Linear.Quaternion:$csinh from linear-1.19.1.3

Percentage Accurate: 99.9% → 99.9%

Time: 10.3s

Alternatives: 23

Speedup: 1.0×

• 48.8% 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

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

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]

Derivation

1. Initial program 99.9%

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

Alternative 2: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ t_1 := \cosh x \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\cosh x \cdot \mathsf{fma}\left(y, y \cdot -0.16666666666666666, 1\right)\\ \mathbf{elif}\;t\_1 \leq 0.9999999999282907:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)) (t_1 (* (cosh x) t_0)))
   (if (<= t_1 (- INFINITY))
     (* (cosh x) (fma y (* y -0.16666666666666666) 1.0))
     (if (<= t_1 0.9999999999282907)
       (*
        t_0
        (fma
         (* x x)
         (fma
          x
          (* x (fma (* x x) 0.001388888888888889 0.041666666666666664))
          0.5)
         1.0))
       (* (cosh x) 1.0)))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double t_1 = cosh(x) * t_0;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = cosh(x) * fma(y, (y * -0.16666666666666666), 1.0);
	} else if (t_1 <= 0.9999999999282907) {
		tmp = t_0 * fma((x * x), fma(x, (x * fma((x * x), 0.001388888888888889, 0.041666666666666664)), 0.5), 1.0);
	} else {
		tmp = cosh(x) * 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(y) / y)
	t_1 = Float64(cosh(x) * t_0)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(cosh(x) * fma(y, Float64(y * -0.16666666666666666), 1.0));
	elseif (t_1 <= 0.9999999999282907)
		tmp = Float64(t_0 * fma(Float64(x * x), fma(x, Float64(x * fma(Float64(x * x), 0.001388888888888889, 0.041666666666666664)), 0.5), 1.0));
	else
		tmp = Float64(cosh(x) * 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cosh[x], $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[Cosh[x], $MachinePrecision] * N[(y * N[(y * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.9999999999282907], N[(t$95$0 * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Cosh[x], $MachinePrecision] * 1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -inf.0 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < 0.99999999992829069

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)} \cdot \frac{\sin y}{y} \]
   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. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 98.6

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

   5. Applied rewrites 98.6%

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

   if 0.99999999992829069 < (*.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 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -\infty:\\ \;\;\;\;\cosh x \cdot \mathsf{fma}\left(y, y \cdot -0.16666666666666666, 1\right)\\ \mathbf{elif}\;\cosh x \cdot \frac{\sin y}{y} \leq 0.9999999999282907:\\ \;\;\;\;\frac{\sin y}{y} \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ t_1 := \cosh x \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\cosh x \cdot \mathsf{fma}\left(y, y \cdot -0.16666666666666666, 1\right)\\ \mathbf{elif}\;t\_1 \leq 0.9999999999282907:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)) (t_1 (* (cosh x) t_0)))
   (if (<= t_1 (- INFINITY))
     (* (cosh x) (fma y (* y -0.16666666666666666) 1.0))
     (if (<= t_1 0.9999999999282907)
       (* t_0 (fma (* x x) (fma (* x x) 0.041666666666666664 0.5) 1.0))
       (* (cosh x) 1.0)))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double t_1 = cosh(x) * t_0;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = cosh(x) * fma(y, (y * -0.16666666666666666), 1.0);
	} else if (t_1 <= 0.9999999999282907) {
		tmp = t_0 * fma((x * x), fma((x * x), 0.041666666666666664, 0.5), 1.0);
	} else {
		tmp = cosh(x) * 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(y) / y)
	t_1 = Float64(cosh(x) * t_0)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(cosh(x) * fma(y, Float64(y * -0.16666666666666666), 1.0));
	elseif (t_1 <= 0.9999999999282907)
		tmp = Float64(t_0 * fma(Float64(x * x), fma(Float64(x * x), 0.041666666666666664, 0.5), 1.0));
	else
		tmp = Float64(cosh(x) * 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cosh[x], $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[Cosh[x], $MachinePrecision] * N[(y * N[(y * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.9999999999282907], N[(t$95$0 * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Cosh[x], $MachinePrecision] * 1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -inf.0 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < 0.99999999992829069

   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. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 98.6

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

   5. Applied rewrites 98.6%

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

   if 0.99999999992829069 < (*.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 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -\infty:\\ \;\;\;\;\cosh x \cdot \mathsf{fma}\left(y, y \cdot -0.16666666666666666, 1\right)\\ \mathbf{elif}\;\cosh x \cdot \frac{\sin y}{y} \leq 0.9999999999282907:\\ \;\;\;\;\frac{\sin y}{y} \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ t_1 := \cosh x \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\cosh x \cdot \mathsf{fma}\left(y, y \cdot -0.16666666666666666, 1\right)\\ \mathbf{elif}\;t\_1 \leq 0.9999999999282907:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(0.5, x \cdot x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)) (t_1 (* (cosh x) t_0)))
   (if (<= t_1 (- INFINITY))
     (* (cosh x) (fma y (* y -0.16666666666666666) 1.0))
     (if (<= t_1 0.9999999999282907)
       (* t_0 (fma 0.5 (* x x) 1.0))
       (* (cosh x) 1.0)))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double t_1 = cosh(x) * t_0;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = cosh(x) * fma(y, (y * -0.16666666666666666), 1.0);
	} else if (t_1 <= 0.9999999999282907) {
		tmp = t_0 * fma(0.5, (x * x), 1.0);
	} else {
		tmp = cosh(x) * 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(y) / y)
	t_1 = Float64(cosh(x) * t_0)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(cosh(x) * fma(y, Float64(y * -0.16666666666666666), 1.0));
	elseif (t_1 <= 0.9999999999282907)
		tmp = Float64(t_0 * fma(0.5, Float64(x * x), 1.0));
	else
		tmp = Float64(cosh(x) * 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cosh[x], $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[Cosh[x], $MachinePrecision] * N[(y * N[(y * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.9999999999282907], N[(t$95$0 * N[(0.5 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Cosh[x], $MachinePrecision] * 1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -inf.0 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < 0.99999999992829069

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 98.3

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

   5. Applied rewrites 98.3%

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

   if 0.99999999992829069 < (*.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 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -\infty:\\ \;\;\;\;\cosh x \cdot \mathsf{fma}\left(y, y \cdot -0.16666666666666666, 1\right)\\ \mathbf{elif}\;\cosh x \cdot \frac{\sin y}{y} \leq 0.9999999999282907:\\ \;\;\;\;\frac{\sin y}{y} \cdot \mathsf{fma}\left(0.5, x \cdot x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ t_1 := \cosh x \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\cosh x \cdot \mathsf{fma}\left(y, y \cdot -0.16666666666666666, 1\right)\\ \mathbf{elif}\;t\_1 \leq 0.9999999999282907:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)) (t_1 (* (cosh x) t_0)))
   (if (<= t_1 (- INFINITY))
     (* (cosh x) (fma y (* y -0.16666666666666666) 1.0))
     (if (<= t_1 0.9999999999282907) t_0 (* (cosh x) 1.0)))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double t_1 = cosh(x) * t_0;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = cosh(x) * fma(y, (y * -0.16666666666666666), 1.0);
	} else if (t_1 <= 0.9999999999282907) {
		tmp = t_0;
	} else {
		tmp = cosh(x) * 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(y) / y)
	t_1 = Float64(cosh(x) * t_0)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(cosh(x) * fma(y, Float64(y * -0.16666666666666666), 1.0));
	elseif (t_1 <= 0.9999999999282907)
		tmp = t_0;
	else
		tmp = Float64(cosh(x) * 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cosh[x], $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[Cosh[x], $MachinePrecision] * N[(y * N[(y * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.9999999999282907], t$95$0, N[(N[Cosh[x], $MachinePrecision] * 1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -inf.0 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < 0.99999999992829069

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 97.1

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

   5. Applied rewrites 97.1%

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

   if 0.99999999992829069 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 100.0%

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

Alternative 6: 99.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ t_1 := \cosh x \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)\\ \mathbf{elif}\;t\_1 \leq 0.9999999999282907:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)) (t_1 (* (cosh x) t_0)))
   (if (<= t_1 (- INFINITY))
     (*
      (fma
       x
       (*
        x
        (fma
         x
         (* x (fma x (* x 0.001388888888888889) 0.041666666666666664))
         0.5))
       1.0)
      (fma
       (* y y)
       (fma
        (* y y)
        (fma (* y y) -0.0001984126984126984 0.008333333333333333)
        -0.16666666666666666)
       1.0))
     (if (<= t_1 0.9999999999282907) t_0 (* (cosh x) 1.0)))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double t_1 = cosh(x) * t_0;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma(x, (x * fma(x, (x * fma(x, (x * 0.001388888888888889), 0.041666666666666664)), 0.5)), 1.0) * fma((y * y), fma((y * y), fma((y * y), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666), 1.0);
	} else if (t_1 <= 0.9999999999282907) {
		tmp = t_0;
	} else {
		tmp = cosh(x) * 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(y) / y)
	t_1 = Float64(cosh(x) * t_0)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(fma(x, Float64(x * fma(x, Float64(x * fma(x, Float64(x * 0.001388888888888889), 0.041666666666666664)), 0.5)), 1.0) * fma(Float64(y * y), fma(Float64(y * y), fma(Float64(y * y), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666), 1.0));
	elseif (t_1 <= 0.9999999999282907)
		tmp = t_0;
	else
		tmp = Float64(cosh(x) * 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cosh[x], $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(x * N[(x * N[(x * N[(x * N[(x * N[(x * 0.001388888888888889), $MachinePrecision] + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.9999999999282907], t$95$0, N[(N[Cosh[x], $MachinePrecision] * 1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 58.4

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

   5. Applied rewrites 58.4%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 94.5

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

   8. Applied rewrites 94.5%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. lower-fma.f64 N/A

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

   11. Applied rewrites 94.5%

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

   if -inf.0 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < 0.99999999992829069

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 97.1

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

   5. Applied rewrites 97.1%

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

   if 0.99999999992829069 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 100.0%

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

Alternative 7: 75.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -5 \cdot 10^{-149}:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cosh x) (/ (sin y) y)) -5e-149)
   (*
    (fma
     x
     (*
      x
      (fma
       x
       (* x (fma x (* x 0.001388888888888889) 0.041666666666666664))
       0.5))
     1.0)
    (fma
     (* y y)
     (fma
      (* y y)
      (fma (* y y) -0.0001984126984126984 0.008333333333333333)
      -0.16666666666666666)
     1.0))
   (* (cosh x) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((cosh(x) * (sin(y) / y)) <= -5e-149) {
		tmp = fma(x, (x * fma(x, (x * fma(x, (x * 0.001388888888888889), 0.041666666666666664)), 0.5)), 1.0) * fma((y * y), fma((y * y), fma((y * y), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666), 1.0);
	} else {
		tmp = cosh(x) * 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(cosh(x) * Float64(sin(y) / y)) <= -5e-149)
		tmp = Float64(fma(x, Float64(x * fma(x, Float64(x * fma(x, Float64(x * 0.001388888888888889), 0.041666666666666664)), 0.5)), 1.0) * fma(Float64(y * y), fma(Float64(y * y), fma(Float64(y * y), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666), 1.0));
	else
		tmp = Float64(cosh(x) * 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Cosh[x], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], -5e-149], N[(N[(x * N[(x * N[(x * N[(x * N[(x * N[(x * 0.001388888888888889), $MachinePrecision] + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Cosh[x], $MachinePrecision] * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < -4.99999999999999968e-149

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 72.6

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

   5. Applied rewrites 72.6%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 59.3

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

   8. Applied rewrites 59.3%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. lower-fma.f64 N/A

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

   11. Applied rewrites 59.3%

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

   if -4.99999999999999968e-149 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y))

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 77.4%

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

Alternative 8: 70.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ t_1 := \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-300}:\\ \;\;\;\;t\_1 \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-65}:\\ \;\;\;\;\mathsf{fma}\left(0.5, x \cdot x, 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.008333333333333333, -0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y))
        (t_1
         (fma
          x
          (*
           x
           (fma
            x
            (* x (fma x (* x 0.001388888888888889) 0.041666666666666664))
            0.5))
          1.0)))
   (if (<= t_0 -2e-300)
     (*
      t_1
      (fma
       (* y y)
       (fma
        (* y y)
        (fma (* y y) -0.0001984126984126984 0.008333333333333333)
        -0.16666666666666666)
       1.0))
     (if (<= t_0 2e-65)
       (*
        (fma 0.5 (* x x) 1.0)
        (fma
         (* y y)
         (fma (* y y) 0.008333333333333333 -0.16666666666666666)
         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 * fma(x, (x * fma(x, (x * 0.001388888888888889), 0.041666666666666664)), 0.5)), 1.0);
	double tmp;
	if (t_0 <= -2e-300) {
		tmp = t_1 * fma((y * y), fma((y * y), fma((y * y), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666), 1.0);
	} else if (t_0 <= 2e-65) {
		tmp = fma(0.5, (x * x), 1.0) * fma((y * y), fma((y * y), 0.008333333333333333, -0.16666666666666666), 1.0);
	} else {
		tmp = 1.0 * t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(y) / y)
	t_1 = fma(x, Float64(x * fma(x, Float64(x * fma(x, Float64(x * 0.001388888888888889), 0.041666666666666664)), 0.5)), 1.0)
	tmp = 0.0
	if (t_0 <= -2e-300)
		tmp = Float64(t_1 * fma(Float64(y * y), fma(Float64(y * y), fma(Float64(y * y), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666), 1.0));
	elseif (t_0 <= 2e-65)
		tmp = Float64(fma(0.5, Float64(x * x), 1.0) * fma(Float64(y * y), fma(Float64(y * y), 0.008333333333333333, -0.16666666666666666), 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[(x * N[(x * N[(x * N[(x * N[(x * N[(x * 0.001388888888888889), $MachinePrecision] + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-300], N[(t$95$1 * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e-65], N[(N[(0.5 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $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) < -2.00000000000000005e-300

   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. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 77.6

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

   5. Applied rewrites 77.6%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 48.7

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

   8. Applied rewrites 48.7%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. lower-fma.f64 N/A

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

   11. Applied rewrites 48.7%

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

   if -2.00000000000000005e-300 < (/.f64 (sin.f64 y) y) < 1.99999999999999985e-65

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 79.1

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

   5. Applied rewrites 79.1%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 51.0

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

   8. Applied rewrites 51.0%

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

   if 1.99999999999999985e-65 < (/.f64 (sin.f64 y) y)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 81.9

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

   5. Applied rewrites 81.9%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 29.5%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 29.5%

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

   12. Taylor expanded in x around 0

       \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)} \cdot 1 \]
   13. 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. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. lower-fma.f64 N/A

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

   14. Applied rewrites 89.1%

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

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq -2 \cdot 10^{-300}:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)\\ \mathbf{elif}\;\frac{\sin y}{y} \leq 2 \cdot 10^{-65}:\\ \;\;\;\;\mathsf{fma}\left(0.5, x \cdot x, 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.008333333333333333, -0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 70.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-300}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right), 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-65}:\\ \;\;\;\;\mathsf{fma}\left(0.5, x \cdot x, 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.008333333333333333, -0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)))
   (if (<= t_0 -2e-300)
     (*
      (fma (* x x) (fma (* x x) 0.041666666666666664 0.5) 1.0)
      (fma
       (* y y)
       (fma
        (* y y)
        (fma (* y y) -0.0001984126984126984 0.008333333333333333)
        -0.16666666666666666)
       1.0))
     (if (<= t_0 2e-65)
       (*
        (fma 0.5 (* x x) 1.0)
        (fma
         (* y y)
         (fma (* y y) 0.008333333333333333 -0.16666666666666666)
         1.0))
       (*
        1.0
        (fma
         x
         (*
          x
          (fma
           x
           (* x (fma x (* x 0.001388888888888889) 0.041666666666666664))
           0.5))
         1.0))))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double tmp;
	if (t_0 <= -2e-300) {
		tmp = fma((x * x), fma((x * x), 0.041666666666666664, 0.5), 1.0) * fma((y * y), fma((y * y), fma((y * y), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666), 1.0);
	} else if (t_0 <= 2e-65) {
		tmp = fma(0.5, (x * x), 1.0) * fma((y * y), fma((y * y), 0.008333333333333333, -0.16666666666666666), 1.0);
	} else {
		tmp = 1.0 * fma(x, (x * fma(x, (x * fma(x, (x * 0.001388888888888889), 0.041666666666666664)), 0.5)), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(y) / y)
	tmp = 0.0
	if (t_0 <= -2e-300)
		tmp = Float64(fma(Float64(x * x), fma(Float64(x * x), 0.041666666666666664, 0.5), 1.0) * fma(Float64(y * y), fma(Float64(y * y), fma(Float64(y * y), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666), 1.0));
	elseif (t_0 <= 2e-65)
		tmp = Float64(fma(0.5, Float64(x * x), 1.0) * fma(Float64(y * y), fma(Float64(y * y), 0.008333333333333333, -0.16666666666666666), 1.0));
	else
		tmp = Float64(1.0 * fma(x, Float64(x * fma(x, Float64(x * fma(x, Float64(x * 0.001388888888888889), 0.041666666666666664)), 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, -2e-300], N[(N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e-65], N[(N[(0.5 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 * N[(x * N[(x * N[(x * N[(x * N[(x * N[(x * 0.001388888888888889), $MachinePrecision] + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 y) y) < -2.00000000000000005e-300

   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. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 83.5

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

   5. Applied rewrites 83.5%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 48.7

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

   8. Applied rewrites 48.7%

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

   if -2.00000000000000005e-300 < (/.f64 (sin.f64 y) y) < 1.99999999999999985e-65

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 79.1

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

   5. Applied rewrites 79.1%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 51.0

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

   8. Applied rewrites 51.0%

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

   if 1.99999999999999985e-65 < (/.f64 (sin.f64 y) y)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 81.9

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

   5. Applied rewrites 81.9%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 29.5%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 29.5%

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

   12. Taylor expanded in x around 0

       \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)} \cdot 1 \]
   13. 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. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. lower-fma.f64 N/A

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

   14. Applied rewrites 89.1%

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

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq -2 \cdot 10^{-300}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right), 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)\\ \mathbf{elif}\;\frac{\sin y}{y} \leq 2 \cdot 10^{-65}:\\ \;\;\;\;\mathsf{fma}\left(0.5, x \cdot x, 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.008333333333333333, -0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 70.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ t_1 := \mathsf{fma}\left(0.5, x \cdot x, 1\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-300}:\\ \;\;\;\;t\_1 \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-65}:\\ \;\;\;\;t\_1 \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.008333333333333333, -0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)) (t_1 (fma 0.5 (* x x) 1.0)))
   (if (<= t_0 -2e-300)
     (*
      t_1
      (fma
       (* y y)
       (fma
        (* y y)
        (fma (* y y) -0.0001984126984126984 0.008333333333333333)
        -0.16666666666666666)
       1.0))
     (if (<= t_0 2e-65)
       (*
        t_1
        (fma
         (* y y)
         (fma (* y y) 0.008333333333333333 -0.16666666666666666)
         1.0))
       (*
        1.0
        (fma
         x
         (*
          x
          (fma
           x
           (* x (fma x (* x 0.001388888888888889) 0.041666666666666664))
           0.5))
         1.0))))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double t_1 = fma(0.5, (x * x), 1.0);
	double tmp;
	if (t_0 <= -2e-300) {
		tmp = t_1 * fma((y * y), fma((y * y), fma((y * y), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666), 1.0);
	} else if (t_0 <= 2e-65) {
		tmp = t_1 * fma((y * y), fma((y * y), 0.008333333333333333, -0.16666666666666666), 1.0);
	} else {
		tmp = 1.0 * fma(x, (x * fma(x, (x * fma(x, (x * 0.001388888888888889), 0.041666666666666664)), 0.5)), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(y) / y)
	t_1 = fma(0.5, Float64(x * x), 1.0)
	tmp = 0.0
	if (t_0 <= -2e-300)
		tmp = Float64(t_1 * fma(Float64(y * y), fma(Float64(y * y), fma(Float64(y * y), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666), 1.0));
	elseif (t_0 <= 2e-65)
		tmp = Float64(t_1 * fma(Float64(y * y), fma(Float64(y * y), 0.008333333333333333, -0.16666666666666666), 1.0));
	else
		tmp = Float64(1.0 * fma(x, Float64(x * fma(x, Float64(x * fma(x, Float64(x * 0.001388888888888889), 0.041666666666666664)), 0.5)), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-300], N[(t$95$1 * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e-65], N[(t$95$1 * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 * N[(x * N[(x * N[(x * N[(x * N[(x * N[(x * 0.001388888888888889), $MachinePrecision] + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 y) y) < -2.00000000000000005e-300

   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. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 77.6

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

   5. Applied rewrites 77.6%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 48.7

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

   8. Applied rewrites 48.7%

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

   if -2.00000000000000005e-300 < (/.f64 (sin.f64 y) y) < 1.99999999999999985e-65

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 79.1

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

   5. Applied rewrites 79.1%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 51.0

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

   8. Applied rewrites 51.0%

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

   if 1.99999999999999985e-65 < (/.f64 (sin.f64 y) y)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 81.9

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

   5. Applied rewrites 81.9%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 29.5%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 29.5%

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

   12. Taylor expanded in x around 0

       \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)} \cdot 1 \]
   13. 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. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. lower-fma.f64 N/A

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

   14. Applied rewrites 89.1%

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

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq -2 \cdot 10^{-300}:\\ \;\;\;\;\mathsf{fma}\left(0.5, x \cdot x, 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)\\ \mathbf{elif}\;\frac{\sin y}{y} \leq 2 \cdot 10^{-65}:\\ \;\;\;\;\mathsf{fma}\left(0.5, x \cdot x, 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.008333333333333333, -0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 70.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ t_1 := \mathsf{fma}\left(0.5, x \cdot x, 1\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-300}:\\ \;\;\;\;t\_1 \cdot \mathsf{fma}\left(y \cdot y, y \cdot \left(-0.0001984126984126984 \cdot \left(y \cdot \left(y \cdot y\right)\right)\right), 1\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-65}:\\ \;\;\;\;t\_1 \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.008333333333333333, -0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)) (t_1 (fma 0.5 (* x x) 1.0)))
   (if (<= t_0 -2e-300)
     (* t_1 (fma (* y y) (* y (* -0.0001984126984126984 (* y (* y y)))) 1.0))
     (if (<= t_0 2e-65)
       (*
        t_1
        (fma
         (* y y)
         (fma (* y y) 0.008333333333333333 -0.16666666666666666)
         1.0))
       (*
        1.0
        (fma
         x
         (*
          x
          (fma
           x
           (* x (fma x (* x 0.001388888888888889) 0.041666666666666664))
           0.5))
         1.0))))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double t_1 = fma(0.5, (x * x), 1.0);
	double tmp;
	if (t_0 <= -2e-300) {
		tmp = t_1 * fma((y * y), (y * (-0.0001984126984126984 * (y * (y * y)))), 1.0);
	} else if (t_0 <= 2e-65) {
		tmp = t_1 * fma((y * y), fma((y * y), 0.008333333333333333, -0.16666666666666666), 1.0);
	} else {
		tmp = 1.0 * fma(x, (x * fma(x, (x * fma(x, (x * 0.001388888888888889), 0.041666666666666664)), 0.5)), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(y) / y)
	t_1 = fma(0.5, Float64(x * x), 1.0)
	tmp = 0.0
	if (t_0 <= -2e-300)
		tmp = Float64(t_1 * fma(Float64(y * y), Float64(y * Float64(-0.0001984126984126984 * Float64(y * Float64(y * y)))), 1.0));
	elseif (t_0 <= 2e-65)
		tmp = Float64(t_1 * fma(Float64(y * y), fma(Float64(y * y), 0.008333333333333333, -0.16666666666666666), 1.0));
	else
		tmp = Float64(1.0 * fma(x, Float64(x * fma(x, Float64(x * fma(x, Float64(x * 0.001388888888888889), 0.041666666666666664)), 0.5)), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-300], N[(t$95$1 * N[(N[(y * y), $MachinePrecision] * N[(y * N[(-0.0001984126984126984 * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e-65], N[(t$95$1 * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 * N[(x * N[(x * N[(x * N[(x * N[(x * N[(x * 0.001388888888888889), $MachinePrecision] + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 y) y) < -2.00000000000000005e-300

   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. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 77.6

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

   5. Applied rewrites 77.6%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 48.7

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

   8. Applied rewrites 48.7%

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

   9. Taylor expanded in y around inf

      \[\leadsto \mathsf{fma}\left(\frac{1}{2}, x \cdot x, 1\right) \cdot \mathsf{fma}\left(y \cdot y, \frac{-1}{5040} \cdot \color{blue}{{y}^{4}}, 1\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 48.7%

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

   if -2.00000000000000005e-300 < (/.f64 (sin.f64 y) y) < 1.99999999999999985e-65

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 79.1

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

   5. Applied rewrites 79.1%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 51.0

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

   8. Applied rewrites 51.0%

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

   if 1.99999999999999985e-65 < (/.f64 (sin.f64 y) y)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 81.9

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

   5. Applied rewrites 81.9%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 29.5%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 29.5%

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

   12. Taylor expanded in x around 0

       \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)} \cdot 1 \]
   13. 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. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. lower-fma.f64 N/A

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

   14. Applied rewrites 89.1%

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

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq -2 \cdot 10^{-300}:\\ \;\;\;\;\mathsf{fma}\left(0.5, x \cdot x, 1\right) \cdot \mathsf{fma}\left(y \cdot y, y \cdot \left(-0.0001984126984126984 \cdot \left(y \cdot \left(y \cdot y\right)\right)\right), 1\right)\\ \mathbf{elif}\;\frac{\sin y}{y} \leq 2 \cdot 10^{-65}:\\ \;\;\;\;\mathsf{fma}\left(0.5, x \cdot x, 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.008333333333333333, -0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 70.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-300}:\\ \;\;\;\;\mathsf{fma}\left(y, y \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right), 1\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-65}:\\ \;\;\;\;\mathsf{fma}\left(0.5, x \cdot x, 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.008333333333333333, -0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)))
   (if (<= t_0 -2e-300)
     (*
      (fma y (* y -0.16666666666666666) 1.0)
      (fma (* x x) (fma (* x x) 0.041666666666666664 0.5) 1.0))
     (if (<= t_0 2e-65)
       (*
        (fma 0.5 (* x x) 1.0)
        (fma
         (* y y)
         (fma (* y y) 0.008333333333333333 -0.16666666666666666)
         1.0))
       (*
        1.0
        (fma
         x
         (*
          x
          (fma
           x
           (* x (fma x (* x 0.001388888888888889) 0.041666666666666664))
           0.5))
         1.0))))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double tmp;
	if (t_0 <= -2e-300) {
		tmp = fma(y, (y * -0.16666666666666666), 1.0) * fma((x * x), fma((x * x), 0.041666666666666664, 0.5), 1.0);
	} else if (t_0 <= 2e-65) {
		tmp = fma(0.5, (x * x), 1.0) * fma((y * y), fma((y * y), 0.008333333333333333, -0.16666666666666666), 1.0);
	} else {
		tmp = 1.0 * fma(x, (x * fma(x, (x * fma(x, (x * 0.001388888888888889), 0.041666666666666664)), 0.5)), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(y) / y)
	tmp = 0.0
	if (t_0 <= -2e-300)
		tmp = Float64(fma(y, Float64(y * -0.16666666666666666), 1.0) * fma(Float64(x * x), fma(Float64(x * x), 0.041666666666666664, 0.5), 1.0));
	elseif (t_0 <= 2e-65)
		tmp = Float64(fma(0.5, Float64(x * x), 1.0) * fma(Float64(y * y), fma(Float64(y * y), 0.008333333333333333, -0.16666666666666666), 1.0));
	else
		tmp = Float64(1.0 * fma(x, Float64(x * fma(x, Float64(x * fma(x, Float64(x * 0.001388888888888889), 0.041666666666666664)), 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, -2e-300], N[(N[(y * N[(y * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e-65], N[(N[(0.5 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 * N[(x * N[(x * N[(x * N[(x * N[(x * N[(x * 0.001388888888888889), $MachinePrecision] + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 y) y) < -2.00000000000000005e-300

   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. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 83.5

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

   5. Applied rewrites 83.5%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 46.4

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

   8. Applied rewrites 46.4%

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

   if -2.00000000000000005e-300 < (/.f64 (sin.f64 y) y) < 1.99999999999999985e-65

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 79.1

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

   5. Applied rewrites 79.1%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 51.0

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

   8. Applied rewrites 51.0%

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

   if 1.99999999999999985e-65 < (/.f64 (sin.f64 y) y)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 81.9

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

   5. Applied rewrites 81.9%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 29.5%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 29.5%

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

   12. Taylor expanded in x around 0

       \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)} \cdot 1 \]
   13. 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. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. lower-fma.f64 N/A

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

   14. Applied rewrites 89.1%

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

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq -2 \cdot 10^{-300}:\\ \;\;\;\;\mathsf{fma}\left(y, y \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right), 1\right)\\ \mathbf{elif}\;\frac{\sin y}{y} \leq 2 \cdot 10^{-65}:\\ \;\;\;\;\mathsf{fma}\left(0.5, x \cdot x, 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.008333333333333333, -0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 69.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ t_1 := \mathsf{fma}\left(0.5, x \cdot x, 1\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-300}:\\ \;\;\;\;\mathsf{fma}\left(y, y \cdot -0.16666666666666666, 1\right) \cdot t\_1\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-65}:\\ \;\;\;\;t\_1 \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.008333333333333333, -0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)) (t_1 (fma 0.5 (* x x) 1.0)))
   (if (<= t_0 -2e-300)
     (* (fma y (* y -0.16666666666666666) 1.0) t_1)
     (if (<= t_0 2e-65)
       (*
        t_1
        (fma
         (* y y)
         (fma (* y y) 0.008333333333333333 -0.16666666666666666)
         1.0))
       (*
        1.0
        (fma
         x
         (*
          x
          (fma
           x
           (* x (fma x (* x 0.001388888888888889) 0.041666666666666664))
           0.5))
         1.0))))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double t_1 = fma(0.5, (x * x), 1.0);
	double tmp;
	if (t_0 <= -2e-300) {
		tmp = fma(y, (y * -0.16666666666666666), 1.0) * t_1;
	} else if (t_0 <= 2e-65) {
		tmp = t_1 * fma((y * y), fma((y * y), 0.008333333333333333, -0.16666666666666666), 1.0);
	} else {
		tmp = 1.0 * fma(x, (x * fma(x, (x * fma(x, (x * 0.001388888888888889), 0.041666666666666664)), 0.5)), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(y) / y)
	t_1 = fma(0.5, Float64(x * x), 1.0)
	tmp = 0.0
	if (t_0 <= -2e-300)
		tmp = Float64(fma(y, Float64(y * -0.16666666666666666), 1.0) * t_1);
	elseif (t_0 <= 2e-65)
		tmp = Float64(t_1 * fma(Float64(y * y), fma(Float64(y * y), 0.008333333333333333, -0.16666666666666666), 1.0));
	else
		tmp = Float64(1.0 * fma(x, Float64(x * fma(x, Float64(x * fma(x, Float64(x * 0.001388888888888889), 0.041666666666666664)), 0.5)), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-300], N[(N[(y * N[(y * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t$95$0, 2e-65], N[(t$95$1 * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 * N[(x * N[(x * N[(x * N[(x * N[(x * N[(x * 0.001388888888888889), $MachinePrecision] + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 y) y) < -2.00000000000000005e-300

   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. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 77.6

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

   5. Applied rewrites 77.6%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 45.1

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

   8. Applied rewrites 45.1%

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

   if -2.00000000000000005e-300 < (/.f64 (sin.f64 y) y) < 1.99999999999999985e-65

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 79.1

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

   5. Applied rewrites 79.1%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 51.0

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

   8. Applied rewrites 51.0%

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

   if 1.99999999999999985e-65 < (/.f64 (sin.f64 y) y)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 81.9

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

   5. Applied rewrites 81.9%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 29.5%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 29.5%

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

   12. Taylor expanded in x around 0

       \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)} \cdot 1 \]
   13. 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. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. lower-fma.f64 N/A

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

   14. Applied rewrites 89.1%

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

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq -2 \cdot 10^{-300}:\\ \;\;\;\;\mathsf{fma}\left(y, y \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(0.5, x \cdot x, 1\right)\\ \mathbf{elif}\;\frac{\sin y}{y} \leq 2 \cdot 10^{-65}:\\ \;\;\;\;\mathsf{fma}\left(0.5, x \cdot x, 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.008333333333333333, -0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 67.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ t_1 := \mathsf{fma}\left(0.5, x \cdot x, 1\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-300}:\\ \;\;\;\;\mathsf{fma}\left(y, y \cdot -0.16666666666666666, 1\right) \cdot t\_1\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-65}:\\ \;\;\;\;t\_1 \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.008333333333333333, -0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)) (t_1 (fma 0.5 (* x x) 1.0)))
   (if (<= t_0 -2e-300)
     (* (fma y (* y -0.16666666666666666) 1.0) t_1)
     (if (<= t_0 2e-65)
       (*
        t_1
        (fma
         (* y y)
         (fma (* y y) 0.008333333333333333 -0.16666666666666666)
         1.0))
       (* 1.0 (fma x (* x (fma x (* x 0.041666666666666664) 0.5)) 1.0))))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double t_1 = fma(0.5, (x * x), 1.0);
	double tmp;
	if (t_0 <= -2e-300) {
		tmp = fma(y, (y * -0.16666666666666666), 1.0) * t_1;
	} else if (t_0 <= 2e-65) {
		tmp = t_1 * fma((y * y), fma((y * y), 0.008333333333333333, -0.16666666666666666), 1.0);
	} else {
		tmp = 1.0 * fma(x, (x * fma(x, (x * 0.041666666666666664), 0.5)), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(y) / y)
	t_1 = fma(0.5, Float64(x * x), 1.0)
	tmp = 0.0
	if (t_0 <= -2e-300)
		tmp = Float64(fma(y, Float64(y * -0.16666666666666666), 1.0) * t_1);
	elseif (t_0 <= 2e-65)
		tmp = Float64(t_1 * fma(Float64(y * y), fma(Float64(y * y), 0.008333333333333333, -0.16666666666666666), 1.0));
	else
		tmp = Float64(1.0 * fma(x, Float64(x * fma(x, Float64(x * 0.041666666666666664), 0.5)), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-300], N[(N[(y * N[(y * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t$95$0, 2e-65], N[(t$95$1 * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 * N[(x * N[(x * N[(x * N[(x * 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 y) y) < -2.00000000000000005e-300

   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. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 77.6

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

   5. Applied rewrites 77.6%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 45.1

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

   8. Applied rewrites 45.1%

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

   if -2.00000000000000005e-300 < (/.f64 (sin.f64 y) y) < 1.99999999999999985e-65

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 79.1

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

   5. Applied rewrites 79.1%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 51.0

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

   8. Applied rewrites 51.0%

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

   if 1.99999999999999985e-65 < (/.f64 (sin.f64 y) y)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 81.9

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

   5. Applied rewrites 81.9%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 29.5%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 29.5%

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

   12. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. lower-fma.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. +-commutative N/A

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

       7. *-commutative N/A

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

       8. unpow2 N/A

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

       9. associate-*l* N/A

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

       10. lower-fma.f64 N/A

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

       11. lower-*.f64 86.9

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

   14. Applied rewrites 86.9%

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

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq -2 \cdot 10^{-300}:\\ \;\;\;\;\mathsf{fma}\left(y, y \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(0.5, x \cdot x, 1\right)\\ \mathbf{elif}\;\frac{\sin y}{y} \leq 2 \cdot 10^{-65}:\\ \;\;\;\;\mathsf{fma}\left(0.5, x \cdot x, 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.008333333333333333, -0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 67.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-300}:\\ \;\;\;\;\mathsf{fma}\left(y, y \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(0.5, x \cdot x, 1\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-65}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.008333333333333333, -0.16666666666666666\right), 1\right) \cdot \left(\left(x \cdot x\right) \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)))
   (if (<= t_0 -2e-300)
     (* (fma y (* y -0.16666666666666666) 1.0) (fma 0.5 (* x x) 1.0))
     (if (<= t_0 2e-65)
       (*
        (fma
         (* y y)
         (fma (* y y) 0.008333333333333333 -0.16666666666666666)
         1.0)
        (* (* x x) 0.5))
       (* 1.0 (fma x (* x (fma x (* x 0.041666666666666664) 0.5)) 1.0))))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double tmp;
	if (t_0 <= -2e-300) {
		tmp = fma(y, (y * -0.16666666666666666), 1.0) * fma(0.5, (x * x), 1.0);
	} else if (t_0 <= 2e-65) {
		tmp = fma((y * y), fma((y * y), 0.008333333333333333, -0.16666666666666666), 1.0) * ((x * x) * 0.5);
	} else {
		tmp = 1.0 * fma(x, (x * fma(x, (x * 0.041666666666666664), 0.5)), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(y) / y)
	tmp = 0.0
	if (t_0 <= -2e-300)
		tmp = Float64(fma(y, Float64(y * -0.16666666666666666), 1.0) * fma(0.5, Float64(x * x), 1.0));
	elseif (t_0 <= 2e-65)
		tmp = Float64(fma(Float64(y * y), fma(Float64(y * y), 0.008333333333333333, -0.16666666666666666), 1.0) * Float64(Float64(x * x) * 0.5));
	else
		tmp = Float64(1.0 * fma(x, Float64(x * fma(x, Float64(x * 0.041666666666666664), 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, -2e-300], N[(N[(y * N[(y * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * N[(0.5 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e-65], N[(N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], N[(1.0 * N[(x * N[(x * N[(x * N[(x * 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 y) y) < -2.00000000000000005e-300

   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. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 77.6

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

   5. Applied rewrites 77.6%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 45.1

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

   8. Applied rewrites 45.1%

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

   if -2.00000000000000005e-300 < (/.f64 (sin.f64 y) y) < 1.99999999999999985e-65

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 79.1

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

   5. Applied rewrites 79.1%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 32.3%

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

   9. Taylor expanded in y around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 N/A

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

       5. sub-neg N/A

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

       6. *-commutative N/A

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

       7. metadata-eval N/A

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

       8. lower-fma.f64 N/A

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

       9. unpow2 N/A

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

       10. lower-*.f64 50.9

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

   11. Applied rewrites 50.9%

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

   if 1.99999999999999985e-65 < (/.f64 (sin.f64 y) y)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 81.9

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

   5. Applied rewrites 81.9%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 29.5%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 29.5%

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

   12. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. lower-fma.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. +-commutative N/A

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

       7. *-commutative N/A

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

       8. unpow2 N/A

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

       9. associate-*l* N/A

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

       10. lower-fma.f64 N/A

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

       11. lower-*.f64 86.9

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

   14. Applied rewrites 86.9%

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

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq -2 \cdot 10^{-300}:\\ \;\;\;\;\mathsf{fma}\left(y, y \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(0.5, x \cdot x, 1\right)\\ \mathbf{elif}\;\frac{\sin y}{y} \leq 2 \cdot 10^{-65}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.008333333333333333, -0.16666666666666666\right), 1\right) \cdot \left(\left(x \cdot x\right) \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 67.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-300}:\\ \;\;\;\;\mathsf{fma}\left(y, y \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(0.5, x \cdot x, 1\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-101}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.008333333333333333, -0.16666666666666666\right), 1\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)))
   (if (<= t_0 -2e-300)
     (* (fma y (* y -0.16666666666666666) 1.0) (fma 0.5 (* x x) 1.0))
     (if (<= t_0 5e-101)
       (*
        (fma
         (* y y)
         (fma (* y y) 0.008333333333333333 -0.16666666666666666)
         1.0)
        1.0)
       (* 1.0 (fma x (* x (fma x (* x 0.041666666666666664) 0.5)) 1.0))))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double tmp;
	if (t_0 <= -2e-300) {
		tmp = fma(y, (y * -0.16666666666666666), 1.0) * fma(0.5, (x * x), 1.0);
	} else if (t_0 <= 5e-101) {
		tmp = fma((y * y), fma((y * y), 0.008333333333333333, -0.16666666666666666), 1.0) * 1.0;
	} else {
		tmp = 1.0 * fma(x, (x * fma(x, (x * 0.041666666666666664), 0.5)), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(y) / y)
	tmp = 0.0
	if (t_0 <= -2e-300)
		tmp = Float64(fma(y, Float64(y * -0.16666666666666666), 1.0) * fma(0.5, Float64(x * x), 1.0));
	elseif (t_0 <= 5e-101)
		tmp = Float64(fma(Float64(y * y), fma(Float64(y * y), 0.008333333333333333, -0.16666666666666666), 1.0) * 1.0);
	else
		tmp = Float64(1.0 * fma(x, Float64(x * fma(x, Float64(x * 0.041666666666666664), 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, -2e-300], N[(N[(y * N[(y * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * N[(0.5 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e-101], N[(N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * 1.0), $MachinePrecision], N[(1.0 * N[(x * N[(x * N[(x * N[(x * 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 y) y) < -2.00000000000000005e-300

   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. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 77.6

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

   5. Applied rewrites 77.6%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 45.1

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

   8. Applied rewrites 45.1%

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

   if -2.00000000000000005e-300 < (/.f64 (sin.f64 y) y) < 5.0000000000000001e-101

   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 45.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 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. lower-fma.f64 N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 N/A

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

       5. sub-neg N/A

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

       6. *-commutative N/A

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

       7. metadata-eval N/A

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

       8. lower-fma.f64 N/A

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

       9. unpow2 N/A

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

       10. lower-*.f64 44.3

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

   11. Applied rewrites 44.3%

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

   if 5.0000000000000001e-101 < (/.f64 (sin.f64 y) y)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 80.7

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

   5. Applied rewrites 80.7%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 30.7%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 30.8%

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

   12. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. lower-fma.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. +-commutative N/A

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

       7. *-commutative N/A

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

       8. unpow2 N/A

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

       9. associate-*l* N/A

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

       10. lower-fma.f64 N/A

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

       11. lower-*.f64 84.7

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

   14. Applied rewrites 84.7%

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

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq -2 \cdot 10^{-300}:\\ \;\;\;\;\mathsf{fma}\left(y, y \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(0.5, x \cdot x, 1\right)\\ \mathbf{elif}\;\frac{\sin y}{y} \leq 5 \cdot 10^{-101}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.008333333333333333, -0.16666666666666666\right), 1\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 61.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ t_1 := \mathsf{fma}\left(0.5, x \cdot x, 1\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-300}:\\ \;\;\;\;\mathsf{fma}\left(y, y \cdot -0.16666666666666666, 1\right) \cdot t\_1\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{-78}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.008333333333333333, -0.16666666666666666\right), 1\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;1 \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)) (t_1 (fma 0.5 (* x x) 1.0)))
   (if (<= t_0 -2e-300)
     (* (fma y (* y -0.16666666666666666) 1.0) t_1)
     (if (<= t_0 4e-78)
       (*
        (fma
         (* y y)
         (fma (* y y) 0.008333333333333333 -0.16666666666666666)
         1.0)
        1.0)
       (* 1.0 t_1)))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double t_1 = fma(0.5, (x * x), 1.0);
	double tmp;
	if (t_0 <= -2e-300) {
		tmp = fma(y, (y * -0.16666666666666666), 1.0) * t_1;
	} else if (t_0 <= 4e-78) {
		tmp = fma((y * y), fma((y * y), 0.008333333333333333, -0.16666666666666666), 1.0) * 1.0;
	} else {
		tmp = 1.0 * t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(y) / y)
	t_1 = fma(0.5, Float64(x * x), 1.0)
	tmp = 0.0
	if (t_0 <= -2e-300)
		tmp = Float64(fma(y, Float64(y * -0.16666666666666666), 1.0) * t_1);
	elseif (t_0 <= 4e-78)
		tmp = Float64(fma(Float64(y * y), fma(Float64(y * y), 0.008333333333333333, -0.16666666666666666), 1.0) * 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[(0.5 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-300], N[(N[(y * N[(y * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t$95$0, 4e-78], N[(N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * 1.0), $MachinePrecision], N[(1.0 * t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 y) y) < -2.00000000000000005e-300

   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. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 77.6

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

   5. Applied rewrites 77.6%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 45.1

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

   8. Applied rewrites 45.1%

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

   if -2.00000000000000005e-300 < (/.f64 (sin.f64 y) y) < 4e-78

   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 50.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 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. lower-fma.f64 N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 N/A

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

       5. sub-neg N/A

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

       6. *-commutative N/A

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

       7. metadata-eval N/A

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

       8. lower-fma.f64 N/A

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

       9. unpow2 N/A

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

       10. lower-*.f64 47.3

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

   11. Applied rewrites 47.3%

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

   if 4e-78 < (/.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 94.5%

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

   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. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 75.7

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

   8. Applied rewrites 75.7%

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

4. Final simplification 61.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq -2 \cdot 10^{-300}:\\ \;\;\;\;\mathsf{fma}\left(y, y \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(0.5, x \cdot x, 1\right)\\ \mathbf{elif}\;\frac{\sin y}{y} \leq 4 \cdot 10^{-78}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.008333333333333333, -0.16666666666666666\right), 1\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(0.5, x \cdot x, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 61.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-300}:\\ \;\;\;\;\mathsf{fma}\left(y, y \cdot -0.16666666666666666, 1\right) \cdot \left(\left(x \cdot x\right) \cdot 0.5\right)\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{-78}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.008333333333333333, -0.16666666666666666\right), 1\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(0.5, x \cdot x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)))
   (if (<= t_0 -2e-300)
     (* (fma y (* y -0.16666666666666666) 1.0) (* (* x x) 0.5))
     (if (<= t_0 4e-78)
       (*
        (fma
         (* y y)
         (fma (* y y) 0.008333333333333333 -0.16666666666666666)
         1.0)
        1.0)
       (* 1.0 (fma 0.5 (* x x) 1.0))))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double tmp;
	if (t_0 <= -2e-300) {
		tmp = fma(y, (y * -0.16666666666666666), 1.0) * ((x * x) * 0.5);
	} else if (t_0 <= 4e-78) {
		tmp = fma((y * y), fma((y * y), 0.008333333333333333, -0.16666666666666666), 1.0) * 1.0;
	} else {
		tmp = 1.0 * fma(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 <= -2e-300)
		tmp = Float64(fma(y, Float64(y * -0.16666666666666666), 1.0) * Float64(Float64(x * x) * 0.5));
	elseif (t_0 <= 4e-78)
		tmp = Float64(fma(Float64(y * y), fma(Float64(y * y), 0.008333333333333333, -0.16666666666666666), 1.0) * 1.0);
	else
		tmp = Float64(1.0 * fma(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, -2e-300], N[(N[(y * N[(y * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 4e-78], N[(N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * 1.0), $MachinePrecision], N[(1.0 * N[(0.5 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 y) y) < -2.00000000000000005e-300

   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. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 77.6

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

   5. Applied rewrites 77.6%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 32.2%

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

   9. Taylor expanded in y around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

       5. lower-fma.f64 N/A

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

       6. lower-*.f64 45.0

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

   11. Applied rewrites 45.0%

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

   if -2.00000000000000005e-300 < (/.f64 (sin.f64 y) y) < 4e-78

   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 50.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 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. lower-fma.f64 N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 N/A

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

       5. sub-neg N/A

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

       6. *-commutative N/A

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

       7. metadata-eval N/A

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

       8. lower-fma.f64 N/A

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

       9. unpow2 N/A

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

       10. lower-*.f64 47.3

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

   11. Applied rewrites 47.3%

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

   if 4e-78 < (/.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 94.5%

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

   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. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 75.7

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

   8. Applied rewrites 75.7%

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

4. Final simplification 61.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq -2 \cdot 10^{-300}:\\ \;\;\;\;\mathsf{fma}\left(y, y \cdot -0.16666666666666666, 1\right) \cdot \left(\left(x \cdot x\right) \cdot 0.5\right)\\ \mathbf{elif}\;\frac{\sin y}{y} \leq 4 \cdot 10^{-78}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.008333333333333333, -0.16666666666666666\right), 1\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(0.5, x \cdot x, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 57.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -5 \cdot 10^{-149}:\\ \;\;\;\;\mathsf{fma}\left(y, y \cdot -0.16666666666666666, 1\right) \cdot \left(\left(x \cdot x\right) \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(0.5, x \cdot x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cosh x) (/ (sin y) y)) -5e-149)
   (* (fma y (* y -0.16666666666666666) 1.0) (* (* x x) 0.5))
   (* 1.0 (fma 0.5 (* x x) 1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((cosh(x) * (sin(y) / y)) <= -5e-149) {
		tmp = fma(y, (y * -0.16666666666666666), 1.0) * ((x * x) * 0.5);
	} else {
		tmp = 1.0 * fma(0.5, (x * x), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(cosh(x) * Float64(sin(y) / y)) <= -5e-149)
		tmp = Float64(fma(y, Float64(y * -0.16666666666666666), 1.0) * Float64(Float64(x * x) * 0.5));
	else
		tmp = Float64(1.0 * fma(0.5, Float64(x * x), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Cosh[x], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], -5e-149], N[(N[(y * N[(y * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], N[(1.0 * N[(0.5 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < -4.99999999999999968e-149

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 72.6

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

   5. Applied rewrites 72.6%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 37.8%

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

   9. Taylor expanded in y around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

       5. lower-fma.f64 N/A

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

       6. lower-*.f64 55.1

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

   11. Applied rewrites 55.1%

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

   if -4.99999999999999968e-149 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y))

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 77.4%

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

   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. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 59.8

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

   8. Applied rewrites 59.8%

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

4. Final simplification 58.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -5 \cdot 10^{-149}:\\ \;\;\;\;\mathsf{fma}\left(y, y \cdot -0.16666666666666666, 1\right) \cdot \left(\left(x \cdot x\right) \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(0.5, x \cdot x, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 52.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -5 \cdot 10^{-149}:\\ \;\;\;\;\mathsf{fma}\left(y, y \cdot -0.16666666666666666, 1\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(0.5, x \cdot x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cosh x) (/ (sin y) y)) -5e-149)
   (* (fma y (* y -0.16666666666666666) 1.0) 1.0)
   (* 1.0 (fma 0.5 (* x x) 1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((cosh(x) * (sin(y) / y)) <= -5e-149) {
		tmp = fma(y, (y * -0.16666666666666666), 1.0) * 1.0;
	} else {
		tmp = 1.0 * fma(0.5, (x * x), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(cosh(x) * Float64(sin(y) / y)) <= -5e-149)
		tmp = Float64(fma(y, Float64(y * -0.16666666666666666), 1.0) * 1.0);
	else
		tmp = Float64(1.0 * fma(0.5, Float64(x * x), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Cosh[x], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], -5e-149], N[(N[(y * N[(y * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * 1.0), $MachinePrecision], N[(1.0 * N[(0.5 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < -4.99999999999999968e-149

   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 0.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 1.1%

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

   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. *-commutative N/A

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

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

       5. lower-fma.f64 N/A

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

       6. lower-*.f64 33.4

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

   11. Applied rewrites 33.4%

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

   if -4.99999999999999968e-149 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y))

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 77.4%

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

   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. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 59.8

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

   8. Applied rewrites 59.8%

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

4. Final simplification 53.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -5 \cdot 10^{-149}:\\ \;\;\;\;\mathsf{fma}\left(y, y \cdot -0.16666666666666666, 1\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(0.5, x \cdot x, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 52.2% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   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 39.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 39.9%

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

   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. *-commutative N/A

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

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

       5. lower-fma.f64 N/A

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

       6. lower-*.f64 50.1

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

   11. Applied rewrites 50.1%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in 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. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 59.6

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

   5. Applied rewrites 59.6%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 59.6%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 59.8%

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

4. Final simplification 53.4%

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

Alternative 22: 46.1% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((cosh(x) * (sin(y) / y)) <= 2.0d0) then
        tmp = 1.0d0 * 1.0d0
    else
        tmp = 1.0d0 * ((x * x) * 0.5d0)
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((cosh(x) * (sin(y) / y)) <= 2.0)
		tmp = 1.0 * 1.0;
	else
		tmp = 1.0 * ((x * x) * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   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 39.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 39.9%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in 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. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 59.6

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

   5. Applied rewrites 59.6%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 59.6%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 59.8%

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

4. Final simplification 46.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq 2:\\ \;\;\;\;1 \cdot 1\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\left(x \cdot x\right) \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 27.8% 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 60.3%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 27.4%

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

9. Final simplification 27.4%

   \[\leadsto 1 \cdot 1 \]
10. 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 2024223 
(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)))