right), y \cdot y, 1\right) \]

       8. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{720}, {x}^{2}, \frac{1}{24}\right)}, {x}^{2}, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right), y \cdot y, \frac{-1}{6}\right), y \cdot y, 1\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, \color{blue}{x \cdot x}, \frac{1}{24}\right), {x}^{2}, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right), y \cdot y, \frac{-1}{6}\right), y \cdot y, 1\right) \]

       10. lower-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, \color{blue}{x \cdot x}, \frac{1}{24}\right), {x}^{2}, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right), y \cdot y, \frac{-1}{6}\right), y \cdot y, 1\right) \]

       11. unpow2 N/A

           \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, x \cdot x, \frac{1}{24}\right), \color{blue}{x \cdot x}, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right), y \cdot y, \frac{-1}{6}\right), y \cdot y, 1\right) \]

       12. lower-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, x \cdot x, \frac{1}{24}\right), \color{blue}{x \cdot x}, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right), y \cdot y, \frac{-1}{6}\right), y \cdot y, 1\right) \]

       13. unpow2 N/A

           \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, x \cdot x, \frac{1}{24}\right), x \cdot x, \frac{1}{2}\right), \color{blue}{x \cdot x}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right), y \cdot y, \frac{-1}{6}\right), y \cdot y, 1\right) \]

       14. lower-*.f64 57.7

           \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right), \color{blue}{x \cdot x}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, y \cdot y, 0.008333333333333333\right), y \cdot y, -0.16666666666666666\right), y \cdot y, 1\right) \]

   11. Applied rewrites 57.7%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, y \cdot y, 0.008333333333333333\right), y \cdot y, -0.16666666666666666\right), y \cdot y, 1\right) \]

   if -1.9999999999999999e-147 < (*.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 73.7%

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

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \cdot \cosh x \leq -2 \cdot 10^{-147}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, y \cdot y, 0.008333333333333333\right), y \cdot y, -0.16666666666666666\right), y \cdot y, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \cosh x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 70.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ t_1 := \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, y \cdot y, 0.008333333333333333\right), y \cdot y, -0.16666666666666666\right), y \cdot y, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right)\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-308}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), y \cdot y, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y))
        (t_1
         (*
          (fma
           (fma
            (fma -0.0001984126984126984 (* y y) 0.008333333333333333)
            (* y y)
            -0.16666666666666666)
           (* y y)
           1.0)
          (fma
           (fma
            (fma 0.001388888888888889 (* x x) 0.041666666666666664)
            (* x x)
            0.5)
           (* x x)
           1.0))))
   (if (<= t_0 -5e-308)
     t_1
     (if (<= t_0 5e-11)
       (*
        (fma
         (fma 0.008333333333333333 (* y y) -0.16666666666666666)
         (* y y)
         1.0)
        (fma (fma 0.041666666666666664 (* x x) 0.5) (* x x) 1.0))
       t_1))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double t_1 = fma(fma(fma(-0.0001984126984126984, (y * y), 0.008333333333333333), (y * y), -0.16666666666666666), (y * y), 1.0) * fma(fma(fma(0.001388888888888889, (x * x), 0.041666666666666664), (x * x), 0.5), (x * x), 1.0);
	double tmp;
	if (t_0 <= -5e-308) {
		tmp = t_1;
	} else if (t_0 <= 5e-11) {
		tmp = fma(fma(0.008333333333333333, (y * y), -0.16666666666666666), (y * y), 1.0) * fma(fma(0.041666666666666664, (x * x), 0.5), (x * x), 1.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(y) / y)
	t_1 = Float64(fma(fma(fma(-0.0001984126984126984, Float64(y * y), 0.008333333333333333), Float64(y * y), -0.16666666666666666), Float64(y * y), 1.0) * fma(fma(fma(0.001388888888888889, Float64(x * x), 0.041666666666666664), Float64(x * x), 0.5), Float64(x * x), 1.0))
	tmp = 0.0
	if (t_0 <= -5e-308)
		tmp = t_1;
	elseif (t_0 <= 5e-11)
		tmp = Float64(fma(fma(0.008333333333333333, Float64(y * y), -0.16666666666666666), Float64(y * y), 1.0) * fma(fma(0.041666666666666664, Float64(x * x), 0.5), Float64(x * x), 1.0));
	else
		tmp = t_1;
	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[(N[(N[(-0.0001984126984126984 * N[(y * y), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] * N[(y * y), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * 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]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-308], t$95$1, If[LessEqual[t$95$0, 5e-11], N[(N[(N[(0.008333333333333333 * N[(y * y), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(0.041666666666666664 * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 y) y) < -4.99999999999999955e-308 or 5.00000000000000018e-11 < (/.f64 (sin.f64 y) y)

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 87.3

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

   5. Applied rewrites 87.3%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 70.6

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

   8. Applied rewrites 70.6%

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

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right), y \cdot y, \frac{-1}{6}\right), y \cdot y, 1\right) \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) + 1\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right), y \cdot y, \frac{-1}{6}\right), y \cdot y, 1\right) \]

       2. *-commutative N/A

          \[\leadsto \left(\color{blue}{\left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) \cdot {x}^{2}} + 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right), y \cdot y, \frac{-1}{6}\right), y \cdot y, 1\right) \]

       3. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right), {x}^{2}, 1\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right), y \cdot y, \frac{-1}{6}\right), y \cdot y, 1\right) \]

       4. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right) + \frac{1}{2}}, {x}^{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right), y \cdot y, \frac{-1}{6}\right), y \cdot y, 1\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right) \cdot {x}^{2}} + \frac{1}{2}, {x}^{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right), y \cdot y, \frac{-1}{6}\right), y \cdot y, 1\right) \]

       6. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}, {x}^{2}, \frac{1}{2}\right)}, {x}^{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right), y \cdot y, \frac{-1}{6}\right), y \cdot y, 1\right) \]

       7. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{720} \cdot {x}^{2} + \frac{1}{24}}, {x}^{2}, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right), y \cdot y, \frac{-1}{6}\right), y \cdot y, 1\right) \]

       8. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{720}, {x}^{2}, \frac{1}{24}\right)}, {x}^{2}, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right), y \cdot y, \frac{-1}{6}\right), y \cdot y, 1\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, \color{blue}{x \cdot x}, \frac{1}{24}\right), {x}^{2}, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right), y \cdot y, \frac{-1}{6}\right), y \cdot y, 1\right) \]

       10. lower-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, \color{blue}{x \cdot x}, \frac{1}{24}\right), {x}^{2}, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right), y \cdot y, \frac{-1}{6}\right), y \cdot y, 1\right) \]

       11. unpow2 N/A

           \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, x \cdot x, \frac{1}{24}\right), \color{blue}{x \cdot x}, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right), y \cdot y, \frac{-1}{6}\right), y \cdot y, 1\right) \]

       12. lower-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, x \cdot x, \frac{1}{24}\right), \color{blue}{x \cdot x}, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right), y \cdot y, \frac{-1}{6}\right), y \cdot y, 1\right) \]

       13. unpow2 N/A

           \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, x \cdot x, \frac{1}{24}\right), x \cdot x, \frac{1}{2}\right), \color{blue}{x \cdot x}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right), y \cdot y, \frac{-1}{6}\right), y \cdot y, 1\right) \]

       14. lower-*.f64 74.0

           \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right), \color{blue}{x \cdot x}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, y \cdot y, 0.008333333333333333\right), y \cdot y, -0.16666666666666666\right), y \cdot y, 1\right) \]

   11. Applied rewrites 74.0%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, y \cdot y, 0.008333333333333333\right), y \cdot y, -0.16666666666666666\right), y \cdot y, 1\right) \]

   if -4.99999999999999955e-308 < (/.f64 (sin.f64 y) y) < 5.00000000000000018e-11

   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

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 83.3

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

   5. Applied rewrites 83.3%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 33.9

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

   8. Applied rewrites 33.9%

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

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq -5 \cdot 10^{-308}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, y \cdot y, 0.008333333333333333\right), y \cdot y, -0.16666666666666666\right), y \cdot y, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right)\\ \mathbf{elif}\;\frac{\sin y}{y} \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), y \cdot y, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, y \cdot y, 0.008333333333333333\right), y \cdot y, -0.16666666666666666\right), y \cdot y, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 70.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right)\\ t_1 := \frac{\sin y}{y}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-308}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left(y \cdot y\right) \cdot -0.0001984126984126984, y \cdot y, -0.16666666666666666\right), y \cdot y, 1\right) \cdot t\_0\\ \mathbf{elif}\;t\_1 \leq 0.99999999995:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), y \cdot y, 1\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), x \cdot x, 0.5\right) \cdot x, x, 1\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma (fma 0.041666666666666664 (* x x) 0.5) (* x x) 1.0))
        (t_1 (/ (sin y) y)))
   (if (<= t_1 -5e-308)
     (*
      (fma
       (fma (* (* y y) -0.0001984126984126984) (* y y) -0.16666666666666666)
       (* y y)
       1.0)
      t_0)
     (if (<= t_1 0.99999999995)
       (*
        (fma
         (fma 0.008333333333333333 (* y y) -0.16666666666666666)
         (* y y)
         1.0)
        t_0)
       (*
        (fma
         (*
          (fma
           (fma (* x x) 0.001388888888888889 0.041666666666666664)
           (* x x)
           0.5)
          x)
         x
         1.0)
        1.0)))))

C

double code(double x, double y) {
	double t_0 = fma(fma(0.041666666666666664, (x * x), 0.5), (x * x), 1.0);
	double t_1 = sin(y) / y;
	double tmp;
	if (t_1 <= -5e-308) {
		tmp = fma(fma(((y * y) * -0.0001984126984126984), (y * y), -0.16666666666666666), (y * y), 1.0) * t_0;
	} else if (t_1 <= 0.99999999995) {
		tmp = fma(fma(0.008333333333333333, (y * y), -0.16666666666666666), (y * y), 1.0) * t_0;
	} else {
		tmp = fma((fma(fma((x * x), 0.001388888888888889, 0.041666666666666664), (x * x), 0.5) * x), x, 1.0) * 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(fma(0.041666666666666664, Float64(x * x), 0.5), Float64(x * x), 1.0)
	t_1 = Float64(sin(y) / y)
	tmp = 0.0
	if (t_1 <= -5e-308)
		tmp = Float64(fma(fma(Float64(Float64(y * y) * -0.0001984126984126984), Float64(y * y), -0.16666666666666666), Float64(y * y), 1.0) * t_0);
	elseif (t_1 <= 0.99999999995)
		tmp = Float64(fma(fma(0.008333333333333333, Float64(y * y), -0.16666666666666666), Float64(y * y), 1.0) * t_0);
	else
		tmp = Float64(fma(Float64(fma(fma(Float64(x * x), 0.001388888888888889, 0.041666666666666664), Float64(x * x), 0.5) * x), x, 1.0) * 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(0.041666666666666664 * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-308], N[(N[(N[(N[(N[(y * y), $MachinePrecision] * -0.0001984126984126984), $MachinePrecision] * N[(y * y), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[t$95$1, 0.99999999995], N[(N[(N[(0.008333333333333333 * N[(y * y), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision], N[(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] * 1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 y) y) < -4.99999999999999955e-308

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 89.1

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

   5. Applied rewrites 89.1%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 42.5

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

   8. Applied rewrites 42.5%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 42.5%

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

   if -4.99999999999999955e-308 < (/.f64 (sin.f64 y) y) < 0.99999999995

   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

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 84.4

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

   5. Applied rewrites 84.4%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 37.5

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

   8. Applied rewrites 37.5%

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

   if 0.99999999995 < (/.f64 (sin.f64 y) y)

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 89.8

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

   8. Applied rewrites 89.8%

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

   10. Applied rewrites 89.8%

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

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq -5 \cdot 10^{-308}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left(y \cdot y\right) \cdot -0.0001984126984126984, y \cdot y, -0.16666666666666666\right), y \cdot y, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right)\\ \mathbf{elif}\;\frac{\sin y}{y} \leq 0.99999999995:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), y \cdot y, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), x \cdot x, 0.5\right) \cdot x, x, 1\right) \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 70.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-308}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right)\\ \mathbf{elif}\;t\_0 \leq 0.99999999995:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), y \cdot y, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), x \cdot x, 0.5\right) \cdot x, x, 1\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)))
   (if (<= t_0 -5e-308)
     (*
      (fma -0.16666666666666666 (* y y) 1.0)
      (fma
       (fma
        (fma 0.001388888888888889 (* x x) 0.041666666666666664)
        (* x x)
        0.5)
       (* x x)
       1.0))
     (if (<= t_0 0.99999999995)
       (*
        (fma
         (fma 0.008333333333333333 (* y y) -0.16666666666666666)
         (* y y)
         1.0)
        (fma (fma 0.041666666666666664 (* x x) 0.5) (* x x) 1.0))
       (*
        (fma
         (*
          (fma
           (fma (* x x) 0.001388888888888889 0.041666666666666664)
           (* x x)
           0.5)
          x)
         x
         1.0)
        1.0)))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double tmp;
	if (t_0 <= -5e-308) {
		tmp = fma(-0.16666666666666666, (y * y), 1.0) * fma(fma(fma(0.001388888888888889, (x * x), 0.041666666666666664), (x * x), 0.5), (x * x), 1.0);
	} else if (t_0 <= 0.99999999995) {
		tmp = fma(fma(0.008333333333333333, (y * y), -0.16666666666666666), (y * y), 1.0) * fma(fma(0.041666666666666664, (x * x), 0.5), (x * x), 1.0);
	} else {
		tmp = fma((fma(fma((x * x), 0.001388888888888889, 0.041666666666666664), (x * x), 0.5) * x), x, 1.0) * 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(y) / y)
	tmp = 0.0
	if (t_0 <= -5e-308)
		tmp = Float64(fma(-0.16666666666666666, Float64(y * y), 1.0) * fma(fma(fma(0.001388888888888889, Float64(x * x), 0.041666666666666664), Float64(x * x), 0.5), Float64(x * x), 1.0));
	elseif (t_0 <= 0.99999999995)
		tmp = Float64(fma(fma(0.008333333333333333, Float64(y * y), -0.16666666666666666), Float64(y * y), 1.0) * fma(fma(0.041666666666666664, Float64(x * x), 0.5), Float64(x * x), 1.0));
	else
		tmp = Float64(fma(Float64(fma(fma(Float64(x * x), 0.001388888888888889, 0.041666666666666664), Float64(x * x), 0.5) * x), x, 1.0) * 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-308], N[(N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * 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]), $MachinePrecision], If[LessEqual[t$95$0, 0.99999999995], N[(N[(N[(0.008333333333333333 * N[(y * y), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(0.041666666666666664 * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(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] * 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 y) y) < -4.99999999999999955e-308

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 1.2%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 1.2

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

   8. Applied rewrites 1.2%

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

   9. Taylor expanded in y around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 41.5

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

   11. Applied rewrites 41.5%

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

   if -4.99999999999999955e-308 < (/.f64 (sin.f64 y) y) < 0.99999999995

   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

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 84.4

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

   5. Applied rewrites 84.4%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 37.5

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

   8. Applied rewrites 37.5%

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

   if 0.99999999995 < (/.f64 (sin.f64 y) y)

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 89.8

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

   8. Applied rewrites 89.8%

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

   10. Applied rewrites 89.8%

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

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq -5 \cdot 10^{-308}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right)\\ \mathbf{elif}\;\frac{\sin y}{y} \leq 0.99999999995:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), y \cdot y, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), x \cdot x, 0.5\right) \cdot x, x, 1\right) \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 70.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-308}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, 0.5, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, y \cdot y, 0.008333333333333333\right), y \cdot y, -0.16666666666666666\right), y \cdot y, 1\right)\\ \mathbf{elif}\;t\_0 \leq 0.99999999995:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), y \cdot y, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), x \cdot x, 0.5\right) \cdot x, x, 1\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)))
   (if (<= t_0 -5e-308)
     (*
      (fma (* x x) 0.5 1.0)
      (fma
       (fma
        (fma -0.0001984126984126984 (* y y) 0.008333333333333333)
        (* y y)
        -0.16666666666666666)
       (* y y)
       1.0))
     (if (<= t_0 0.99999999995)
       (*
        (fma
         (fma 0.008333333333333333 (* y y) -0.16666666666666666)
         (* y y)
         1.0)
        (fma (fma 0.041666666666666664 (* x x) 0.5) (* x x) 1.0))
       (*
        (fma
         (*
          (fma
           (fma (* x x) 0.001388888888888889 0.041666666666666664)
           (* x x)
           0.5)
          x)
         x
         1.0)
        1.0)))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double tmp;
	if (t_0 <= -5e-308) {
		tmp = fma((x * x), 0.5, 1.0) * fma(fma(fma(-0.0001984126984126984, (y * y), 0.008333333333333333), (y * y), -0.16666666666666666), (y * y), 1.0);
	} else if (t_0 <= 0.99999999995) {
		tmp = fma(fma(0.008333333333333333, (y * y), -0.16666666666666666), (y * y), 1.0) * fma(fma(0.041666666666666664, (x * x), 0.5), (x * x), 1.0);
	} else {
		tmp = fma((fma(fma((x * x), 0.001388888888888889, 0.041666666666666664), (x * x), 0.5) * x), x, 1.0) * 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(y) / y)
	tmp = 0.0
	if (t_0 <= -5e-308)
		tmp = Float64(fma(Float64(x * x), 0.5, 1.0) * fma(fma(fma(-0.0001984126984126984, Float64(y * y), 0.008333333333333333), Float64(y * y), -0.16666666666666666), Float64(y * y), 1.0));
	elseif (t_0 <= 0.99999999995)
		tmp = Float64(fma(fma(0.008333333333333333, Float64(y * y), -0.16666666666666666), Float64(y * y), 1.0) * fma(fma(0.041666666666666664, Float64(x * x), 0.5), Float64(x * x), 1.0));
	else
		tmp = Float64(fma(Float64(fma(fma(Float64(x * x), 0.001388888888888889, 0.041666666666666664), Float64(x * x), 0.5) * x), x, 1.0) * 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-308], N[(N[(N[(x * x), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] * N[(N[(N[(-0.0001984126984126984 * N[(y * y), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] * N[(y * y), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.99999999995], N[(N[(N[(0.008333333333333333 * N[(y * y), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(0.041666666666666664 * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(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] * 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 y) y) < -4.99999999999999955e-308

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 89.1

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

   5. Applied rewrites 89.1%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 42.5

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

   8. Applied rewrites 42.5%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. unpow2 N/A

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

       5. lower-*.f64 41.2

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

   11. Applied rewrites 41.2%

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

   if -4.99999999999999955e-308 < (/.f64 (sin.f64 y) y) < 0.99999999995

   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

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 84.4

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

   5. Applied rewrites 84.4%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 37.5

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

   8. Applied rewrites 37.5%

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

   if 0.99999999995 < (/.f64 (sin.f64 y) y)

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 89.8

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

   8. Applied rewrites 89.8%

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

   10. Applied rewrites 89.8%

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

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq -5 \cdot 10^{-308}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, 0.5, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, y \cdot y, 0.008333333333333333\right), y \cdot y, -0.16666666666666666\right), y \cdot y, 1\right)\\ \mathbf{elif}\;\frac{\sin y}{y} \leq 0.99999999995:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), y \cdot y, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), x \cdot x, 0.5\right) \cdot x, x, 1\right) \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 70.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-308}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, 0.5, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, y \cdot y, 0.008333333333333333\right), y \cdot y, -0.16666666666666666\right), y \cdot y, 1\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-123}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), x \cdot x, 0.5\right) \cdot x, x, 1\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)))
   (if (<= t_0 -5e-308)
     (*
      (fma (* x x) 0.5 1.0)
      (fma
       (fma
        (fma -0.0001984126984126984 (* y y) 0.008333333333333333)
        (* y y)
        -0.16666666666666666)
       (* y y)
       1.0))
     (if (<= t_0 2e-123)
       (*
        1.0
        (fma
         (fma 0.008333333333333333 (* y y) -0.16666666666666666)
         (* y y)
         1.0))
       (*
        (fma
         (*
          (fma
           (fma (* x x) 0.001388888888888889 0.041666666666666664)
           (* x x)
           0.5)
          x)
         x
         1.0)
        1.0)))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double tmp;
	if (t_0 <= -5e-308) {
		tmp = fma((x * x), 0.5, 1.0) * fma(fma(fma(-0.0001984126984126984, (y * y), 0.008333333333333333), (y * y), -0.16666666666666666), (y * y), 1.0);
	} else if (t_0 <= 2e-123) {
		tmp = 1.0 * fma(fma(0.008333333333333333, (y * y), -0.16666666666666666), (y * y), 1.0);
	} else {
		tmp = fma((fma(fma((x * x), 0.001388888888888889, 0.041666666666666664), (x * x), 0.5) * x), x, 1.0) * 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(y) / y)
	tmp = 0.0
	if (t_0 <= -5e-308)
		tmp = Float64(fma(Float64(x * x), 0.5, 1.0) * fma(fma(fma(-0.0001984126984126984, Float64(y * y), 0.008333333333333333), Float64(y * y), -0.16666666666666666), Float64(y * y), 1.0));
	elseif (t_0 <= 2e-123)
		tmp = Float64(1.0 * fma(fma(0.008333333333333333, Float64(y * y), -0.16666666666666666), Float64(y * y), 1.0));
	else
		tmp = Float64(fma(Float64(fma(fma(Float64(x * x), 0.001388888888888889, 0.041666666666666664), Float64(x * x), 0.5) * x), x, 1.0) * 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-308], N[(N[(N[(x * x), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] * N[(N[(N[(-0.0001984126984126984 * N[(y * y), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] * N[(y * y), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e-123], N[(1.0 * N[(N[(0.008333333333333333 * N[(y * y), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(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] * 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 y) y) < -4.99999999999999955e-308

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 89.1

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

   5. Applied rewrites 89.1%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 42.5

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

   8. Applied rewrites 42.5%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. unpow2 N/A

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

       5. lower-*.f64 41.2

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

   11. Applied rewrites 41.2%

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

   if -4.99999999999999955e-308 < (/.f64 (sin.f64 y) y) < 2.0000000000000001e-123

   1. Initial program 99.7%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   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 38.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 3.7%

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

   9. Taylor expanded in y around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. sub-neg N/A

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

       5. metadata-eval N/A

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

       6. lower-fma.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 N/A

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

       9. unpow2 N/A

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

       10. lower-*.f64 37.6

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

   11. Applied rewrites 37.6%

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

   if 2.0000000000000001e-123 < (/.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 88.7%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 80.4

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

   8. Applied rewrites 80.4%

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

   10. Applied rewrites 80.4%

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

Alternative 12: 70.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-308}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-123}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), x \cdot x, 0.5\right) \cdot x, x, 1\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)))
   (if (<= t_0 -5e-308)
     (*
      (fma -0.16666666666666666 (* y y) 1.0)
      (fma (fma 0.041666666666666664 (* x x) 0.5) (* x x) 1.0))
     (if (<= t_0 2e-123)
       (*
        1.0
        (fma
         (fma 0.008333333333333333 (* y y) -0.16666666666666666)
         (* y y)
         1.0))
       (*
        (fma
         (*
          (fma
           (fma (* x x) 0.001388888888888889 0.041666666666666664)
           (* x x)
           0.5)
          x)
         x
         1.0)
        1.0)))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double tmp;
	if (t_0 <= -5e-308) {
		tmp = fma(-0.16666666666666666, (y * y), 1.0) * fma(fma(0.041666666666666664, (x * x), 0.5), (x * x), 1.0);
	} else if (t_0 <= 2e-123) {
		tmp = 1.0 * fma(fma(0.008333333333333333, (y * y), -0.16666666666666666), (y * y), 1.0);
	} else {
		tmp = fma((fma(fma((x * x), 0.001388888888888889, 0.041666666666666664), (x * x), 0.5) * x), x, 1.0) * 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(y) / y)
	tmp = 0.0
	if (t_0 <= -5e-308)
		tmp = Float64(fma(-0.16666666666666666, Float64(y * y), 1.0) * fma(fma(0.041666666666666664, Float64(x * x), 0.5), Float64(x * x), 1.0));
	elseif (t_0 <= 2e-123)
		tmp = Float64(1.0 * fma(fma(0.008333333333333333, Float64(y * y), -0.16666666666666666), Float64(y * y), 1.0));
	else
		tmp = Float64(fma(Float64(fma(fma(Float64(x * x), 0.001388888888888889, 0.041666666666666664), Float64(x * x), 0.5) * x), x, 1.0) * 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-308], N[(N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(0.041666666666666664 * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e-123], N[(1.0 * N[(N[(0.008333333333333333 * N[(y * y), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(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] * 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 y) y) < -4.99999999999999955e-308

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 89.1

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

   5. Applied rewrites 89.1%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 40.1

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

   8. Applied rewrites 40.1%

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

   if -4.99999999999999955e-308 < (/.f64 (sin.f64 y) y) < 2.0000000000000001e-123

   1. Initial program 99.7%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   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 38.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 3.7%

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

   9. Taylor expanded in y around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. sub-neg N/A

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

       5. metadata-eval N/A

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

       6. lower-fma.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 N/A

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

       9. unpow2 N/A

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

       10. lower-*.f64 37.6

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

   11. Applied rewrites 37.6%

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

   if 2.0000000000000001e-123 < (/.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 88.7%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 80.4

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

   8. Applied rewrites 80.4%

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

   10. Applied rewrites 80.4%

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

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq -5 \cdot 10^{-308}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right)\\ \mathbf{elif}\;\frac{\sin y}{y} \leq 2 \cdot 10^{-123}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), x \cdot x, 0.5\right) \cdot x, x, 1\right) \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 69.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-308}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right) \cdot \mathsf{fma}\left(x \cdot x, 0.5, 1\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-123}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), x \cdot x, 0.5\right) \cdot x, x, 1\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)))
   (if (<= t_0 -5e-308)
     (* (fma -0.16666666666666666 (* y y) 1.0) (fma (* x x) 0.5 1.0))
     (if (<= t_0 2e-123)
       (*
        1.0
        (fma
         (fma 0.008333333333333333 (* y y) -0.16666666666666666)
         (* y y)
         1.0))
       (*
        (fma
         (*
          (fma
           (fma (* x x) 0.001388888888888889 0.041666666666666664)
           (* x x)
           0.5)
          x)
         x
         1.0)
        1.0)))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double tmp;
	if (t_0 <= -5e-308) {
		tmp = fma(-0.16666666666666666, (y * y), 1.0) * fma((x * x), 0.5, 1.0);
	} else if (t_0 <= 2e-123) {
		tmp = 1.0 * fma(fma(0.008333333333333333, (y * y), -0.16666666666666666), (y * y), 1.0);
	} else {
		tmp = fma((fma(fma((x * x), 0.001388888888888889, 0.041666666666666664), (x * x), 0.5) * x), x, 1.0) * 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(y) / y)
	tmp = 0.0
	if (t_0 <= -5e-308)
		tmp = Float64(fma(-0.16666666666666666, Float64(y * y), 1.0) * fma(Float64(x * x), 0.5, 1.0));
	elseif (t_0 <= 2e-123)
		tmp = Float64(1.0 * fma(fma(0.008333333333333333, Float64(y * y), -0.16666666666666666), Float64(y * y), 1.0));
	else
		tmp = Float64(fma(Float64(fma(fma(Float64(x * x), 0.001388888888888889, 0.041666666666666664), Float64(x * x), 0.5) * x), x, 1.0) * 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-308], N[(N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e-123], N[(1.0 * N[(N[(0.008333333333333333 * N[(y * y), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(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] * 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 y) y) < -4.99999999999999955e-308

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 1.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 1.5%

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

   9. Taylor expanded in y around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 25.3

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

   11. Applied rewrites 25.3%

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

   12. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. unpow2 N/A

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

       5. lower-*.f64 40.1

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

   14. Applied rewrites 40.1%

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

   if -4.99999999999999955e-308 < (/.f64 (sin.f64 y) y) < 2.0000000000000001e-123

   1. Initial program 99.7%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   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 38.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 3.7%

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

   9. Taylor expanded in y around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. sub-neg N/A

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

       5. metadata-eval N/A

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

       6. lower-fma.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 N/A

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

       9. unpow2 N/A

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

       10. lower-*.f64 37.6

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

   11. Applied rewrites 37.6%

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

   if 2.0000000000000001e-123 < (/.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 88.7%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 80.4

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

   8. Applied rewrites 80.4%

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

   10. Applied rewrites 80.4%

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

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq -5 \cdot 10^{-308}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right) \cdot \mathsf{fma}\left(x \cdot x, 0.5, 1\right)\\ \mathbf{elif}\;\frac{\sin y}{y} \leq 2 \cdot 10^{-123}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), x \cdot x, 0.5\right) \cdot x, x, 1\right) \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 69.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-308}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right) \cdot \mathsf{fma}\left(x \cdot x, 0.5, 1\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-123}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left(x \cdot x\right) \cdot 0.001388888888888889, x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)))
   (if (<= t_0 -5e-308)
     (* (fma -0.16666666666666666 (* y y) 1.0) (fma (* x x) 0.5 1.0))
     (if (<= t_0 2e-123)
       (*
        1.0
        (fma
         (fma 0.008333333333333333 (* y y) -0.16666666666666666)
         (* y y)
         1.0))
       (*
        (fma (fma (* (* x x) 0.001388888888888889) (* x x) 0.5) (* x x) 1.0)
        1.0)))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double tmp;
	if (t_0 <= -5e-308) {
		tmp = fma(-0.16666666666666666, (y * y), 1.0) * fma((x * x), 0.5, 1.0);
	} else if (t_0 <= 2e-123) {
		tmp = 1.0 * fma(fma(0.008333333333333333, (y * y), -0.16666666666666666), (y * y), 1.0);
	} else {
		tmp = fma(fma(((x * x) * 0.001388888888888889), (x * x), 0.5), (x * x), 1.0) * 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(y) / y)
	tmp = 0.0
	if (t_0 <= -5e-308)
		tmp = Float64(fma(-0.16666666666666666, Float64(y * y), 1.0) * fma(Float64(x * x), 0.5, 1.0));
	elseif (t_0 <= 2e-123)
		tmp = Float64(1.0 * fma(fma(0.008333333333333333, Float64(y * y), -0.16666666666666666), Float64(y * y), 1.0));
	else
		tmp = Float64(fma(fma(Float64(Float64(x * x) * 0.001388888888888889), Float64(x * x), 0.5), Float64(x * x), 1.0) * 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-308], N[(N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e-123], N[(1.0 * N[(N[(0.008333333333333333 * N[(y * y), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(x * x), $MachinePrecision] * 0.001388888888888889), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 y) y) < -4.99999999999999955e-308

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 1.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 1.5%

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

   9. Taylor expanded in y around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 25.3

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

   11. Applied rewrites 25.3%

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

   12. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. unpow2 N/A

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

       5. lower-*.f64 40.1

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

   14. Applied rewrites 40.1%

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

   if -4.99999999999999955e-308 < (/.f64 (sin.f64 y) y) < 2.0000000000000001e-123

   1. Initial program 99.7%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   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 38.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 3.7%

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

   9. Taylor expanded in y around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. sub-neg N/A

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

       5. metadata-eval N/A

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

       6. lower-fma.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 N/A

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

       9. unpow2 N/A

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

       10. lower-*.f64 37.6

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

   11. Applied rewrites 37.6%

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

   if 2.0000000000000001e-123 < (/.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 88.7%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 80.4

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

   8. Applied rewrites 80.4%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 80.1%

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

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq -5 \cdot 10^{-308}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right) \cdot \mathsf{fma}\left(x \cdot x, 0.5, 1\right)\\ \mathbf{elif}\;\frac{\sin y}{y} \leq 2 \cdot 10^{-123}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left(x \cdot x\right) \cdot 0.001388888888888889, x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 69.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-308}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right) \cdot \mathsf{fma}\left(x \cdot x, 0.5, 1\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-123}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right) \cdot x\right) \cdot x, x \cdot x, 1\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)))
   (if (<= t_0 -5e-308)
     (* (fma -0.16666666666666666 (* y y) 1.0) (fma (* x x) 0.5 1.0))
     (if (<= t_0 2e-123)
       (*
        1.0
        (fma
         (fma 0.008333333333333333 (* y y) -0.16666666666666666)
         (* y y)
         1.0))
       (*
        (fma
         (* (* (fma 0.001388888888888889 (* x x) 0.041666666666666664) x) x)
         (* x x)
         1.0)
        1.0)))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double tmp;
	if (t_0 <= -5e-308) {
		tmp = fma(-0.16666666666666666, (y * y), 1.0) * fma((x * x), 0.5, 1.0);
	} else if (t_0 <= 2e-123) {
		tmp = 1.0 * fma(fma(0.008333333333333333, (y * y), -0.16666666666666666), (y * y), 1.0);
	} else {
		tmp = fma(((fma(0.001388888888888889, (x * x), 0.041666666666666664) * x) * x), (x * x), 1.0) * 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(y) / y)
	tmp = 0.0
	if (t_0 <= -5e-308)
		tmp = Float64(fma(-0.16666666666666666, Float64(y * y), 1.0) * fma(Float64(x * x), 0.5, 1.0));
	elseif (t_0 <= 2e-123)
		tmp = Float64(1.0 * fma(fma(0.008333333333333333, Float64(y * y), -0.16666666666666666), Float64(y * y), 1.0));
	else
		tmp = Float64(fma(Float64(Float64(fma(0.001388888888888889, Float64(x * x), 0.041666666666666664) * x) * x), Float64(x * x), 1.0) * 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-308], N[(N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e-123], N[(1.0 * N[(N[(0.008333333333333333 * N[(y * y), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(0.001388888888888889 * N[(x * x), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 y) y) < -4.99999999999999955e-308

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 1.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 1.5%

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

   9. Taylor expanded in y around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 25.3

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

   11. Applied rewrites 25.3%

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

   12. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. unpow2 N/A

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

       5. lower-*.f64 40.1

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

   14. Applied rewrites 40.1%

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

   if -4.99999999999999955e-308 < (/.f64 (sin.f64 y) y) < 2.0000000000000001e-123

   1. Initial program 99.7%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   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 38.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 3.7%

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

   9. Taylor expanded in y around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. sub-neg N/A

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

       5. metadata-eval N/A

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

       6. lower-fma.f64 N/A

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

       7. unpow2 N/A

          \[\leadsto 1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, \color{blue}{y \cdot y}, \frac{-1}{6}\right), {y}^{2}, 1\right) \]

       8. lower-*.f64 N/A

          \[\leadsto 1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, \color{blue}{y \cdot y}, \frac{-1}{6}\right), {y}^{2}, 1\right) \]

       9. unpow2 N/A

          \[\leadsto 1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), \color{blue}{y \cdot y}, 1\right) \]

       10. lower-*.f64 37.6

           \[\leadsto 1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), \color{blue}{y \cdot y}, 1\right) \]

   11. Applied rewrites 37.6%

       \[\leadsto 1 \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), y \cdot y, 1\right)} \]

   if 2.0000000000000001e-123 < (/.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 88.7%

      \[\leadsto \cosh x \cdot \color{blue}{1} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)} \cdot 1 \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) + 1\right)} \cdot 1 \]

      2. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) \cdot {x}^{2}} + 1\right) \cdot 1 \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right), {x}^{2}, 1\right)} \cdot 1 \]

      4. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right) + \frac{1}{2}}, {x}^{2}, 1\right) \cdot 1 \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right) \cdot {x}^{2}} + \frac{1}{2}, {x}^{2}, 1\right) \cdot 1 \]

      6. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}, {x}^{2}, \frac{1}{2}\right)}, {x}^{2}, 1\right) \cdot 1 \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{720} \cdot {x}^{2} + \frac{1}{24}}, {x}^{2}, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot 1 \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{720}, {x}^{2}, \frac{1}{24}\right)}, {x}^{2}, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot 1 \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, \color{blue}{x \cdot x}, \frac{1}{24}\right), {x}^{2}, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot 1 \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, \color{blue}{x \cdot x}, \frac{1}{24}\right), {x}^{2}, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot 1 \]

      11. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, x \cdot x, \frac{1}{24}\right), \color{blue}{x \cdot x}, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot 1 \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, x \cdot x, \frac{1}{24}\right), \color{blue}{x \cdot x}, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot 1 \]

      13. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, x \cdot x, \frac{1}{24}\right), x \cdot x, \frac{1}{2}\right), \color{blue}{x \cdot x}, 1\right) \cdot 1 \]

      14. lower-*.f64 80.4

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right), \color{blue}{x \cdot x}, 1\right) \cdot 1 \]

   8. Applied rewrites 80.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right)} \cdot 1 \]

   9. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left({x}^{4} \cdot \left(\frac{1}{720} + \frac{1}{24} \cdot \frac{1}{{x}^{2}}\right), \color{blue}{x} \cdot x, 1\right) \cdot 1 \]
   10. Step-by-step derivation

   11. Applied rewrites 79.8%

       \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right) \cdot x\right) \cdot x, \color{blue}{x} \cdot x, 1\right) \cdot 1 \]
3. Recombined 3 regimes into one program.

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq -5 \cdot 10^{-308}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right) \cdot \mathsf{fma}\left(x \cdot x, 0.5, 1\right)\\ \mathbf{elif}\;\frac{\sin y}{y} \leq 2 \cdot 10^{-123}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right) \cdot x\right) \cdot x, x \cdot x, 1\right) \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 67.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-308}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right) \cdot \mathsf{fma}\left(x \cdot x, 0.5, 1\right)\\ \mathbf{elif}\;t\_0 \leq 10^{-99}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)))
   (if (<= t_0 -5e-308)
     (* (fma -0.16666666666666666 (* y y) 1.0) (fma (* x x) 0.5 1.0))
     (if (<= t_0 1e-99)
       (*
        1.0
        (fma
         (fma 0.008333333333333333 (* y y) -0.16666666666666666)
         (* y y)
         1.0))
       (* 1.0 (fma (fma 0.041666666666666664 (* x x) 0.5) (* x x) 1.0))))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double tmp;
	if (t_0 <= -5e-308) {
		tmp = fma(-0.16666666666666666, (y * y), 1.0) * fma((x * x), 0.5, 1.0);
	} else if (t_0 <= 1e-99) {
		tmp = 1.0 * fma(fma(0.008333333333333333, (y * y), -0.16666666666666666), (y * y), 1.0);
	} else {
		tmp = 1.0 * fma(fma(0.041666666666666664, (x * x), 0.5), (x * x), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(y) / y)
	tmp = 0.0
	if (t_0 <= -5e-308)
		tmp = Float64(fma(-0.16666666666666666, Float64(y * y), 1.0) * fma(Float64(x * x), 0.5, 1.0));
	elseif (t_0 <= 1e-99)
		tmp = Float64(1.0 * fma(fma(0.008333333333333333, Float64(y * y), -0.16666666666666666), Float64(y * y), 1.0));
	else
		tmp = Float64(1.0 * fma(fma(0.041666666666666664, Float64(x * x), 0.5), Float64(x * x), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-308], N[(N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e-99], N[(1.0 * N[(N[(0.008333333333333333 * N[(y * y), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 * N[(N[(0.041666666666666664 * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 y) y) < -4.99999999999999955e-308

   1. Initial program 99.8%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \cosh x \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 1.2%

      \[\leadsto \cosh x \cdot \color{blue}{1} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \cdot 1 \]
   7. Step-by-step derivation

   8. Applied rewrites 1.5%

      \[\leadsto \color{blue}{1} \cdot 1 \]

   9. Taylor expanded in y around 0

      \[\leadsto 1 \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {y}^{2}\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto 1 \cdot \color{blue}{\left(\frac{-1}{6} \cdot {y}^{2} + 1\right)} \]

       2. lower-fma.f64 N/A

          \[\leadsto 1 \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{6}, {y}^{2}, 1\right)} \]

       3. unpow2 N/A

          \[\leadsto 1 \cdot \mathsf{fma}\left(\frac{-1}{6}, \color{blue}{y \cdot y}, 1\right) \]

       4. lower-*.f64 25.3

          \[\leadsto 1 \cdot \mathsf{fma}\left(-0.16666666666666666, \color{blue}{y \cdot y}, 1\right) \]

   11. Applied rewrites 25.3%

       \[\leadsto 1 \cdot \color{blue}{\mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)} \]

   12. Taylor expanded in x around 0

       \[\leadsto \color{blue}{\left(1 + \frac{1}{2} \cdot {x}^{2}\right)} \cdot \mathsf{fma}\left(\frac{-1}{6}, y \cdot y, 1\right) \]
   13. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot {x}^{2} + 1\right)} \cdot \mathsf{fma}\left(\frac{-1}{6}, y \cdot y, 1\right) \]

       2. *-commutative N/A

          \[\leadsto \left(\color{blue}{{x}^{2} \cdot \frac{1}{2}} + 1\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, y \cdot y, 1\right) \]

       3. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{2}, 1\right)} \cdot \mathsf{fma}\left(\frac{-1}{6}, y \cdot y, 1\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2}, 1\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, y \cdot y, 1\right) \]

       5. lower-*.f64 40.1

          \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.5, 1\right) \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right) \]

   14. Applied rewrites 40.1%

       \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, 0.5, 1\right)} \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right) \]

   if -4.99999999999999955e-308 < (/.f64 (sin.f64 y) y) < 1e-99

   1. Initial program 99.7%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   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 41.4%

      \[\leadsto \cosh x \cdot \color{blue}{1} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \cdot 1 \]
   7. Step-by-step derivation

   8. Applied rewrites 3.7%

      \[\leadsto \color{blue}{1} \cdot 1 \]

   9. Taylor expanded in y around 0

      \[\leadsto 1 \cdot \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto 1 \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right) + 1\right)} \]

       2. *-commutative N/A

          \[\leadsto 1 \cdot \left(\color{blue}{\left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right) \cdot {y}^{2}} + 1\right) \]

       3. lower-fma.f64 N/A

          \[\leadsto 1 \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}, {y}^{2}, 1\right)} \]

       4. sub-neg N/A

          \[\leadsto 1 \cdot \mathsf{fma}\left(\color{blue}{\frac{1}{120} \cdot {y}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}, {y}^{2}, 1\right) \]

       5. metadata-eval N/A

          \[\leadsto 1 \cdot \mathsf{fma}\left(\frac{1}{120} \cdot {y}^{2} + \color{blue}{\frac{-1}{6}}, {y}^{2}, 1\right) \]

       6. lower-fma.f64 N/A

          \[\leadsto 1 \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{120}, {y}^{2}, \frac{-1}{6}\right)}, {y}^{2}, 1\right) \]

       7. unpow2 N/A

          \[\leadsto 1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, \color{blue}{y \cdot y}, \frac{-1}{6}\right), {y}^{2}, 1\right) \]

       8. lower-*.f64 N/A

          \[\leadsto 1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, \color{blue}{y \cdot y}, \frac{-1}{6}\right), {y}^{2}, 1\right) \]

       9. unpow2 N/A

          \[\leadsto 1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), \color{blue}{y \cdot y}, 1\right) \]

       10. lower-*.f64 40.1

           \[\leadsto 1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), \color{blue}{y \cdot y}, 1\right) \]

   11. Applied rewrites 40.1%

       \[\leadsto 1 \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), y \cdot y, 1\right)} \]

   if 1e-99 < (/.f64 (sin.f64 y) y)

   1. Initial program 99.9%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)} \cdot \frac{\sin y}{y} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) + 1\right)} \cdot \frac{\sin y}{y} \]

      2. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) \cdot {x}^{2}} + 1\right) \cdot \frac{\sin y}{y} \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, {x}^{2}, 1\right)} \cdot \frac{\sin y}{y} \]

      4. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{24} \cdot {x}^{2} + \frac{1}{2}}, {x}^{2}, 1\right) \cdot \frac{\sin y}{y} \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24}, {x}^{2}, \frac{1}{2}\right)}, {x}^{2}, 1\right) \cdot \frac{\sin y}{y} \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{x \cdot x}, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot \frac{\sin y}{y} \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{x \cdot x}, \frac{1}{2}\right), {x}^{2}, 1\right) \cdot \frac{\sin y}{y} \]

      8. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), \color{blue}{x \cdot x}, 1\right) \cdot \frac{\sin y}{y} \]

      9. lower-*.f64 87.1

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), \color{blue}{x \cdot x}, 1\right) \cdot \frac{\sin y}{y} \]

   5. Applied rewrites 87.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right)} \cdot \frac{\sin y}{y} \]

   6. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 76.7%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot \color{blue}{1} \]
3. Recombined 3 regimes into one program.

4. Final simplification 61.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq -5 \cdot 10^{-308}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right) \cdot \mathsf{fma}\left(x \cdot x, 0.5, 1\right)\\ \mathbf{elif}\;\frac{\sin y}{y} \leq 10^{-99}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 60.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ t_1 := \mathsf{fma}\left(x \cdot x, 0.5, 1\right)\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-308}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right) \cdot t\_1\\ \mathbf{elif}\;t\_0 \leq 10^{-99}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)) (t_1 (fma (* x x) 0.5 1.0)))
   (if (<= t_0 -5e-308)
     (* (fma -0.16666666666666666 (* y y) 1.0) t_1)
     (if (<= t_0 1e-99)
       (*
        1.0
        (fma
         (fma 0.008333333333333333 (* y y) -0.16666666666666666)
         (* y y)
         1.0))
       (* 1.0 t_1)))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double t_1 = fma((x * x), 0.5, 1.0);
	double tmp;
	if (t_0 <= -5e-308) {
		tmp = fma(-0.16666666666666666, (y * y), 1.0) * t_1;
	} else if (t_0 <= 1e-99) {
		tmp = 1.0 * fma(fma(0.008333333333333333, (y * y), -0.16666666666666666), (y * y), 1.0);
	} else {
		tmp = 1.0 * t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(y) / y)
	t_1 = fma(Float64(x * x), 0.5, 1.0)
	tmp = 0.0
	if (t_0 <= -5e-308)
		tmp = Float64(fma(-0.16666666666666666, Float64(y * y), 1.0) * t_1);
	elseif (t_0 <= 1e-99)
		tmp = Float64(1.0 * fma(fma(0.008333333333333333, Float64(y * y), -0.16666666666666666), Float64(y * y), 1.0));
	else
		tmp = Float64(1.0 * t_1);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * x), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-308], N[(N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t$95$0, 1e-99], N[(1.0 * N[(N[(0.008333333333333333 * N[(y * y), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 * t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 y) y) < -4.99999999999999955e-308

   1. Initial program 99.8%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \cosh x \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 1.2%

      \[\leadsto \cosh x \cdot \color{blue}{1} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \cdot 1 \]
   7. Step-by-step derivation

   8. Applied rewrites 1.5%

      \[\leadsto \color{blue}{1} \cdot 1 \]

   9. Taylor expanded in y around 0

      \[\leadsto 1 \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {y}^{2}\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto 1 \cdot \color{blue}{\left(\frac{-1}{6} \cdot {y}^{2} + 1\right)} \]

       2. lower-fma.f64 N/A

          \[\leadsto 1 \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{6}, {y}^{2}, 1\right)} \]

       3. unpow2 N/A

          \[\leadsto 1 \cdot \mathsf{fma}\left(\frac{-1}{6}, \color{blue}{y \cdot y}, 1\right) \]

       4. lower-*.f64 25.3

          \[\leadsto 1 \cdot \mathsf{fma}\left(-0.16666666666666666, \color{blue}{y \cdot y}, 1\right) \]

   11. Applied rewrites 25.3%

       \[\leadsto 1 \cdot \color{blue}{\mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)} \]

   12. Taylor expanded in x around 0

       \[\leadsto \color{blue}{\left(1 + \frac{1}{2} \cdot {x}^{2}\right)} \cdot \mathsf{fma}\left(\frac{-1}{6}, y \cdot y, 1\right) \]
   13. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot {x}^{2} + 1\right)} \cdot \mathsf{fma}\left(\frac{-1}{6}, y \cdot y, 1\right) \]

       2. *-commutative N/A

          \[\leadsto \left(\color{blue}{{x}^{2} \cdot \frac{1}{2}} + 1\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, y \cdot y, 1\right) \]

       3. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{2}, 1\right)} \cdot \mathsf{fma}\left(\frac{-1}{6}, y \cdot y, 1\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2}, 1\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, y \cdot y, 1\right) \]

       5. lower-*.f64 40.1

          \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.5, 1\right) \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right) \]

   14. Applied rewrites 40.1%

       \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, 0.5, 1\right)} \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right) \]

   if -4.99999999999999955e-308 < (/.f64 (sin.f64 y) y) < 1e-99

   1. Initial program 99.7%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   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 41.4%

      \[\leadsto \cosh x \cdot \color{blue}{1} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \cdot 1 \]
   7. Step-by-step derivation

   8. Applied rewrites 3.7%

      \[\leadsto \color{blue}{1} \cdot 1 \]

   9. Taylor expanded in y around 0

      \[\leadsto 1 \cdot \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto 1 \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right) + 1\right)} \]

       2. *-commutative N/A

          \[\leadsto 1 \cdot \left(\color{blue}{\left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right) \cdot {y}^{2}} + 1\right) \]

       3. lower-fma.f64 N/A

          \[\leadsto 1 \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}, {y}^{2}, 1\right)} \]

       4. sub-neg N/A

          \[\leadsto 1 \cdot \mathsf{fma}\left(\color{blue}{\frac{1}{120} \cdot {y}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}, {y}^{2}, 1\right) \]

       5. metadata-eval N/A

          \[\leadsto 1 \cdot \mathsf{fma}\left(\frac{1}{120} \cdot {y}^{2} + \color{blue}{\frac{-1}{6}}, {y}^{2}, 1\right) \]

       6. lower-fma.f64 N/A

          \[\leadsto 1 \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{120}, {y}^{2}, \frac{-1}{6}\right)}, {y}^{2}, 1\right) \]

       7. unpow2 N/A

          \[\leadsto 1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, \color{blue}{y \cdot y}, \frac{-1}{6}\right), {y}^{2}, 1\right) \]

       8. lower-*.f64 N/A

          \[\leadsto 1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, \color{blue}{y \cdot y}, \frac{-1}{6}\right), {y}^{2}, 1\right) \]

       9. unpow2 N/A

          \[\leadsto 1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), \color{blue}{y \cdot y}, 1\right) \]

       10. lower-*.f64 40.1

           \[\leadsto 1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), \color{blue}{y \cdot y}, 1\right) \]

   11. Applied rewrites 40.1%

       \[\leadsto 1 \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), y \cdot y, 1\right)} \]

   if 1e-99 < (/.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 89.1%

      \[\leadsto \cosh x \cdot \color{blue}{1} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + \frac{1}{2} \cdot {x}^{2}\right)} \cdot 1 \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot {x}^{2} + 1\right)} \cdot 1 \]

      2. *-commutative N/A

         \[\leadsto \left(\color{blue}{{x}^{2} \cdot \frac{1}{2}} + 1\right) \cdot 1 \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{2}, 1\right)} \cdot 1 \]

      4. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2}, 1\right) \cdot 1 \]

      5. lower-*.f64 66.3

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.5, 1\right) \cdot 1 \]

   8. Applied rewrites 66.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, 0.5, 1\right)} \cdot 1 \]
3. Recombined 3 regimes into one program.

4. Final simplification 55.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq -5 \cdot 10^{-308}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right) \cdot \mathsf{fma}\left(x \cdot x, 0.5, 1\right)\\ \mathbf{elif}\;\frac{\sin y}{y} \leq 10^{-99}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(x \cdot x, 0.5, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 55.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-308}:\\ \;\;\;\;\left(-0.16666666666666666 \cdot \left(y \cdot y\right)\right) \cdot 1\\ \mathbf{elif}\;t\_0 \leq 10^{-99}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(x \cdot x, 0.5, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)))
   (if (<= t_0 -5e-308)
     (* (* -0.16666666666666666 (* y y)) 1.0)
     (if (<= t_0 1e-99)
       (*
        1.0
        (fma
         (fma 0.008333333333333333 (* y y) -0.16666666666666666)
         (* y y)
         1.0))
       (* 1.0 (fma (* x x) 0.5 1.0))))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double tmp;
	if (t_0 <= -5e-308) {
		tmp = (-0.16666666666666666 * (y * y)) * 1.0;
	} else if (t_0 <= 1e-99) {
		tmp = 1.0 * fma(fma(0.008333333333333333, (y * y), -0.16666666666666666), (y * y), 1.0);
	} else {
		tmp = 1.0 * fma((x * x), 0.5, 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(y) / y)
	tmp = 0.0
	if (t_0 <= -5e-308)
		tmp = Float64(Float64(-0.16666666666666666 * Float64(y * y)) * 1.0);
	elseif (t_0 <= 1e-99)
		tmp = Float64(1.0 * fma(fma(0.008333333333333333, Float64(y * y), -0.16666666666666666), Float64(y * y), 1.0));
	else
		tmp = Float64(1.0 * fma(Float64(x * x), 0.5, 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-308], N[(N[(-0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[t$95$0, 1e-99], N[(1.0 * N[(N[(0.008333333333333333 * N[(y * y), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 * N[(N[(x * x), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 y) y) < -4.99999999999999955e-308

   1. Initial program 99.8%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \cosh x \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 1.2%

      \[\leadsto \cosh x \cdot \color{blue}{1} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \cdot 1 \]
   7. Step-by-step derivation

   8. Applied rewrites 1.5%

      \[\leadsto \color{blue}{1} \cdot 1 \]

   9. Taylor expanded in y around 0

      \[\leadsto 1 \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {y}^{2}\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto 1 \cdot \color{blue}{\left(\frac{-1}{6} \cdot {y}^{2} + 1\right)} \]

       2. lower-fma.f64 N/A

          \[\leadsto 1 \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{6}, {y}^{2}, 1\right)} \]

       3. unpow2 N/A

          \[\leadsto 1 \cdot \mathsf{fma}\left(\frac{-1}{6}, \color{blue}{y \cdot y}, 1\right) \]

       4. lower-*.f64 25.3

          \[\leadsto 1 \cdot \mathsf{fma}\left(-0.16666666666666666, \color{blue}{y \cdot y}, 1\right) \]

   11. Applied rewrites 25.3%

       \[\leadsto 1 \cdot \color{blue}{\mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)} \]

   12. Taylor expanded in y around inf

       \[\leadsto 1 \cdot \left(\frac{-1}{6} \cdot \color{blue}{{y}^{2}}\right) \]
   13. Step-by-step derivation

   14. Applied rewrites 25.3%

       \[\leadsto 1 \cdot \left(-0.16666666666666666 \cdot \color{blue}{\left(y \cdot y\right)}\right) \]

   if -4.99999999999999955e-308 < (/.f64 (sin.f64 y) y) < 1e-99

   1. Initial program 99.7%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   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 41.4%

      \[\leadsto \cosh x \cdot \color{blue}{1} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \cdot 1 \]
   7. Step-by-step derivation

   8. Applied rewrites 3.7%

      \[\leadsto \color{blue}{1} \cdot 1 \]

   9. Taylor expanded in y around 0

      \[\leadsto 1 \cdot \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto 1 \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right) + 1\right)} \]

       2. *-commutative N/A

          \[\leadsto 1 \cdot \left(\color{blue}{\left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right) \cdot {y}^{2}} + 1\right) \]

       3. lower-fma.f64 N/A

          \[\leadsto 1 \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}, {y}^{2}, 1\right)} \]

       4. sub-neg N/A

          \[\leadsto 1 \cdot \mathsf{fma}\left(\color{blue}{\frac{1}{120} \cdot {y}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}, {y}^{2}, 1\right) \]

       5. metadata-eval N/A

          \[\leadsto 1 \cdot \mathsf{fma}\left(\frac{1}{120} \cdot {y}^{2} + \color{blue}{\frac{-1}{6}}, {y}^{2}, 1\right) \]

       6. lower-fma.f64 N/A

          \[\leadsto 1 \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{120}, {y}^{2}, \frac{-1}{6}\right)}, {y}^{2}, 1\right) \]

       7. unpow2 N/A

          \[\leadsto 1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, \color{blue}{y \cdot y}, \frac{-1}{6}\right), {y}^{2}, 1\right) \]

       8. lower-*.f64 N/A

          \[\leadsto 1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, \color{blue}{y \cdot y}, \frac{-1}{6}\right), {y}^{2}, 1\right) \]

       9. unpow2 N/A

          \[\leadsto 1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), \color{blue}{y \cdot y}, 1\right) \]

       10. lower-*.f64 40.1

           \[\leadsto 1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), \color{blue}{y \cdot y}, 1\right) \]

   11. Applied rewrites 40.1%

       \[\leadsto 1 \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), y \cdot y, 1\right)} \]

   if 1e-99 < (/.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 89.1%

      \[\leadsto \cosh x \cdot \color{blue}{1} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + \frac{1}{2} \cdot {x}^{2}\right)} \cdot 1 \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot {x}^{2} + 1\right)} \cdot 1 \]

      2. *-commutative N/A

         \[\leadsto \left(\color{blue}{{x}^{2} \cdot \frac{1}{2}} + 1\right) \cdot 1 \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{2}, 1\right)} \cdot 1 \]

      4. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2}, 1\right) \cdot 1 \]

      5. lower-*.f64 66.3

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.5, 1\right) \cdot 1 \]

   8. Applied rewrites 66.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, 0.5, 1\right)} \cdot 1 \]
3. Recombined 3 regimes into one program.

4. Final simplification 51.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq -5 \cdot 10^{-308}:\\ \;\;\;\;\left(-0.16666666666666666 \cdot \left(y \cdot y\right)\right) \cdot 1\\ \mathbf{elif}\;\frac{\sin y}{y} \leq 10^{-99}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(x \cdot x, 0.5, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 51.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \cdot \cosh x \leq -2 \cdot 10^{-147}:\\ \;\;\;\;\left(-0.16666666666666666 \cdot \left(y \cdot y\right)\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(x \cdot x, 0.5, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (/ (sin y) y) (cosh x)) -2e-147)
   (* (* -0.16666666666666666 (* y y)) 1.0)
   (* 1.0 (fma (* x x) 0.5 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (((sin(y) / y) * cosh(x)) <= -2e-147) {
		tmp = (-0.16666666666666666 * (y * y)) * 1.0;
	} else {
		tmp = 1.0 * fma((x * x), 0.5, 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(sin(y) / y) * cosh(x)) <= -2e-147)
		tmp = Float64(Float64(-0.16666666666666666 * Float64(y * y)) * 1.0);
	else
		tmp = Float64(1.0 * fma(Float64(x * x), 0.5, 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision] * N[Cosh[x], $MachinePrecision]), $MachinePrecision], -2e-147], N[(N[(-0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], N[(1.0 * N[(N[(x * x), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < -1.9999999999999999e-147

   1. Initial program 99.7%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   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 0.8%

      \[\leadsto \cosh x \cdot \color{blue}{1} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \cdot 1 \]
   7. Step-by-step derivation

   8. Applied rewrites 1.2%

      \[\leadsto \color{blue}{1} \cdot 1 \]

   9. Taylor expanded in y around 0

      \[\leadsto 1 \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {y}^{2}\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto 1 \cdot \color{blue}{\left(\frac{-1}{6} \cdot {y}^{2} + 1\right)} \]

       2. lower-fma.f64 N/A

          \[\leadsto 1 \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{6}, {y}^{2}, 1\right)} \]

       3. unpow2 N/A

          \[\leadsto 1 \cdot \mathsf{fma}\left(\frac{-1}{6}, \color{blue}{y \cdot y}, 1\right) \]

       4. lower-*.f64 32.1

          \[\leadsto 1 \cdot \mathsf{fma}\left(-0.16666666666666666, \color{blue}{y \cdot y}, 1\right) \]

   11. Applied rewrites 32.1%

       \[\leadsto 1 \cdot \color{blue}{\mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)} \]

   12. Taylor expanded in y around inf

       \[\leadsto 1 \cdot \left(\frac{-1}{6} \cdot \color{blue}{{y}^{2}}\right) \]
   13. Step-by-step derivation

   14. Applied rewrites 32.1%

       \[\leadsto 1 \cdot \left(-0.16666666666666666 \cdot \color{blue}{\left(y \cdot y\right)}\right) \]

   if -1.9999999999999999e-147 < (*.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 73.7%

      \[\leadsto \cosh x \cdot \color{blue}{1} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + \frac{1}{2} \cdot {x}^{2}\right)} \cdot 1 \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot {x}^{2} + 1\right)} \cdot 1 \]

      2. *-commutative N/A

         \[\leadsto \left(\color{blue}{{x}^{2} \cdot \frac{1}{2}} + 1\right) \cdot 1 \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{2}, 1\right)} \cdot 1 \]

      4. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2}, 1\right) \cdot 1 \]

      5. lower-*.f64 52.5

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.5, 1\right) \cdot 1 \]

   8. Applied rewrites 52.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, 0.5, 1\right)} \cdot 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 48.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \cdot \cosh x \leq -2 \cdot 10^{-147}:\\ \;\;\;\;\left(-0.16666666666666666 \cdot \left(y \cdot y\right)\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(x \cdot x, 0.5, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 33.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \cdot \cosh x \leq -2 \cdot 10^{-147}:\\ \;\;\;\;\left(-0.16666666666666666 \cdot \left(y \cdot y\right)\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;1 \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (/ (sin y) y) (cosh x)) -2e-147)
   (* (* -0.16666666666666666 (* y y)) 1.0)
   (* 1.0 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (((sin(y) / y) * cosh(x)) <= -2e-147) {
		tmp = (-0.16666666666666666 * (y * y)) * 1.0;
	} else {
		tmp = 1.0 * 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (((sin(y) / y) * cosh(x)) <= (-2d-147)) then
        tmp = ((-0.16666666666666666d0) * (y * y)) * 1.0d0
    else
        tmp = 1.0d0 * 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (((Math.sin(y) / y) * Math.cosh(x)) <= -2e-147) {
		tmp = (-0.16666666666666666 * (y * y)) * 1.0;
	} else {
		tmp = 1.0 * 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((math.sin(y) / y) * math.cosh(x)) <= -2e-147:
		tmp = (-0.16666666666666666 * (y * y)) * 1.0
	else:
		tmp = 1.0 * 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(sin(y) / y) * cosh(x)) <= -2e-147)
		tmp = Float64(Float64(-0.16666666666666666 * Float64(y * y)) * 1.0);
	else
		tmp = Float64(1.0 * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((sin(y) / y) * cosh(x)) <= -2e-147)
		tmp = (-0.16666666666666666 * (y * y)) * 1.0;
	else
		tmp = 1.0 * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision] * N[Cosh[x], $MachinePrecision]), $MachinePrecision], -2e-147], N[(N[(-0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], N[(1.0 * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < -1.9999999999999999e-147

   1. Initial program 99.7%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   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 0.8%

      \[\leadsto \cosh x \cdot \color{blue}{1} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \cdot 1 \]
   7. Step-by-step derivation

   8. Applied rewrites 1.2%

      \[\leadsto \color{blue}{1} \cdot 1 \]

   9. Taylor expanded in y around 0

      \[\leadsto 1 \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {y}^{2}\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto 1 \cdot \color{blue}{\left(\frac{-1}{6} \cdot {y}^{2} + 1\right)} \]

       2. lower-fma.f64 N/A

          \[\leadsto 1 \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{6}, {y}^{2}, 1\right)} \]

       3. unpow2 N/A

          \[\leadsto 1 \cdot \mathsf{fma}\left(\frac{-1}{6}, \color{blue}{y \cdot y}, 1\right) \]

       4. lower-*.f64 32.1

          \[\leadsto 1 \cdot \mathsf{fma}\left(-0.16666666666666666, \color{blue}{y \cdot y}, 1\right) \]

   11. Applied rewrites 32.1%

       \[\leadsto 1 \cdot \color{blue}{\mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)} \]

   12. Taylor expanded in y around inf

       \[\leadsto 1 \cdot \left(\frac{-1}{6} \cdot \color{blue}{{y}^{2}}\right) \]
   13. Step-by-step derivation

   14. Applied rewrites 32.1%

       \[\leadsto 1 \cdot \left(-0.16666666666666666 \cdot \color{blue}{\left(y \cdot y\right)}\right) \]

   if -1.9999999999999999e-147 < (*.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 73.7%

      \[\leadsto \cosh x \cdot \color{blue}{1} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \cdot 1 \]
   7. Step-by-step derivation

   8. Applied rewrites 37.4%

      \[\leadsto \color{blue}{1} \cdot 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 36.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \cdot \cosh x \leq -2 \cdot 10^{-147}:\\ \;\;\;\;\left(-0.16666666666666666 \cdot \left(y \cdot y\right)\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;1 \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 32.7% accurate, 12.8× speedup

Math

\[\begin{array}{l} \\ 1 \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right) \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (* 1.0 (fma -0.16666666666666666 (* y y) 1.0)))

C

double code(double x, double y) {
	return 1.0 * fma(-0.16666666666666666, (y * y), 1.0);
}

Julia

function code(x, y)
	return Float64(1.0 * fma(-0.16666666666666666, Float64(y * y), 1.0))
end

Wolfram

code[x_, y_] := N[(1.0 * N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\cosh x \cdot \frac{\sin y}{y} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \cosh x \cdot \color{blue}{1} \]
4. Step-by-step derivation

5. Applied rewrites 58.0%

   \[\leadsto \cosh x \cdot \color{blue}{1} \]

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1} \cdot 1 \]
7. Step-by-step derivation

8. Applied rewrites 29.6%

   \[\leadsto \color{blue}{1} \cdot 1 \]

9. Taylor expanded in y around 0

   \[\leadsto 1 \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {y}^{2}\right)} \]
10. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto 1 \cdot \color{blue}{\left(\frac{-1}{6} \cdot {y}^{2} + 1\right)} \]

    2. lower-fma.f64 N/A

       \[\leadsto 1 \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{6}, {y}^{2}, 1\right)} \]

    3. unpow2 N/A

       \[\leadsto 1 \cdot \mathsf{fma}\left(\frac{-1}{6}, \color{blue}{y \cdot y}, 1\right) \]

    4. lower-*.f64 35.7

       \[\leadsto 1 \cdot \mathsf{fma}\left(-0.16666666666666666, \color{blue}{y \cdot y}, 1\right) \]

11. Applied rewrites 35.7%

    \[\leadsto 1 \cdot \color{blue}{\mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)} \]
12. Add Preprocessing

Alternative 22: 27.1% accurate, 36.2× speedup

Math

\[\begin{array}{l} \\ 1 \cdot 1 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* 1.0 1.0))

C

double code(double x, double y) {
	return 1.0 * 1.0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 * 1.0d0
end function

Java

public static double code(double x, double y) {
	return 1.0 * 1.0;
}

Python

def code(x, y):
	return 1.0 * 1.0

Julia

function code(x, y)
	return Float64(1.0 * 1.0)
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0 * 1.0;
end

Wolfram

code[x_, y_] := N[(1.0 * 1.0), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\cosh x \cdot \frac{\sin y}{y} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \cosh x \cdot \color{blue}{1} \]
4. Step-by-step derivation

5. Applied rewrites 58.0%

   \[\leadsto \cosh x \cdot \color{blue}{1} \]

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1} \cdot 1 \]
7. Step-by-step derivation

8. Applied rewrites 29.6%

   \[\leadsto \color{blue}{1} \cdot 1 \]
9. Add Preprocessing

Developer Target 1: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\cosh x \cdot \sin y}{y} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ (* (cosh x) (sin y)) y))

C

double code(double x, double y) {
	return (cosh(x) * sin(y)) / y;
}

Fortran

real(8) function code(x, y)
    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 2024276 
(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)))