Linear.Quaternion:$csinh from linear-1.19.1.3

Percentage Accurate: 99.9% → 99.9%

Time: 12.6s

Alternatives: 23

Speedup: 1.0×

• 49.5% 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: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ t_1 := \cosh x \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\cosh x \cdot \left(y \cdot \left(y \cdot -0.16666666666666666\right)\right)\\ \mathbf{elif}\;t\_1 \leq 0.9999999999709752:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)) (t_1 (* (cosh x) t_0)))
   (if (<= t_1 (- INFINITY))
     (* (cosh x) (* y (* y -0.16666666666666666)))
     (if (<= t_1 0.9999999999709752)
       (* t_0 (fma (* x x) (fma (* x x) 0.041666666666666664 0.5) 1.0))
       (* (cosh x) 1.0)))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double t_1 = cosh(x) * t_0;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = cosh(x) * (y * (y * -0.16666666666666666));
	} else if (t_1 <= 0.9999999999709752) {
		tmp = t_0 * fma((x * x), fma((x * x), 0.041666666666666664, 0.5), 1.0);
	} else {
		tmp = cosh(x) * 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(y) / y)
	t_1 = Float64(cosh(x) * t_0)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(cosh(x) * Float64(y * Float64(y * -0.16666666666666666)));
	elseif (t_1 <= 0.9999999999709752)
		tmp = Float64(t_0 * fma(Float64(x * x), fma(Float64(x * x), 0.041666666666666664, 0.5), 1.0));
	else
		tmp = Float64(cosh(x) * 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cosh[x], $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[Cosh[x], $MachinePrecision] * N[(y * N[(y * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.9999999999709752], N[(t$95$0 * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Cosh[x], $MachinePrecision] * 1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 100.0%

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

   if -inf.0 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < 0.99999999997097522

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 96.8

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

   5. Applied rewrites 96.8%

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

   if 0.99999999997097522 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -\infty:\\ \;\;\;\;\cosh x \cdot \left(y \cdot \left(y \cdot -0.16666666666666666\right)\right)\\ \mathbf{elif}\;\cosh x \cdot \frac{\sin y}{y} \leq 0.9999999999709752:\\ \;\;\;\;\frac{\sin y}{y} \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ t_1 := \cosh x \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\cosh x \cdot \left(y \cdot \left(y \cdot -0.16666666666666666\right)\right)\\ \mathbf{elif}\;t\_1 \leq 0.9999999999709752:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot 0.041666666666666664, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)) (t_1 (* (cosh x) t_0)))
   (if (<= t_1 (- INFINITY))
     (* (cosh x) (* y (* y -0.16666666666666666)))
     (if (<= t_1 0.9999999999709752)
       (* t_0 (fma (* x x) (* (* x x) 0.041666666666666664) 1.0))
       (* (cosh x) 1.0)))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double t_1 = cosh(x) * t_0;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = cosh(x) * (y * (y * -0.16666666666666666));
	} else if (t_1 <= 0.9999999999709752) {
		tmp = t_0 * fma((x * x), ((x * x) * 0.041666666666666664), 1.0);
	} else {
		tmp = cosh(x) * 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(y) / y)
	t_1 = Float64(cosh(x) * t_0)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(cosh(x) * Float64(y * Float64(y * -0.16666666666666666)));
	elseif (t_1 <= 0.9999999999709752)
		tmp = Float64(t_0 * fma(Float64(x * x), Float64(Float64(x * x) * 0.041666666666666664), 1.0));
	else
		tmp = Float64(cosh(x) * 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cosh[x], $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[Cosh[x], $MachinePrecision] * N[(y * N[(y * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.9999999999709752], N[(t$95$0 * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.041666666666666664), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Cosh[x], $MachinePrecision] * 1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 100.0%

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

   if -inf.0 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < 0.99999999997097522

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 96.8

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

   5. Applied rewrites 96.8%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 96.8%

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

   if 0.99999999997097522 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -\infty:\\ \;\;\;\;\cosh x \cdot \left(y \cdot \left(y \cdot -0.16666666666666666\right)\right)\\ \mathbf{elif}\;\cosh x \cdot \frac{\sin y}{y} \leq 0.9999999999709752:\\ \;\;\;\;\frac{\sin y}{y} \cdot \mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot 0.041666666666666664, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ t_1 := \cosh x \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\cosh x \cdot \left(y \cdot \left(y \cdot -0.16666666666666666\right)\right)\\ \mathbf{elif}\;t\_1 \leq 0.9999999999709752:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(0.5, x \cdot x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)) (t_1 (* (cosh x) t_0)))
   (if (<= t_1 (- INFINITY))
     (* (cosh x) (* y (* y -0.16666666666666666)))
     (if (<= t_1 0.9999999999709752)
       (* t_0 (fma 0.5 (* x x) 1.0))
       (* (cosh x) 1.0)))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double t_1 = cosh(x) * t_0;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = cosh(x) * (y * (y * -0.16666666666666666));
	} else if (t_1 <= 0.9999999999709752) {
		tmp = t_0 * fma(0.5, (x * x), 1.0);
	} else {
		tmp = cosh(x) * 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(y) / y)
	t_1 = Float64(cosh(x) * t_0)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(cosh(x) * Float64(y * Float64(y * -0.16666666666666666)));
	elseif (t_1 <= 0.9999999999709752)
		tmp = Float64(t_0 * fma(0.5, Float64(x * x), 1.0));
	else
		tmp = Float64(cosh(x) * 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cosh[x], $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[Cosh[x], $MachinePrecision] * N[(y * N[(y * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.9999999999709752], N[(t$95$0 * N[(0.5 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Cosh[x], $MachinePrecision] * 1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 100.0%

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

   if -inf.0 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < 0.99999999997097522

   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. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 96.8

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

   5. Applied rewrites 96.8%

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

   if 0.99999999997097522 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -\infty:\\ \;\;\;\;\cosh x \cdot \left(y \cdot \left(y \cdot -0.16666666666666666\right)\right)\\ \mathbf{elif}\;\cosh x \cdot \frac{\sin y}{y} \leq 0.9999999999709752:\\ \;\;\;\;\frac{\sin y}{y} \cdot \mathsf{fma}\left(0.5, x \cdot x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ t_1 := \cosh x \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\cosh x \cdot \left(y \cdot \left(y \cdot -0.16666666666666666\right)\right)\\ \mathbf{elif}\;t\_1 \leq 0.9999999999709752:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)) (t_1 (* (cosh x) t_0)))
   (if (<= t_1 (- INFINITY))
     (* (cosh x) (* y (* y -0.16666666666666666)))
     (if (<= t_1 0.9999999999709752) t_0 (* (cosh x) 1.0)))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double t_1 = cosh(x) * t_0;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = cosh(x) * (y * (y * -0.16666666666666666));
	} else if (t_1 <= 0.9999999999709752) {
		tmp = t_0;
	} else {
		tmp = cosh(x) * 1.0;
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double t_0 = Math.sin(y) / y;
	double t_1 = Math.cosh(x) * t_0;
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = Math.cosh(x) * (y * (y * -0.16666666666666666));
	} else if (t_1 <= 0.9999999999709752) {
		tmp = t_0;
	} else {
		tmp = Math.cosh(x) * 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sin(y) / y
	t_1 = math.cosh(x) * t_0
	tmp = 0
	if t_1 <= -math.inf:
		tmp = math.cosh(x) * (y * (y * -0.16666666666666666))
	elif t_1 <= 0.9999999999709752:
		tmp = t_0
	else:
		tmp = math.cosh(x) * 1.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sin(y) / y)
	t_1 = Float64(cosh(x) * t_0)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(cosh(x) * Float64(y * Float64(y * -0.16666666666666666)));
	elseif (t_1 <= 0.9999999999709752)
		tmp = t_0;
	else
		tmp = Float64(cosh(x) * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sin(y) / y;
	t_1 = cosh(x) * t_0;
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = cosh(x) * (y * (y * -0.16666666666666666));
	elseif (t_1 <= 0.9999999999709752)
		tmp = t_0;
	else
		tmp = cosh(x) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cosh[x], $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[Cosh[x], $MachinePrecision] * N[(y * N[(y * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.9999999999709752], t$95$0, N[(N[Cosh[x], $MachinePrecision] * 1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 100.0%

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

   if -inf.0 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < 0.99999999997097522

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 96.7

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

   5. Applied rewrites 96.7%

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

   if 0.99999999997097522 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 100.0%

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

Alternative 6: 99.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ t_1 := \cosh x \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)\\ \mathbf{elif}\;t\_1 \leq 0.9999999999709752:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)) (t_1 (* (cosh x) t_0)))
   (if (<= t_1 (- INFINITY))
     (*
      (fma
       x
       (*
        x
        (fma
         (* x x)
         (fma (* x x) 0.001388888888888889 0.041666666666666664)
         0.5))
       1.0)
      (fma
       (* y y)
       (fma
        (* y y)
        (fma (* y y) -0.0001984126984126984 0.008333333333333333)
        -0.16666666666666666)
       1.0))
     (if (<= t_1 0.9999999999709752) t_0 (* (cosh x) 1.0)))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double t_1 = cosh(x) * t_0;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma(x, (x * fma((x * x), fma((x * x), 0.001388888888888889, 0.041666666666666664), 0.5)), 1.0) * fma((y * y), fma((y * y), fma((y * y), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666), 1.0);
	} else if (t_1 <= 0.9999999999709752) {
		tmp = t_0;
	} else {
		tmp = cosh(x) * 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(y) / y)
	t_1 = Float64(cosh(x) * t_0)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(fma(x, Float64(x * fma(Float64(x * x), fma(Float64(x * x), 0.001388888888888889, 0.041666666666666664), 0.5)), 1.0) * fma(Float64(y * y), fma(Float64(y * y), fma(Float64(y * y), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666), 1.0));
	elseif (t_1 <= 0.9999999999709752)
		tmp = t_0;
	else
		tmp = Float64(cosh(x) * 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cosh[x], $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.9999999999709752], t$95$0, N[(N[Cosh[x], $MachinePrecision] * 1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 0.0%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 0.0

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

   8. Applied rewrites 0.0%

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

   9. Taylor expanded in y around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 N/A

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

       5. sub-neg N/A

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

       6. metadata-eval N/A

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

       7. lower-fma.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 N/A

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

       10. +-commutative N/A

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

       11. *-commutative N/A

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

       12. lower-fma.f64 N/A

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

       13. unpow2 N/A

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

       14. lower-*.f64 97.0

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

   11. Applied rewrites 97.0%

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

   if -inf.0 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < 0.99999999997097522

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 96.7

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

   5. Applied rewrites 96.7%

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

   if 0.99999999997097522 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 100.0%

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

Alternative 7: 97.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ \mathbf{if}\;\cosh x \cdot t\_0 \leq 0.9999999999709752:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)))
   (if (<= (* (cosh x) t_0) 0.9999999999709752)
     (*
      t_0
      (fma
       (* x x)
       (fma
        x
        (* x (fma (* x x) 0.001388888888888889 0.041666666666666664))
        0.5)
       1.0))
     (* (cosh x) 1.0))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double tmp;
	if ((cosh(x) * t_0) <= 0.9999999999709752) {
		tmp = t_0 * fma((x * x), fma(x, (x * fma((x * x), 0.001388888888888889, 0.041666666666666664)), 0.5), 1.0);
	} else {
		tmp = cosh(x) * 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(y) / y)
	tmp = 0.0
	if (Float64(cosh(x) * t_0) <= 0.9999999999709752)
		tmp = Float64(t_0 * fma(Float64(x * x), fma(x, Float64(x * fma(Float64(x * x), 0.001388888888888889, 0.041666666666666664)), 0.5), 1.0));
	else
		tmp = Float64(cosh(x) * 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[N[(N[Cosh[x], $MachinePrecision] * t$95$0), $MachinePrecision], 0.9999999999709752], N[(t$95$0 * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Cosh[x], $MachinePrecision] * 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < 0.99999999997097522

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 93.7

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

   5. Applied rewrites 93.7%

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

   if 0.99999999997097522 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq 0.9999999999709752:\\ \;\;\;\;\frac{\sin y}{y} \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq 0.9999999999709752:\\ \;\;\;\;\sin y \cdot \frac{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot 0.001388888888888889, 0.5\right), 1\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cosh x) (/ (sin y) y)) 0.9999999999709752)
   (*
    (sin y)
    (/ (fma x (* x (fma (* x x) (* (* x x) 0.001388888888888889) 0.5)) 1.0) y))
   (* (cosh x) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((cosh(x) * (sin(y) / y)) <= 0.9999999999709752) {
		tmp = sin(y) * (fma(x, (x * fma((x * x), ((x * x) * 0.001388888888888889), 0.5)), 1.0) / y);
	} else {
		tmp = cosh(x) * 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(cosh(x) * Float64(sin(y) / y)) <= 0.9999999999709752)
		tmp = Float64(sin(y) * Float64(fma(x, Float64(x * fma(Float64(x * x), Float64(Float64(x * x) * 0.001388888888888889), 0.5)), 1.0) / y));
	else
		tmp = Float64(cosh(x) * 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Cosh[x], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], 0.9999999999709752], N[(N[Sin[y], $MachinePrecision] * N[(N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.001388888888888889), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(N[Cosh[x], $MachinePrecision] * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < 0.99999999997097522

   1. Initial program 99.7%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. associate-/r/ N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. div-inv N/A

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

      10. lower-/.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 93.7

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

   7. Applied rewrites 93.7%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 93.7%

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

   if 0.99999999997097522 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq 0.9999999999709752:\\ \;\;\;\;\sin y \cdot \frac{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot 0.001388888888888889, 0.5\right), 1\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 97.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq 0.9999999999709752:\\ \;\;\;\;\sin y \cdot \frac{\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.001388888888888889\right)\right), 1\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cosh x) (/ (sin y) y)) 0.9999999999709752)
   (*
    (sin y)
    (/ (fma x (* (* x x) (* x (* (* x x) 0.001388888888888889))) 1.0) y))
   (* (cosh x) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((cosh(x) * (sin(y) / y)) <= 0.9999999999709752) {
		tmp = sin(y) * (fma(x, ((x * x) * (x * ((x * x) * 0.001388888888888889))), 1.0) / y);
	} else {
		tmp = cosh(x) * 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(cosh(x) * Float64(sin(y) / y)) <= 0.9999999999709752)
		tmp = Float64(sin(y) * Float64(fma(x, Float64(Float64(x * x) * Float64(x * Float64(Float64(x * x) * 0.001388888888888889))), 1.0) / y));
	else
		tmp = Float64(cosh(x) * 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Cosh[x], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], 0.9999999999709752], N[(N[Sin[y], $MachinePrecision] * N[(N[(x * N[(N[(x * x), $MachinePrecision] * N[(x * N[(N[(x * x), $MachinePrecision] * 0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(N[Cosh[x], $MachinePrecision] * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < 0.99999999997097522

   1. Initial program 99.7%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. associate-/r/ N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. div-inv N/A

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

      10. lower-/.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 93.7

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

   7. Applied rewrites 93.7%

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

   8. Taylor expanded in x around inf

      \[\leadsto \frac{\mathsf{fma}\left(x, \frac{1}{720} \cdot \color{blue}{{x}^{5}}, 1\right)}{y} \cdot \sin y \]
   9. Step-by-step derivation

   10. Applied rewrites 93.7%

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

   if 0.99999999997097522 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq 0.9999999999709752:\\ \;\;\;\;\sin y \cdot \frac{\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.001388888888888889\right)\right), 1\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 75.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -5 \cdot 10^{-148}:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot -0.16666666666666666, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cosh x) (/ (sin y) y)) -5e-148)
   (*
    (fma
     x
     (*
      x
      (fma
       (* x x)
       (fma (* x x) 0.001388888888888889 0.041666666666666664)
       0.5))
     1.0)
    (fma y (* y -0.16666666666666666) 1.0))
   (* (cosh x) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((cosh(x) * (sin(y) / y)) <= -5e-148) {
		tmp = fma(x, (x * fma((x * x), fma((x * x), 0.001388888888888889, 0.041666666666666664), 0.5)), 1.0) * fma(y, (y * -0.16666666666666666), 1.0);
	} else {
		tmp = cosh(x) * 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(cosh(x) * Float64(sin(y) / y)) <= -5e-148)
		tmp = Float64(fma(x, Float64(x * fma(Float64(x * x), fma(Float64(x * x), 0.001388888888888889, 0.041666666666666664), 0.5)), 1.0) * fma(y, Float64(y * -0.16666666666666666), 1.0));
	else
		tmp = Float64(cosh(x) * 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Cosh[x], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], -5e-148], N[(N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(y * N[(y * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Cosh[x], $MachinePrecision] * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < -4.9999999999999999e-148

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 0.7%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 0.7

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

   8. Applied rewrites 0.7%

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

   9. Taylor expanded in y around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. lower-fma.f64 N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 63.0

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

   11. Applied rewrites 63.0%

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

   if -4.9999999999999999e-148 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y))

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 79.7%

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

Alternative 11: 70.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ t_1 := \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-307}:\\ \;\;\;\;t\_1 \cdot \mathsf{fma}\left(y, y \cdot -0.16666666666666666, 1\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(0.5, x \cdot x, 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.008333333333333333, -0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y))
        (t_1
         (fma
          x
          (*
           x
           (fma
            (* x x)
            (fma (* x x) 0.001388888888888889 0.041666666666666664)
            0.5))
          1.0)))
   (if (<= t_0 -1e-307)
     (* t_1 (fma y (* y -0.16666666666666666) 1.0))
     (if (<= t_0 5e-11)
       (*
        (fma 0.5 (* x x) 1.0)
        (fma
         (* y y)
         (fma (* y y) 0.008333333333333333 -0.16666666666666666)
         1.0))
       (*
        t_1
        (fma
         (* y y)
         (fma
          (* y y)
          (fma (* y y) -0.0001984126984126984 0.008333333333333333)
          -0.16666666666666666)
         1.0))))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double t_1 = fma(x, (x * fma((x * x), fma((x * x), 0.001388888888888889, 0.041666666666666664), 0.5)), 1.0);
	double tmp;
	if (t_0 <= -1e-307) {
		tmp = t_1 * fma(y, (y * -0.16666666666666666), 1.0);
	} else if (t_0 <= 5e-11) {
		tmp = fma(0.5, (x * x), 1.0) * fma((y * y), fma((y * y), 0.008333333333333333, -0.16666666666666666), 1.0);
	} else {
		tmp = t_1 * fma((y * y), fma((y * y), fma((y * y), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(y) / y)
	t_1 = fma(x, Float64(x * fma(Float64(x * x), fma(Float64(x * x), 0.001388888888888889, 0.041666666666666664), 0.5)), 1.0)
	tmp = 0.0
	if (t_0 <= -1e-307)
		tmp = Float64(t_1 * fma(y, Float64(y * -0.16666666666666666), 1.0));
	elseif (t_0 <= 5e-11)
		tmp = Float64(fma(0.5, Float64(x * x), 1.0) * fma(Float64(y * y), fma(Float64(y * y), 0.008333333333333333, -0.16666666666666666), 1.0));
	else
		tmp = Float64(t_1 * fma(Float64(y * y), fma(Float64(y * y), fma(Float64(y * y), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666), 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[(x * N[(x * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-307], N[(t$95$1 * N[(y * N[(y * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e-11], N[(N[(0.5 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 y) y) < -9.99999999999999909e-308

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 1.1%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 1.1

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

   8. Applied rewrites 1.1%

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

   9. Taylor expanded in y around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. lower-fma.f64 N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 51.8

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

   11. Applied rewrites 51.8%

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

   if -9.99999999999999909e-308 < (/.f64 (sin.f64 y) y) < 5.00000000000000018e-11

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 69.1

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

   5. Applied rewrites 69.1%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 49.6

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

   8. Applied rewrites 49.6%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 98.6%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 89.6

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

   8. Applied rewrites 89.6%

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

   9. Taylor expanded in y around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 N/A

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

       5. sub-neg N/A

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

       6. metadata-eval N/A

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

       7. lower-fma.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 N/A

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

       10. +-commutative N/A

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

       11. *-commutative N/A

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

       12. lower-fma.f64 N/A

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

       13. unpow2 N/A

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

       14. lower-*.f64 90.1

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

   11. Applied rewrites 90.1%

       \[\leadsto \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right) \cdot \color{blue}{\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 70.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ t_1 := \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot -0.16666666666666666, 1\right)\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-307}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0.8:\\ \;\;\;\;\mathsf{fma}\left(0.5, x \cdot x, 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.008333333333333333, -0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y))
        (t_1
         (*
          (fma
           x
           (*
            x
            (fma
             (* x x)
             (fma (* x x) 0.001388888888888889 0.041666666666666664)
             0.5))
           1.0)
          (fma y (* y -0.16666666666666666) 1.0))))
   (if (<= t_0 -1e-307)
     t_1
     (if (<= t_0 0.8)
       (*
        (fma 0.5 (* x x) 1.0)
        (fma
         (* y y)
         (fma (* y y) 0.008333333333333333 -0.16666666666666666)
         1.0))
       t_1))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double t_1 = fma(x, (x * fma((x * x), fma((x * x), 0.001388888888888889, 0.041666666666666664), 0.5)), 1.0) * fma(y, (y * -0.16666666666666666), 1.0);
	double tmp;
	if (t_0 <= -1e-307) {
		tmp = t_1;
	} else if (t_0 <= 0.8) {
		tmp = fma(0.5, (x * x), 1.0) * fma((y * y), fma((y * y), 0.008333333333333333, -0.16666666666666666), 1.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(y) / y)
	t_1 = Float64(fma(x, Float64(x * fma(Float64(x * x), fma(Float64(x * x), 0.001388888888888889, 0.041666666666666664), 0.5)), 1.0) * fma(y, Float64(y * -0.16666666666666666), 1.0))
	tmp = 0.0
	if (t_0 <= -1e-307)
		tmp = t_1;
	elseif (t_0 <= 0.8)
		tmp = Float64(fma(0.5, Float64(x * x), 1.0) * fma(Float64(y * y), fma(Float64(y * y), 0.008333333333333333, -0.16666666666666666), 1.0));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(y * N[(y * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-307], t$95$1, If[LessEqual[t$95$0, 0.8], N[(N[(0.5 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 y) y) < -9.99999999999999909e-308 or 0.80000000000000004 < (/.f64 (sin.f64 y) y)

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 69.3%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 62.9

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

   8. Applied rewrites 62.9%

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

   9. Taylor expanded in y around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. lower-fma.f64 N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 78.8

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

   11. Applied rewrites 78.8%

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

   if -9.99999999999999909e-308 < (/.f64 (sin.f64 y) y) < 0.80000000000000004

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 70.1

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

   5. Applied rewrites 70.1%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 49.0

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

   8. Applied rewrites 49.0%

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

Alternative 13: 70.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-307}:\\ \;\;\;\;\mathsf{fma}\left(y, y \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right), 1\right)\\ \mathbf{elif}\;t\_0 \leq 0.9999999999709752:\\ \;\;\;\;\mathsf{fma}\left(0.5, x \cdot x, 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.008333333333333333, -0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)))
   (if (<= t_0 -1e-307)
     (*
      (fma y (* y -0.16666666666666666) 1.0)
      (fma x (* x (fma (* x x) 0.041666666666666664 0.5)) 1.0))
     (if (<= t_0 0.9999999999709752)
       (*
        (fma 0.5 (* x x) 1.0)
        (fma
         (* y y)
         (fma (* y y) 0.008333333333333333 -0.16666666666666666)
         1.0))
       (*
        1.0
        (fma
         x
         (*
          x
          (fma
           (* x x)
           (fma (* x x) 0.001388888888888889 0.041666666666666664)
           0.5))
         1.0))))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double tmp;
	if (t_0 <= -1e-307) {
		tmp = fma(y, (y * -0.16666666666666666), 1.0) * fma(x, (x * fma((x * x), 0.041666666666666664, 0.5)), 1.0);
	} else if (t_0 <= 0.9999999999709752) {
		tmp = fma(0.5, (x * x), 1.0) * fma((y * y), fma((y * y), 0.008333333333333333, -0.16666666666666666), 1.0);
	} else {
		tmp = 1.0 * fma(x, (x * fma((x * x), fma((x * x), 0.001388888888888889, 0.041666666666666664), 0.5)), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(y) / y)
	tmp = 0.0
	if (t_0 <= -1e-307)
		tmp = Float64(fma(y, Float64(y * -0.16666666666666666), 1.0) * fma(x, Float64(x * fma(Float64(x * x), 0.041666666666666664, 0.5)), 1.0));
	elseif (t_0 <= 0.9999999999709752)
		tmp = Float64(fma(0.5, Float64(x * x), 1.0) * fma(Float64(y * y), fma(Float64(y * y), 0.008333333333333333, -0.16666666666666666), 1.0));
	else
		tmp = Float64(1.0 * fma(x, Float64(x * fma(Float64(x * x), fma(Float64(x * x), 0.001388888888888889, 0.041666666666666664), 0.5)), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-307], N[(N[(y * N[(y * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.9999999999709752], N[(N[(0.5 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 * N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 y) y) < -9.99999999999999909e-308

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 53.4

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

   5. Applied rewrites 53.4%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 51.8

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

   8. Applied rewrites 51.8%

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

   if -9.99999999999999909e-308 < (/.f64 (sin.f64 y) y) < 0.99999999997097522

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 69.5

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

   5. Applied rewrites 69.5%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 49.1

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

   8. Applied rewrites 49.1%

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

   if 0.99999999997097522 < (/.f64 (sin.f64 y) y)

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 91.4

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

   8. Applied rewrites 91.4%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq -1 \cdot 10^{-307}:\\ \;\;\;\;\mathsf{fma}\left(y, y \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right), 1\right)\\ \mathbf{elif}\;\frac{\sin y}{y} \leq 0.9999999999709752:\\ \;\;\;\;\mathsf{fma}\left(0.5, x \cdot x, 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.008333333333333333, -0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 70.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-307}:\\ \;\;\;\;\mathsf{fma}\left(y, y \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right), 1\right)\\ \mathbf{elif}\;t\_0 \leq 0.9999999999709752:\\ \;\;\;\;\mathsf{fma}\left(0.5, x \cdot x, 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.008333333333333333, -0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot 0.001388888888888889, 0.5\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)))
   (if (<= t_0 -1e-307)
     (*
      (fma y (* y -0.16666666666666666) 1.0)
      (fma x (* x (fma (* x x) 0.041666666666666664 0.5)) 1.0))
     (if (<= t_0 0.9999999999709752)
       (*
        (fma 0.5 (* x x) 1.0)
        (fma
         (* y y)
         (fma (* y y) 0.008333333333333333 -0.16666666666666666)
         1.0))
       (*
        1.0
        (fma
         x
         (* x (fma (* x x) (* (* x x) 0.001388888888888889) 0.5))
         1.0))))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double tmp;
	if (t_0 <= -1e-307) {
		tmp = fma(y, (y * -0.16666666666666666), 1.0) * fma(x, (x * fma((x * x), 0.041666666666666664, 0.5)), 1.0);
	} else if (t_0 <= 0.9999999999709752) {
		tmp = fma(0.5, (x * x), 1.0) * fma((y * y), fma((y * y), 0.008333333333333333, -0.16666666666666666), 1.0);
	} else {
		tmp = 1.0 * fma(x, (x * fma((x * x), ((x * x) * 0.001388888888888889), 0.5)), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(y) / y)
	tmp = 0.0
	if (t_0 <= -1e-307)
		tmp = Float64(fma(y, Float64(y * -0.16666666666666666), 1.0) * fma(x, Float64(x * fma(Float64(x * x), 0.041666666666666664, 0.5)), 1.0));
	elseif (t_0 <= 0.9999999999709752)
		tmp = Float64(fma(0.5, Float64(x * x), 1.0) * fma(Float64(y * y), fma(Float64(y * y), 0.008333333333333333, -0.16666666666666666), 1.0));
	else
		tmp = Float64(1.0 * fma(x, Float64(x * fma(Float64(x * x), Float64(Float64(x * x) * 0.001388888888888889), 0.5)), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-307], N[(N[(y * N[(y * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.9999999999709752], N[(N[(0.5 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 * N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.001388888888888889), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 y) y) < -9.99999999999999909e-308

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 53.4

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

   5. Applied rewrites 53.4%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 51.8

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

   8. Applied rewrites 51.8%

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

   if -9.99999999999999909e-308 < (/.f64 (sin.f64 y) y) < 0.99999999997097522

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 69.5

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

   5. Applied rewrites 69.5%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 49.1

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

   8. Applied rewrites 49.1%

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

   if 0.99999999997097522 < (/.f64 (sin.f64 y) y)

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 91.4

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

   8. Applied rewrites 91.4%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 91.4%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq -1 \cdot 10^{-307}:\\ \;\;\;\;\mathsf{fma}\left(y, y \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right), 1\right)\\ \mathbf{elif}\;\frac{\sin y}{y} \leq 0.9999999999709752:\\ \;\;\;\;\mathsf{fma}\left(0.5, x \cdot x, 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.008333333333333333, -0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot 0.001388888888888889, 0.5\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 69.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ t_1 := \mathsf{fma}\left(0.5, x \cdot x, 1\right)\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-307}:\\ \;\;\;\;t\_1 \cdot \mathsf{fma}\left(y, y \cdot -0.16666666666666666, 1\right)\\ \mathbf{elif}\;t\_0 \leq 0.9999999999709752:\\ \;\;\;\;t\_1 \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.008333333333333333, -0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot 0.001388888888888889, 0.5\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)) (t_1 (fma 0.5 (* x x) 1.0)))
   (if (<= t_0 -1e-307)
     (* t_1 (fma y (* y -0.16666666666666666) 1.0))
     (if (<= t_0 0.9999999999709752)
       (*
        t_1
        (fma
         (* y y)
         (fma (* y y) 0.008333333333333333 -0.16666666666666666)
         1.0))
       (*
        1.0
        (fma
         x
         (* x (fma (* x x) (* (* x x) 0.001388888888888889) 0.5))
         1.0))))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double t_1 = fma(0.5, (x * x), 1.0);
	double tmp;
	if (t_0 <= -1e-307) {
		tmp = t_1 * fma(y, (y * -0.16666666666666666), 1.0);
	} else if (t_0 <= 0.9999999999709752) {
		tmp = t_1 * fma((y * y), fma((y * y), 0.008333333333333333, -0.16666666666666666), 1.0);
	} else {
		tmp = 1.0 * fma(x, (x * fma((x * x), ((x * x) * 0.001388888888888889), 0.5)), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(y) / y)
	t_1 = fma(0.5, Float64(x * x), 1.0)
	tmp = 0.0
	if (t_0 <= -1e-307)
		tmp = Float64(t_1 * fma(y, Float64(y * -0.16666666666666666), 1.0));
	elseif (t_0 <= 0.9999999999709752)
		tmp = Float64(t_1 * fma(Float64(y * y), fma(Float64(y * y), 0.008333333333333333, -0.16666666666666666), 1.0));
	else
		tmp = Float64(1.0 * fma(x, Float64(x * fma(Float64(x * x), Float64(Float64(x * x) * 0.001388888888888889), 0.5)), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-307], N[(t$95$1 * N[(y * N[(y * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.9999999999709752], N[(t$95$1 * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 * N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.001388888888888889), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 y) y) < -9.99999999999999909e-308

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 74.3

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

   5. Applied rewrites 74.3%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 48.7

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

   8. Applied rewrites 48.7%

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

   if -9.99999999999999909e-308 < (/.f64 (sin.f64 y) y) < 0.99999999997097522

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 69.5

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

   5. Applied rewrites 69.5%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 49.1

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

   8. Applied rewrites 49.1%

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

   if 0.99999999997097522 < (/.f64 (sin.f64 y) y)

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 91.4

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

   8. Applied rewrites 91.4%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 91.4%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq -1 \cdot 10^{-307}:\\ \;\;\;\;\mathsf{fma}\left(0.5, x \cdot x, 1\right) \cdot \mathsf{fma}\left(y, y \cdot -0.16666666666666666, 1\right)\\ \mathbf{elif}\;\frac{\sin y}{y} \leq 0.9999999999709752:\\ \;\;\;\;\mathsf{fma}\left(0.5, x \cdot x, 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.008333333333333333, -0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot 0.001388888888888889, 0.5\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 69.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-307}:\\ \;\;\;\;\mathsf{fma}\left(0.5, x \cdot x, 1\right) \cdot \mathsf{fma}\left(y, y \cdot -0.16666666666666666, 1\right)\\ \mathbf{elif}\;t\_0 \leq 10^{-86}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.008333333333333333, -0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot 0.001388888888888889, 0.5\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)))
   (if (<= t_0 -1e-307)
     (* (fma 0.5 (* x x) 1.0) (fma y (* y -0.16666666666666666) 1.0))
     (if (<= t_0 1e-86)
       (*
        1.0
        (fma
         y
         (* y (fma (* y y) 0.008333333333333333 -0.16666666666666666))
         1.0))
       (*
        1.0
        (fma
         x
         (* x (fma (* x x) (* (* x x) 0.001388888888888889) 0.5))
         1.0))))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double tmp;
	if (t_0 <= -1e-307) {
		tmp = fma(0.5, (x * x), 1.0) * fma(y, (y * -0.16666666666666666), 1.0);
	} else if (t_0 <= 1e-86) {
		tmp = 1.0 * fma(y, (y * fma((y * y), 0.008333333333333333, -0.16666666666666666)), 1.0);
	} else {
		tmp = 1.0 * fma(x, (x * fma((x * x), ((x * x) * 0.001388888888888889), 0.5)), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(y) / y)
	tmp = 0.0
	if (t_0 <= -1e-307)
		tmp = Float64(fma(0.5, Float64(x * x), 1.0) * fma(y, Float64(y * -0.16666666666666666), 1.0));
	elseif (t_0 <= 1e-86)
		tmp = Float64(1.0 * fma(y, Float64(y * fma(Float64(y * y), 0.008333333333333333, -0.16666666666666666)), 1.0));
	else
		tmp = Float64(1.0 * fma(x, Float64(x * fma(Float64(x * x), Float64(Float64(x * x) * 0.001388888888888889), 0.5)), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-307], N[(N[(0.5 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(y * N[(y * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e-86], N[(1.0 * N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 * N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.001388888888888889), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 y) y) < -9.99999999999999909e-308

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 74.3

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

   5. Applied rewrites 74.3%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 48.7

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

   8. Applied rewrites 48.7%

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

   if -9.99999999999999909e-308 < (/.f64 (sin.f64 y) y) < 1.00000000000000008e-86

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 52.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 3.7%

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

   9. Taylor expanded in y around 0

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

       1. +-commutative N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. lower-fma.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. sub-neg N/A

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

       7. *-commutative N/A

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

       8. metadata-eval N/A

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

       9. lower-fma.f64 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f64 51.0

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

   11. Applied rewrites 51.0%

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

   if 1.00000000000000008e-86 < (/.f64 (sin.f64 y) y)

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 94.9%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 86.5

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

   8. Applied rewrites 86.5%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 86.5%

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

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq -1 \cdot 10^{-307}:\\ \;\;\;\;\mathsf{fma}\left(0.5, x \cdot x, 1\right) \cdot \mathsf{fma}\left(y, y \cdot -0.16666666666666666, 1\right)\\ \mathbf{elif}\;\frac{\sin y}{y} \leq 10^{-86}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.008333333333333333, -0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot 0.001388888888888889, 0.5\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 67.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-307}:\\ \;\;\;\;\mathsf{fma}\left(0.5, x \cdot x, 1\right) \cdot \mathsf{fma}\left(y, y \cdot -0.16666666666666666, 1\right)\\ \mathbf{elif}\;t\_0 \leq 10^{-86}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.008333333333333333, -0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)))
   (if (<= t_0 -1e-307)
     (* (fma 0.5 (* x x) 1.0) (fma y (* y -0.16666666666666666) 1.0))
     (if (<= t_0 1e-86)
       (*
        1.0
        (fma
         y
         (* y (fma (* y y) 0.008333333333333333 -0.16666666666666666))
         1.0))
       (* 1.0 (fma x (* x (fma (* x x) 0.041666666666666664 0.5)) 1.0))))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double tmp;
	if (t_0 <= -1e-307) {
		tmp = fma(0.5, (x * x), 1.0) * fma(y, (y * -0.16666666666666666), 1.0);
	} else if (t_0 <= 1e-86) {
		tmp = 1.0 * fma(y, (y * fma((y * y), 0.008333333333333333, -0.16666666666666666)), 1.0);
	} else {
		tmp = 1.0 * fma(x, (x * fma((x * x), 0.041666666666666664, 0.5)), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(y) / y)
	tmp = 0.0
	if (t_0 <= -1e-307)
		tmp = Float64(fma(0.5, Float64(x * x), 1.0) * fma(y, Float64(y * -0.16666666666666666), 1.0));
	elseif (t_0 <= 1e-86)
		tmp = Float64(1.0 * fma(y, Float64(y * fma(Float64(y * y), 0.008333333333333333, -0.16666666666666666)), 1.0));
	else
		tmp = Float64(1.0 * fma(x, Float64(x * fma(Float64(x * x), 0.041666666666666664, 0.5)), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-307], N[(N[(0.5 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(y * N[(y * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e-86], N[(1.0 * N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 * N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 y) y) < -9.99999999999999909e-308

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 74.3

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

   5. Applied rewrites 74.3%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 48.7

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

   8. Applied rewrites 48.7%

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

   if -9.99999999999999909e-308 < (/.f64 (sin.f64 y) y) < 1.00000000000000008e-86

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 52.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 3.7%

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

   9. Taylor expanded in y around 0

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

       1. +-commutative N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. lower-fma.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. sub-neg N/A

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

       7. *-commutative N/A

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

       8. metadata-eval N/A

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

       9. lower-fma.f64 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f64 51.0

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

   11. Applied rewrites 51.0%

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

   if 1.00000000000000008e-86 < (/.f64 (sin.f64 y) y)

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 94.9%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 81.4

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

   8. Applied rewrites 81.4%

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

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq -1 \cdot 10^{-307}:\\ \;\;\;\;\mathsf{fma}\left(0.5, x \cdot x, 1\right) \cdot \mathsf{fma}\left(y, y \cdot -0.16666666666666666, 1\right)\\ \mathbf{elif}\;\frac{\sin y}{y} \leq 10^{-86}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.008333333333333333, -0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 61.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ t_1 := \mathsf{fma}\left(0.5, x \cdot x, 1\right)\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-307}:\\ \;\;\;\;t\_1 \cdot \mathsf{fma}\left(y, y \cdot -0.16666666666666666, 1\right)\\ \mathbf{elif}\;t\_0 \leq 10^{-86}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.008333333333333333, -0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)) (t_1 (fma 0.5 (* x x) 1.0)))
   (if (<= t_0 -1e-307)
     (* t_1 (fma y (* y -0.16666666666666666) 1.0))
     (if (<= t_0 1e-86)
       (*
        1.0
        (fma
         y
         (* y (fma (* y y) 0.008333333333333333 -0.16666666666666666))
         1.0))
       (* 1.0 t_1)))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double t_1 = fma(0.5, (x * x), 1.0);
	double tmp;
	if (t_0 <= -1e-307) {
		tmp = t_1 * fma(y, (y * -0.16666666666666666), 1.0);
	} else if (t_0 <= 1e-86) {
		tmp = 1.0 * fma(y, (y * fma((y * y), 0.008333333333333333, -0.16666666666666666)), 1.0);
	} else {
		tmp = 1.0 * t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(y) / y)
	t_1 = fma(0.5, Float64(x * x), 1.0)
	tmp = 0.0
	if (t_0 <= -1e-307)
		tmp = Float64(t_1 * fma(y, Float64(y * -0.16666666666666666), 1.0));
	elseif (t_0 <= 1e-86)
		tmp = Float64(1.0 * fma(y, Float64(y * fma(Float64(y * y), 0.008333333333333333, -0.16666666666666666)), 1.0));
	else
		tmp = Float64(1.0 * t_1);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-307], N[(t$95$1 * N[(y * N[(y * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e-86], N[(1.0 * N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 * t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 y) y) < -9.99999999999999909e-308

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 74.3

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

   5. Applied rewrites 74.3%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 48.7

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

   8. Applied rewrites 48.7%

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

   if -9.99999999999999909e-308 < (/.f64 (sin.f64 y) y) < 1.00000000000000008e-86

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 52.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 3.7%

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

   9. Taylor expanded in y around 0

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

       1. +-commutative N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. lower-fma.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. sub-neg N/A

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

       7. *-commutative N/A

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

       8. metadata-eval N/A

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

       9. lower-fma.f64 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f64 51.0

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

   11. Applied rewrites 51.0%

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

   if 1.00000000000000008e-86 < (/.f64 (sin.f64 y) y)

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 94.9%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 72.9

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

   8. Applied rewrites 72.9%

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

4. Final simplification 62.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq -1 \cdot 10^{-307}:\\ \;\;\;\;\mathsf{fma}\left(0.5, x \cdot x, 1\right) \cdot \mathsf{fma}\left(y, y \cdot -0.16666666666666666, 1\right)\\ \mathbf{elif}\;\frac{\sin y}{y} \leq 10^{-86}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.008333333333333333, -0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(0.5, x \cdot x, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 56.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-307}:\\ \;\;\;\;\mathsf{fma}\left(y, y \cdot -0.16666666666666666, 1\right) \cdot 1\\ \mathbf{elif}\;t\_0 \leq 10^{-86}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.008333333333333333, -0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(0.5, x \cdot x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)))
   (if (<= t_0 -1e-307)
     (* (fma y (* y -0.16666666666666666) 1.0) 1.0)
     (if (<= t_0 1e-86)
       (*
        1.0
        (fma
         y
         (* y (fma (* y y) 0.008333333333333333 -0.16666666666666666))
         1.0))
       (* 1.0 (fma 0.5 (* x x) 1.0))))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double tmp;
	if (t_0 <= -1e-307) {
		tmp = fma(y, (y * -0.16666666666666666), 1.0) * 1.0;
	} else if (t_0 <= 1e-86) {
		tmp = 1.0 * fma(y, (y * fma((y * y), 0.008333333333333333, -0.16666666666666666)), 1.0);
	} else {
		tmp = 1.0 * fma(0.5, (x * x), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(y) / y)
	tmp = 0.0
	if (t_0 <= -1e-307)
		tmp = Float64(fma(y, Float64(y * -0.16666666666666666), 1.0) * 1.0);
	elseif (t_0 <= 1e-86)
		tmp = Float64(1.0 * fma(y, Float64(y * fma(Float64(y * y), 0.008333333333333333, -0.16666666666666666)), 1.0));
	else
		tmp = Float64(1.0 * fma(0.5, Float64(x * x), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-307], N[(N[(y * N[(y * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[t$95$0, 1e-86], N[(1.0 * N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 * N[(0.5 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 y) y) < -9.99999999999999909e-308

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 1.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 1.4%

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

   9. Taylor expanded in y around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. lower-fma.f64 N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 32.6

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

   11. Applied rewrites 32.6%

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

   if -9.99999999999999909e-308 < (/.f64 (sin.f64 y) y) < 1.00000000000000008e-86

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 52.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 3.7%

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

   9. Taylor expanded in y around 0

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

       1. +-commutative N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. lower-fma.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. sub-neg N/A

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

       7. *-commutative N/A

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

       8. metadata-eval N/A

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

       9. lower-fma.f64 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f64 51.0

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

   11. Applied rewrites 51.0%

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

   if 1.00000000000000008e-86 < (/.f64 (sin.f64 y) y)

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 94.9%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 72.9

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

   8. Applied rewrites 72.9%

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

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq -1 \cdot 10^{-307}:\\ \;\;\;\;\mathsf{fma}\left(y, y \cdot -0.16666666666666666, 1\right) \cdot 1\\ \mathbf{elif}\;\frac{\sin y}{y} \leq 10^{-86}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.008333333333333333, -0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(0.5, x \cdot x, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 52.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -5 \cdot 10^{-148}:\\ \;\;\;\;\mathsf{fma}\left(y, y \cdot -0.16666666666666666, 1\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(0.5, x \cdot x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cosh x) (/ (sin y) y)) -5e-148)
   (* (fma y (* y -0.16666666666666666) 1.0) 1.0)
   (* 1.0 (fma 0.5 (* x x) 1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((cosh(x) * (sin(y) / y)) <= -5e-148) {
		tmp = fma(y, (y * -0.16666666666666666), 1.0) * 1.0;
	} else {
		tmp = 1.0 * fma(0.5, (x * x), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(cosh(x) * Float64(sin(y) / y)) <= -5e-148)
		tmp = Float64(fma(y, Float64(y * -0.16666666666666666), 1.0) * 1.0);
	else
		tmp = Float64(1.0 * fma(0.5, Float64(x * x), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Cosh[x], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], -5e-148], N[(N[(y * N[(y * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * 1.0), $MachinePrecision], N[(1.0 * N[(0.5 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < -4.9999999999999999e-148

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 0.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 1.1%

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

   9. Taylor expanded in y around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. lower-fma.f64 N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 39.5

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

   11. Applied rewrites 39.5%

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

   if -4.9999999999999999e-148 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y))

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 79.7%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 55.9

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

   8. Applied rewrites 55.9%

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

4. Final simplification 52.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -5 \cdot 10^{-148}:\\ \;\;\;\;\mathsf{fma}\left(y, y \cdot -0.16666666666666666, 1\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(0.5, x \cdot x, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 34.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -5 \cdot 10^{-148}:\\ \;\;\;\;1 \cdot \left(-0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cosh x) (/ (sin y) y)) -5e-148)
   (* 1.0 (* -0.16666666666666666 (* y y)))
   (* 1.0 1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((cosh(x) * (sin(y) / y)) <= -5e-148) {
		tmp = 1.0 * (-0.16666666666666666 * (y * y));
	} else {
		tmp = 1.0 * 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((cosh(x) * (sin(y) / y)) <= (-5d-148)) then
        tmp = 1.0d0 * ((-0.16666666666666666d0) * (y * y))
    else
        tmp = 1.0d0 * 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((Math.cosh(x) * (Math.sin(y) / y)) <= -5e-148) {
		tmp = 1.0 * (-0.16666666666666666 * (y * y));
	} else {
		tmp = 1.0 * 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (math.cosh(x) * (math.sin(y) / y)) <= -5e-148:
		tmp = 1.0 * (-0.16666666666666666 * (y * y))
	else:
		tmp = 1.0 * 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(cosh(x) * Float64(sin(y) / y)) <= -5e-148)
		tmp = Float64(1.0 * Float64(-0.16666666666666666 * Float64(y * y)));
	else
		tmp = Float64(1.0 * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((cosh(x) * (sin(y) / y)) <= -5e-148)
		tmp = 1.0 * (-0.16666666666666666 * (y * y));
	else
		tmp = 1.0 * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Cosh[x], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], -5e-148], N[(1.0 * N[(-0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < -4.9999999999999999e-148

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 0.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 1.1%

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

   9. Taylor expanded in y around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. lower-fma.f64 N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 39.5

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

   11. Applied rewrites 39.5%

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

   12. Taylor expanded in y around inf

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

   14. Applied rewrites 39.5%

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

   if -4.9999999999999999e-148 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y))

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 79.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 32.9%

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

4. Final simplification 34.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -5 \cdot 10^{-148}:\\ \;\;\;\;1 \cdot \left(-0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 33.4% accurate, 12.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in y around 0

   \[\leadsto \cosh x \cdot \color{blue}{1} \]
4. Step-by-step derivation

5. Applied rewrites 64.6%

   \[\leadsto \cosh x \cdot \color{blue}{1} \]

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1} \cdot 1 \]
7. Step-by-step derivation

8. Applied rewrites 26.8%

   \[\leadsto \color{blue}{1} \cdot 1 \]

9. Taylor expanded in y around 0

   \[\leadsto 1 \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {y}^{2}\right)} \]
10. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto 1 \cdot \color{blue}{\left(\frac{-1}{6} \cdot {y}^{2} + 1\right)} \]

    2. *-commutative N/A

       \[\leadsto 1 \cdot \left(\color{blue}{{y}^{2} \cdot \frac{-1}{6}} + 1\right) \]

    3. unpow2 N/A

       \[\leadsto 1 \cdot \left(\color{blue}{\left(y \cdot y\right)} \cdot \frac{-1}{6} + 1\right) \]

    4. associate-*l* N/A

       \[\leadsto 1 \cdot \left(\color{blue}{y \cdot \left(y \cdot \frac{-1}{6}\right)} + 1\right) \]

    5. *-commutative N/A

       \[\leadsto 1 \cdot \left(y \cdot \color{blue}{\left(\frac{-1}{6} \cdot y\right)} + 1\right) \]

    6. lower-fma.f64 N/A

       \[\leadsto 1 \cdot \color{blue}{\mathsf{fma}\left(y, \frac{-1}{6} \cdot y, 1\right)} \]

    7. *-commutative N/A

       \[\leadsto 1 \cdot \mathsf{fma}\left(y, \color{blue}{y \cdot \frac{-1}{6}}, 1\right) \]

    8. lower-*.f64 33.5

       \[\leadsto 1 \cdot \mathsf{fma}\left(y, \color{blue}{y \cdot -0.16666666666666666}, 1\right) \]

11. Applied rewrites 33.5%

    \[\leadsto 1 \cdot \color{blue}{\mathsf{fma}\left(y, y \cdot -0.16666666666666666, 1\right)} \]

12. Final simplification 33.5%

    \[\leadsto \mathsf{fma}\left(y, y \cdot -0.16666666666666666, 1\right) \cdot 1 \]
13. Add Preprocessing

Alternative 23: 27.2% accurate, 36.2× speedup

Math

\[\begin{array}{l} \\ 1 \cdot 1 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* 1.0 1.0))

C

double code(double x, double y) {
	return 1.0 * 1.0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 * 1.0d0
end function

Java

public static double code(double x, double y) {
	return 1.0 * 1.0;
}

Python

def code(x, y):
	return 1.0 * 1.0

Julia

function code(x, y)
	return Float64(1.0 * 1.0)
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0 * 1.0;
end

Wolfram

code[x_, y_] := N[(1.0 * 1.0), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\cosh x \cdot \frac{\sin y}{y} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \cosh x \cdot \color{blue}{1} \]
4. Step-by-step derivation

5. Applied rewrites 64.6%

   \[\leadsto \cosh x \cdot \color{blue}{1} \]

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1} \cdot 1 \]
7. Step-by-step derivation

8. Applied rewrites 26.8%

   \[\leadsto \color{blue}{1} \cdot 1 \]

9. Final simplification 26.8%

   \[\leadsto 1 \cdot 1 \]
10. Add Preprocessing

Developer Target 1: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\cosh x \cdot \sin y}{y} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ (* (cosh x) (sin y)) y))

C

double code(double x, double y) {
	return (cosh(x) * sin(y)) / y;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (cosh(x) * sin(y)) / y
end function

Java

public static double code(double x, double y) {
	return (Math.cosh(x) * Math.sin(y)) / y;
}

Python

def code(x, y):
	return (math.cosh(x) * math.sin(y)) / y

Julia

function code(x, y)
	return Float64(Float64(cosh(x) * sin(y)) / y)
end

MATLAB

function tmp = code(x, y)
	tmp = (cosh(x) * sin(y)) / y;
end

Wolfram

code[x_, y_] := N[(N[(N[Cosh[x], $MachinePrecision] * N[Sin[y], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]

Reproduce

herbie shell --seed 2024221 
(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)))