Linear.Quaternion:$csinh from linear-1.19.1.3

Percentage Accurate: 99.9% → 99.9%

Time: 7.1s

Alternatives: 15

Speedup: 1.0×

• 47.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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: 99.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ t_1 := \cosh x \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\cosh x \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{elif}\;t\_1 \leq 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, 0.5, 1\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot \mathsf{fma}\left(0.008333333333333333 \cdot \left(y \cdot y\right) - 0.16666666666666666, y \cdot y, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)) (t_1 (* (cosh x) t_0)))
   (if (<= t_1 (- INFINITY))
     (* (cosh x) (fma -0.16666666666666666 (* y y) 1.0))
     (if (<= t_1 1e-13)
       (* (fma (* x x) 0.5 1.0) t_0)
       (*
        (cosh x)
        (fma
         (- (* 0.008333333333333333 (* y y)) 0.16666666666666666)
         (* y y)
         1.0))))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double t_1 = cosh(x) * t_0;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = cosh(x) * fma(-0.16666666666666666, (y * y), 1.0);
	} else if (t_1 <= 1e-13) {
		tmp = fma((x * x), 0.5, 1.0) * t_0;
	} else {
		tmp = cosh(x) * fma(((0.008333333333333333 * (y * y)) - 0.16666666666666666), (y * y), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(y) / y)
	t_1 = Float64(cosh(x) * t_0)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(cosh(x) * fma(-0.16666666666666666, Float64(y * y), 1.0));
	elseif (t_1 <= 1e-13)
		tmp = Float64(fma(Float64(x * x), 0.5, 1.0) * t_0);
	else
		tmp = Float64(cosh(x) * fma(Float64(Float64(0.008333333333333333 * Float64(y * y)) - 0.16666666666666666), Float64(y * y), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cosh[x], $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[Cosh[x], $MachinePrecision] * N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e-13], N[(N[(N[(x * x), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[Cosh[x], $MachinePrecision] * N[(N[(N[(0.008333333333333333 * N[(y * y), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in 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. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -inf.0 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < 1e-13

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 99.6

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

   5. Applied rewrites 99.6%

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

   if 1e-13 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

Alternative 3: 91.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ t_1 := \cosh x \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\cosh x \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{elif}\;t\_1 \leq 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, 0.5, 1\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot y}{y}\\ \end{array} \end{array} \]

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 -0.16666666666666666 (* y y) 1.0))
     (if (<= t_1 1e-13)
       (* (fma (* x x) 0.5 1.0) t_0)
       (/ (* (fma (fma 0.041666666666666664 (* x x) 0.5) (* x x) 1.0) y) y)))))

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(-0.16666666666666666, (y * y), 1.0);
	} else if (t_1 <= 1e-13) {
		tmp = fma((x * x), 0.5, 1.0) * t_0;
	} else {
		tmp = (fma(fma(0.041666666666666664, (x * x), 0.5), (x * x), 1.0) * y) / y;
	}
	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(-0.16666666666666666, Float64(y * y), 1.0));
	elseif (t_1 <= 1e-13)
		tmp = Float64(fma(Float64(x * x), 0.5, 1.0) * t_0);
	else
		tmp = Float64(Float64(fma(fma(0.041666666666666664, Float64(x * x), 0.5), Float64(x * x), 1.0) * y) / y);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cosh[x], $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[Cosh[x], $MachinePrecision] * N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e-13], N[(N[(N[(x * x), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(N[(N[(0.041666666666666664 * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision] / y), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in 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. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -inf.0 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < 1e-13

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 99.6

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

   5. Applied rewrites 99.6%

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

   if 1e-13 < (*.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

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-*r/ N/A

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

      4. associate-*r/ N/A

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

      5. div-add-rev N/A

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

      6. associate-*l/ N/A

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

      7. div-add-rev N/A

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

      8. lower-/.f64 N/A

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

   5. Applied rewrites 84.1%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 88.3%

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

Alternative 4: 91.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ t_1 := \cosh x \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\cosh x \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{elif}\;t\_1 \leq 10^{-13}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot y}{y}\\ \end{array} \end{array} \]

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 -0.16666666666666666 (* y y) 1.0))
     (if (<= t_1 1e-13)
       t_0
       (/ (* (fma (fma 0.041666666666666664 (* x x) 0.5) (* x x) 1.0) y) y)))))

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(-0.16666666666666666, (y * y), 1.0);
	} else if (t_1 <= 1e-13) {
		tmp = t_0;
	} else {
		tmp = (fma(fma(0.041666666666666664, (x * x), 0.5), (x * x), 1.0) * y) / y;
	}
	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(-0.16666666666666666, Float64(y * y), 1.0));
	elseif (t_1 <= 1e-13)
		tmp = t_0;
	else
		tmp = Float64(Float64(fma(fma(0.041666666666666664, Float64(x * x), 0.5), Float64(x * x), 1.0) * y) / y);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cosh[x], $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[Cosh[x], $MachinePrecision] * N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e-13], t$95$0, N[(N[(N[(N[(0.041666666666666664 * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision] / y), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in 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. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -inf.0 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < 1e-13

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 98.5

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

   5. Applied rewrites 98.5%

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

   if 1e-13 < (*.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

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-*r/ N/A

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

      4. associate-*r/ N/A

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

      5. div-add-rev N/A

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

      6. associate-*l/ N/A

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

      7. div-add-rev N/A

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

      8. lower-/.f64 N/A

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

   5. Applied rewrites 84.1%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 88.3%

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

Alternative 5: 91.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ t_1 := \cosh x \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{elif}\;t\_1 \leq 10^{-13}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot y}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)) (t_1 (* (cosh x) t_0)))
   (if (<= t_1 (- INFINITY))
     (*
      (fma
       (fma
        (fma 0.001388888888888889 (* x x) 0.041666666666666664)
        (* x x)
        0.5)
       (* x x)
       1.0)
      (fma -0.16666666666666666 (* y y) 1.0))
     (if (<= t_1 1e-13)
       t_0
       (/ (* (fma (fma 0.041666666666666664 (* x x) 0.5) (* x x) 1.0) y) y)))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double t_1 = cosh(x) * t_0;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma(fma(fma(0.001388888888888889, (x * x), 0.041666666666666664), (x * x), 0.5), (x * x), 1.0) * fma(-0.16666666666666666, (y * y), 1.0);
	} else if (t_1 <= 1e-13) {
		tmp = t_0;
	} else {
		tmp = (fma(fma(0.041666666666666664, (x * x), 0.5), (x * x), 1.0) * y) / y;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(y) / y)
	t_1 = Float64(cosh(x) * t_0)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(fma(fma(fma(0.001388888888888889, Float64(x * x), 0.041666666666666664), Float64(x * x), 0.5), Float64(x * x), 1.0) * fma(-0.16666666666666666, Float64(y * y), 1.0));
	elseif (t_1 <= 1e-13)
		tmp = t_0;
	else
		tmp = Float64(Float64(fma(fma(0.041666666666666664, Float64(x * x), 0.5), Float64(x * x), 1.0) * y) / y);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cosh[x], $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(N[(0.001388888888888889 * N[(x * x), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e-13], t$95$0, N[(N[(N[(N[(0.041666666666666664 * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision] / y), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in 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. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 96.9

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

   8. Applied rewrites 96.9%

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

   if -inf.0 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < 1e-13

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 98.5

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

   5. Applied rewrites 98.5%

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

   if 1e-13 < (*.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

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-*r/ N/A

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

      4. associate-*r/ N/A

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

      5. div-add-rev N/A

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

      6. associate-*l/ N/A

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

      7. div-add-rev N/A

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

      8. lower-/.f64 N/A

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

   5. Applied rewrites 84.1%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 88.3%

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

Alternative 6: 69.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right)\\ t_1 := \frac{\sin y}{y}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-306} \lor \neg \left(t\_1 \leq 0.02\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333 \cdot y, y, \left(\left(t\_0 \cdot x\right) \cdot x\right) \cdot \mathsf{fma}\left(0.008333333333333333 \cdot y, y, -0.16666666666666666\right)\right) - 0.16666666666666666, y \cdot y, \mathsf{fma}\left(t\_0, x \cdot x, 1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma (* x x) 0.041666666666666664 0.5)) (t_1 (/ (sin y) y)))
   (if (or (<= t_1 -1e-306) (not (<= t_1 0.02)))
     (*
      (fma
       (fma
        (fma 0.001388888888888889 (* x x) 0.041666666666666664)
        (* x x)
        0.5)
       (* x x)
       1.0)
      (fma -0.16666666666666666 (* y y) 1.0))
     (fma
      (-
       (fma
        (* 0.008333333333333333 y)
        y
        (*
         (* (* t_0 x) x)
         (fma (* 0.008333333333333333 y) y -0.16666666666666666)))
       0.16666666666666666)
      (* y y)
      (fma t_0 (* x x) 1.0)))))

C

double code(double x, double y) {
	double t_0 = fma((x * x), 0.041666666666666664, 0.5);
	double t_1 = sin(y) / y;
	double tmp;
	if ((t_1 <= -1e-306) || !(t_1 <= 0.02)) {
		tmp = fma(fma(fma(0.001388888888888889, (x * x), 0.041666666666666664), (x * x), 0.5), (x * x), 1.0) * fma(-0.16666666666666666, (y * y), 1.0);
	} else {
		tmp = fma((fma((0.008333333333333333 * y), y, (((t_0 * x) * x) * fma((0.008333333333333333 * y), y, -0.16666666666666666))) - 0.16666666666666666), (y * y), fma(t_0, (x * x), 1.0));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(Float64(x * x), 0.041666666666666664, 0.5)
	t_1 = Float64(sin(y) / y)
	tmp = 0.0
	if ((t_1 <= -1e-306) || !(t_1 <= 0.02))
		tmp = Float64(fma(fma(fma(0.001388888888888889, Float64(x * x), 0.041666666666666664), Float64(x * x), 0.5), Float64(x * x), 1.0) * fma(-0.16666666666666666, Float64(y * y), 1.0));
	else
		tmp = fma(Float64(fma(Float64(0.008333333333333333 * y), y, Float64(Float64(Float64(t_0 * x) * x) * fma(Float64(0.008333333333333333 * y), y, -0.16666666666666666))) - 0.16666666666666666), Float64(y * y), fma(t_0, Float64(x * x), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -1e-306], N[Not[LessEqual[t$95$1, 0.02]], $MachinePrecision]], N[(N[(N[(N[(0.001388888888888889 * N[(x * x), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.008333333333333333 * y), $MachinePrecision] * y + N[(N[(N[(t$95$0 * x), $MachinePrecision] * x), $MachinePrecision] * N[(N[(0.008333333333333333 * y), $MachinePrecision] * y + -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + N[(t$95$0 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 y) y) < -1.00000000000000003e-306 or 0.0200000000000000004 < (/.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

      \[\leadsto \cosh x \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {y}^{2}\right)} \]
   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. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 81.8

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

   5. Applied rewrites 81.8%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 76.9

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

   8. Applied rewrites 76.9%

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

   if -1.00000000000000003e-306 < (/.f64 (sin.f64 y) y) < 0.0200000000000000004

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-*r/ N/A

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

      4. associate-*r/ N/A

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

      5. div-add-rev N/A

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

      6. associate-*l/ N/A

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

      7. div-add-rev N/A

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

      8. lower-/.f64 N/A

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

   5. Applied rewrites 79.1%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 0.5%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.006944444444444444, x \cdot x, -0.08333333333333333\right) \cdot \left(x \cdot x\right) - 0.16666666666666666, y \cdot y, \left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right) \cdot x\right) \cdot x\right), y, y\right)}{y} \]

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 0.6%

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

   12. Taylor expanded in y around 0

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

   14. Applied rewrites 59.5%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq -1 \cdot 10^{-306} \lor \neg \left(\frac{\sin y}{y} \leq 0.02\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333 \cdot y, y, \left(\left(\mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right) \cdot x\right) \cdot x\right) \cdot \mathsf{fma}\left(0.008333333333333333 \cdot y, y, -0.16666666666666666\right)\right) - 0.16666666666666666, y \cdot y, \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right), x \cdot x, 1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 69.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-306} \lor \neg \left(t\_0 \leq 0.02\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, 0.5, 1\right) \cdot \mathsf{fma}\left(0.008333333333333333 \cdot \left(y \cdot y\right) - 0.16666666666666666, y \cdot y, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)))
   (if (or (<= t_0 -1e-306) (not (<= t_0 0.02)))
     (*
      (fma
       (fma
        (fma 0.001388888888888889 (* x x) 0.041666666666666664)
        (* x x)
        0.5)
       (* x x)
       1.0)
      (fma -0.16666666666666666 (* y y) 1.0))
     (*
      (fma (* x x) 0.5 1.0)
      (fma
       (- (* 0.008333333333333333 (* y y)) 0.16666666666666666)
       (* y y)
       1.0)))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double tmp;
	if ((t_0 <= -1e-306) || !(t_0 <= 0.02)) {
		tmp = fma(fma(fma(0.001388888888888889, (x * x), 0.041666666666666664), (x * x), 0.5), (x * x), 1.0) * fma(-0.16666666666666666, (y * y), 1.0);
	} else {
		tmp = fma((x * x), 0.5, 1.0) * fma(((0.008333333333333333 * (y * y)) - 0.16666666666666666), (y * y), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(y) / y)
	tmp = 0.0
	if ((t_0 <= -1e-306) || !(t_0 <= 0.02))
		tmp = Float64(fma(fma(fma(0.001388888888888889, Float64(x * x), 0.041666666666666664), Float64(x * x), 0.5), Float64(x * x), 1.0) * fma(-0.16666666666666666, Float64(y * y), 1.0));
	else
		tmp = Float64(fma(Float64(x * x), 0.5, 1.0) * fma(Float64(Float64(0.008333333333333333 * Float64(y * y)) - 0.16666666666666666), Float64(y * y), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -1e-306], N[Not[LessEqual[t$95$0, 0.02]], $MachinePrecision]], N[(N[(N[(N[(0.001388888888888889 * N[(x * x), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * x), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] * N[(N[(N[(0.008333333333333333 * N[(y * y), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 y) y) < -1.00000000000000003e-306 or 0.0200000000000000004 < (/.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

      \[\leadsto \cosh x \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {y}^{2}\right)} \]
   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. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 81.8

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

   5. Applied rewrites 81.8%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 76.9

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

   8. Applied rewrites 76.9%

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

   if -1.00000000000000003e-306 < (/.f64 (sin.f64 y) y) < 0.0200000000000000004

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 72.1

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

   5. Applied rewrites 72.1%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 55.6

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

   8. Applied rewrites 55.6%

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

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq -1 \cdot 10^{-306} \lor \neg \left(\frac{\sin y}{y} \leq 0.02\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, 0.5, 1\right) \cdot \mathsf{fma}\left(0.008333333333333333 \cdot \left(y \cdot y\right) - 0.16666666666666666, y \cdot y, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 66.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ t_1 := \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right), x \cdot x, 1\right)\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-306}:\\ \;\;\;\;t\_1 \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-61}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, 0.5, 1\right) \cdot \mathsf{fma}\left(0.008333333333333333 \cdot \left(y \cdot y\right) - 0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y))
        (t_1 (fma (fma (* x x) 0.041666666666666664 0.5) (* x x) 1.0)))
   (if (<= t_0 -1e-306)
     (* t_1 (fma -0.16666666666666666 (* y y) 1.0))
     (if (<= t_0 2e-61)
       (*
        (fma (* x x) 0.5 1.0)
        (fma
         (- (* 0.008333333333333333 (* y y)) 0.16666666666666666)
         (* y y)
         1.0))
       t_1))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double t_1 = fma(fma((x * x), 0.041666666666666664, 0.5), (x * x), 1.0);
	double tmp;
	if (t_0 <= -1e-306) {
		tmp = t_1 * fma(-0.16666666666666666, (y * y), 1.0);
	} else if (t_0 <= 2e-61) {
		tmp = fma((x * x), 0.5, 1.0) * fma(((0.008333333333333333 * (y * y)) - 0.16666666666666666), (y * y), 1.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(y) / y)
	t_1 = fma(fma(Float64(x * x), 0.041666666666666664, 0.5), Float64(x * x), 1.0)
	tmp = 0.0
	if (t_0 <= -1e-306)
		tmp = Float64(t_1 * fma(-0.16666666666666666, Float64(y * y), 1.0));
	elseif (t_0 <= 2e-61)
		tmp = Float64(fma(Float64(x * x), 0.5, 1.0) * fma(Float64(Float64(0.008333333333333333 * Float64(y * y)) - 0.16666666666666666), Float64(y * y), 1.0));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(x * x), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-306], N[(t$95$1 * N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e-61], N[(N[(N[(x * x), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] * N[(N[(N[(0.008333333333333333 * N[(y * y), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 y) y) < -1.00000000000000003e-306

   1. Initial program 99.7%

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

   3. Taylor expanded in y around 0

      \[\leadsto \cosh x \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {y}^{2}\right)} \]
   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. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 48.0

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

   5. Applied rewrites 48.0%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 45.1

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

   8. Applied rewrites 45.1%

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

   if -1.00000000000000003e-306 < (/.f64 (sin.f64 y) y) < 2.0000000000000001e-61

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 72.7

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

   5. Applied rewrites 72.7%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 61.5

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

   8. Applied rewrites 61.5%

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

   if 2.0000000000000001e-61 < (/.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

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-*r/ N/A

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

      4. associate-*r/ N/A

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

      5. div-add-rev N/A

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

      6. associate-*l/ N/A

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

      7. div-add-rev N/A

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

      8. lower-/.f64 N/A

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

   5. Applied rewrites 90.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 81.1%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 87.1%

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

Alternative 9: 66.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ t_1 := \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right), x \cdot x, 1\right)\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-306}:\\ \;\;\;\;t\_1 \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-100}:\\ \;\;\;\;\mathsf{fma}\left(0.008333333333333333 \cdot \left(y \cdot y\right) - 0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y))
        (t_1 (fma (fma (* x x) 0.041666666666666664 0.5) (* x x) 1.0)))
   (if (<= t_0 -1e-306)
     (* t_1 (fma -0.16666666666666666 (* y y) 1.0))
     (if (<= t_0 5e-100)
       (fma
        (- (* 0.008333333333333333 (* y y)) 0.16666666666666666)
        (* y y)
        1.0)
       t_1))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double t_1 = fma(fma((x * x), 0.041666666666666664, 0.5), (x * x), 1.0);
	double tmp;
	if (t_0 <= -1e-306) {
		tmp = t_1 * fma(-0.16666666666666666, (y * y), 1.0);
	} else if (t_0 <= 5e-100) {
		tmp = fma(((0.008333333333333333 * (y * y)) - 0.16666666666666666), (y * y), 1.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(y) / y)
	t_1 = fma(fma(Float64(x * x), 0.041666666666666664, 0.5), Float64(x * x), 1.0)
	tmp = 0.0
	if (t_0 <= -1e-306)
		tmp = Float64(t_1 * fma(-0.16666666666666666, Float64(y * y), 1.0));
	elseif (t_0 <= 5e-100)
		tmp = fma(Float64(Float64(0.008333333333333333 * Float64(y * y)) - 0.16666666666666666), Float64(y * y), 1.0);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(x * x), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-306], N[(t$95$1 * N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e-100], N[(N[(N[(0.008333333333333333 * N[(y * y), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 y) y) < -1.00000000000000003e-306

   1. Initial program 99.7%

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

   3. Taylor expanded in y around 0

      \[\leadsto \cosh x \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {y}^{2}\right)} \]
   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. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 48.0

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

   5. Applied rewrites 48.0%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 45.1

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

   8. Applied rewrites 45.1%

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

   if -1.00000000000000003e-306 < (/.f64 (sin.f64 y) y) < 5.0000000000000001e-100

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 38.3

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

   5. Applied rewrites 38.3%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 63.5%

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

   if 5.0000000000000001e-100 < (/.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

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-*r/ N/A

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

      4. associate-*r/ N/A

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

      5. div-add-rev N/A

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

      6. associate-*l/ N/A

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

      7. div-add-rev N/A

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

      8. lower-/.f64 N/A

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

   5. Applied rewrites 90.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 81.8%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 84.0%

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

Alternative 10: 65.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-306}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, 0.5, 1\right) \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-100}:\\ \;\;\;\;\mathsf{fma}\left(0.008333333333333333 \cdot \left(y \cdot y\right) - 0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right), x \cdot x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)))
   (if (<= t_0 -1e-306)
     (* (fma (* x x) 0.5 1.0) (fma -0.16666666666666666 (* y y) 1.0))
     (if (<= t_0 5e-100)
       (fma
        (- (* 0.008333333333333333 (* y y)) 0.16666666666666666)
        (* y y)
        1.0)
       (fma (fma (* x x) 0.041666666666666664 0.5) (* x x) 1.0)))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double tmp;
	if (t_0 <= -1e-306) {
		tmp = fma((x * x), 0.5, 1.0) * fma(-0.16666666666666666, (y * y), 1.0);
	} else if (t_0 <= 5e-100) {
		tmp = fma(((0.008333333333333333 * (y * y)) - 0.16666666666666666), (y * y), 1.0);
	} else {
		tmp = fma(fma((x * x), 0.041666666666666664, 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 <= -1e-306)
		tmp = Float64(fma(Float64(x * x), 0.5, 1.0) * fma(-0.16666666666666666, Float64(y * y), 1.0));
	elseif (t_0 <= 5e-100)
		tmp = fma(Float64(Float64(0.008333333333333333 * Float64(y * y)) - 0.16666666666666666), Float64(y * y), 1.0);
	else
		tmp = fma(fma(Float64(x * x), 0.041666666666666664, 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, -1e-306], N[(N[(N[(x * x), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] * N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e-100], N[(N[(N[(0.008333333333333333 * N[(y * y), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(N[(x * x), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 y) y) < -1.00000000000000003e-306

   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 76.0

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

   5. Applied rewrites 76.0%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 42.2

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

   8. Applied rewrites 42.2%

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

   if -1.00000000000000003e-306 < (/.f64 (sin.f64 y) y) < 5.0000000000000001e-100

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 38.3

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

   5. Applied rewrites 38.3%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 63.5%

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

   if 5.0000000000000001e-100 < (/.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

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-*r/ N/A

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

      4. associate-*r/ N/A

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

      5. div-add-rev N/A

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

      6. associate-*l/ N/A

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

      7. div-add-rev N/A

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

      8. lower-/.f64 N/A

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

   5. Applied rewrites 90.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 81.8%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 84.0%

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

Alternative 11: 63.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-306}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(y \cdot y\right) \cdot -0.16666666666666666, y, y\right)}{y}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-100}:\\ \;\;\;\;\mathsf{fma}\left(0.008333333333333333 \cdot \left(y \cdot y\right) - 0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right), x \cdot x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)))
   (if (<= t_0 -1e-306)
     (/ (fma (* (* y y) -0.16666666666666666) y y) y)
     (if (<= t_0 5e-100)
       (fma
        (- (* 0.008333333333333333 (* y y)) 0.16666666666666666)
        (* y y)
        1.0)
       (fma (fma (* x x) 0.041666666666666664 0.5) (* x x) 1.0)))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double tmp;
	if (t_0 <= -1e-306) {
		tmp = fma(((y * y) * -0.16666666666666666), y, y) / y;
	} else if (t_0 <= 5e-100) {
		tmp = fma(((0.008333333333333333 * (y * y)) - 0.16666666666666666), (y * y), 1.0);
	} else {
		tmp = fma(fma((x * x), 0.041666666666666664, 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 <= -1e-306)
		tmp = Float64(fma(Float64(Float64(y * y) * -0.16666666666666666), y, y) / y);
	elseif (t_0 <= 5e-100)
		tmp = fma(Float64(Float64(0.008333333333333333 * Float64(y * y)) - 0.16666666666666666), Float64(y * y), 1.0);
	else
		tmp = fma(fma(Float64(x * x), 0.041666666666666664, 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, -1e-306], N[(N[(N[(N[(y * y), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * y + y), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[t$95$0, 5e-100], N[(N[(N[(0.008333333333333333 * N[(y * y), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(N[(x * x), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 y) y) < -1.00000000000000003e-306

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-*r/ N/A

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

      4. associate-*r/ N/A

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

      5. div-add-rev N/A

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

      6. associate-*l/ N/A

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

      7. div-add-rev N/A

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

      8. lower-/.f64 N/A

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

   5. Applied rewrites 84.8%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 15.7%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.006944444444444444, x \cdot x, -0.08333333333333333\right) \cdot \left(x \cdot x\right) - 0.16666666666666666, y \cdot y, \left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right) \cdot x\right) \cdot x\right), y, y\right)}{y} \]

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 31.7%

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

   if -1.00000000000000003e-306 < (/.f64 (sin.f64 y) y) < 5.0000000000000001e-100

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 38.3

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

   5. Applied rewrites 38.3%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 63.5%

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

   if 5.0000000000000001e-100 < (/.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

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-*r/ N/A

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

      4. associate-*r/ N/A

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

      5. div-add-rev N/A

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

      6. associate-*l/ N/A

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

      7. div-add-rev N/A

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

      8. lower-/.f64 N/A

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

   5. Applied rewrites 90.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 81.8%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 84.0%

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

Alternative 12: 61.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-306}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-100}:\\ \;\;\;\;\mathsf{fma}\left(0.008333333333333333 \cdot \left(y \cdot y\right) - 0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right), x \cdot x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)))
   (if (<= t_0 -1e-306)
     (fma -0.16666666666666666 (* y y) 1.0)
     (if (<= t_0 5e-100)
       (fma
        (- (* 0.008333333333333333 (* y y)) 0.16666666666666666)
        (* y y)
        1.0)
       (fma (fma (* x x) 0.041666666666666664 0.5) (* x x) 1.0)))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double tmp;
	if (t_0 <= -1e-306) {
		tmp = fma(-0.16666666666666666, (y * y), 1.0);
	} else if (t_0 <= 5e-100) {
		tmp = fma(((0.008333333333333333 * (y * y)) - 0.16666666666666666), (y * y), 1.0);
	} else {
		tmp = fma(fma((x * x), 0.041666666666666664, 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 <= -1e-306)
		tmp = fma(-0.16666666666666666, Float64(y * y), 1.0);
	elseif (t_0 <= 5e-100)
		tmp = fma(Float64(Float64(0.008333333333333333 * Float64(y * y)) - 0.16666666666666666), Float64(y * y), 1.0);
	else
		tmp = fma(fma(Float64(x * x), 0.041666666666666664, 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, -1e-306], N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[t$95$0, 5e-100], N[(N[(N[(0.008333333333333333 * N[(y * y), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(N[(x * x), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 y) y) < -1.00000000000000003e-306

   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 53.5

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

   5. Applied rewrites 53.5%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 28.9%

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

   if -1.00000000000000003e-306 < (/.f64 (sin.f64 y) y) < 5.0000000000000001e-100

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 38.3

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

   5. Applied rewrites 38.3%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 63.5%

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

   if 5.0000000000000001e-100 < (/.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

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-*r/ N/A

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

      4. associate-*r/ N/A

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

      5. div-add-rev N/A

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

      6. associate-*l/ N/A

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

      7. div-add-rev N/A

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

      8. lower-/.f64 N/A

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

   5. Applied rewrites 90.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 81.8%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 84.0%

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

Alternative 13: 60.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -4 \cdot 10^{-147}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right), x \cdot x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cosh x) (/ (sin y) y)) -4e-147)
   (fma -0.16666666666666666 (* y y) 1.0)
   (fma (fma (* x x) 0.041666666666666664 0.5) (* x x) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((cosh(x) * (sin(y) / y)) <= -4e-147) {
		tmp = fma(-0.16666666666666666, (y * y), 1.0);
	} else {
		tmp = fma(fma((x * x), 0.041666666666666664, 0.5), (x * x), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(cosh(x) * Float64(sin(y) / y)) <= -4e-147)
		tmp = fma(-0.16666666666666666, Float64(y * y), 1.0);
	else
		tmp = fma(fma(Float64(x * x), 0.041666666666666664, 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], -4e-147], N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(N[(x * x), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < -3.9999999999999999e-147

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 34.6

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

   5. Applied rewrites 34.6%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 40.9%

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

   if -3.9999999999999999e-147 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y))

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-*r/ N/A

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

      4. associate-*r/ N/A

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

      5. div-add-rev N/A

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

      6. associate-*l/ N/A

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

      7. div-add-rev N/A

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

      8. lower-/.f64 N/A

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

   5. Applied rewrites 87.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 80.5%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 65.7%

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

Alternative 14: 32.9% accurate, 18.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (fma -0.16666666666666666 (* y y) 1.0))

C

double code(double x, double y) {
	return fma(-0.16666666666666666, (y * y), 1.0);
}

Julia

function code(x, y)
	return fma(-0.16666666666666666, Float64(y * y), 1.0)
end

Wolfram

code[x_, y_] := N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in x around 0

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

   1. lower-/.f64 N/A

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

   2. lower-sin.f64 54.6

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

5. Applied rewrites 54.6%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 36.9%

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

Alternative 15: 27.3% accurate, 217.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

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

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in x around 0

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

   1. lower-/.f64 N/A

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

   2. lower-sin.f64 54.6

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

5. Applied rewrites 54.6%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 36.9%

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

9. Taylor expanded in y around 0

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

11. Applied rewrites 30.8%

    \[\leadsto 1 \]
12. Add Preprocessing

Developer Target 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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 2024338 
(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)))