Linear.Quaternion:$csinh from linear-1.19.1.3

Percentage Accurate: 99.9% → 98.7%

Time: 7.1s

Alternatives: 15

Speedup: 1.0×

• 50.4% 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: 98.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ t_1 := \cosh x \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{elif}\;t\_1 \leq 0.995:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot \mathsf{fma}\left(0.008333333333333333 \cdot \left(y \cdot y\right) - 0.16666666666666666, y \cdot y, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)) (t_1 (* (cosh x) t_0)))
   (if (<= t_1 (- INFINITY))
     (*
      (fma (fma 0.041666666666666664 (* x x) 0.5) (* x x) 1.0)
      (fma -0.16666666666666666 (* y y) 1.0))
     (if (<= t_1 0.995)
       t_0
       (*
        (cosh x)
        (fma
         (- (* 0.008333333333333333 (* y y)) 0.16666666666666666)
         (* y y)
         1.0))))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double t_1 = cosh(x) * t_0;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma(fma(0.041666666666666664, (x * x), 0.5), (x * x), 1.0) * fma(-0.16666666666666666, (y * y), 1.0);
	} else if (t_1 <= 0.995) {
		tmp = t_0;
	} else {
		tmp = cosh(x) * fma(((0.008333333333333333 * (y * y)) - 0.16666666666666666), (y * y), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(y) / y)
	t_1 = Float64(cosh(x) * t_0)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(fma(fma(0.041666666666666664, Float64(x * x), 0.5), Float64(x * x), 1.0) * fma(-0.16666666666666666, Float64(y * y), 1.0));
	elseif (t_1 <= 0.995)
		tmp = t_0;
	else
		tmp = Float64(cosh(x) * fma(Float64(Float64(0.008333333333333333 * Float64(y * y)) - 0.16666666666666666), Float64(y * y), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cosh[x], $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(0.041666666666666664 * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.995], t$95$0, N[(N[Cosh[x], $MachinePrecision] * N[(N[(N[(0.008333333333333333 * N[(y * y), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 100.0

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

   8. Applied rewrites 100.0%

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

   if -inf.0 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < 0.994999999999999996

   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 99.6

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

   5. Applied rewrites 99.6%

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

   if 0.994999999999999996 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

Alternative 2: 69.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)\\ t_1 := \frac{\sin y}{y}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-293}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot t\_0\\ \mathbf{elif}\;t\_1 \leq 0.9999995:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, 0.5, 1\right) \cdot \mathsf{fma}\left(0.008333333333333333 \cdot \left(y \cdot y\right) - 0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right) \cdot x, x, 0.5\right), x \cdot x, 1\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma -0.16666666666666666 (* y y) 1.0)) (t_1 (/ (sin y) y)))
   (if (<= t_1 -1e-293)
     (* (fma (fma 0.041666666666666664 (* x x) 0.5) (* x x) 1.0) t_0)
     (if (<= t_1 0.9999995)
       (*
        (fma (* x x) 0.5 1.0)
        (fma
         (- (* 0.008333333333333333 (* y y)) 0.16666666666666666)
         (* y y)
         1.0))
       (*
        (fma
         (fma
          (* (fma (* x x) 0.001388888888888889 0.041666666666666664) x)
          x
          0.5)
         (* x x)
         1.0)
        t_0)))))

C

double code(double x, double y) {
	double t_0 = fma(-0.16666666666666666, (y * y), 1.0);
	double t_1 = sin(y) / y;
	double tmp;
	if (t_1 <= -1e-293) {
		tmp = fma(fma(0.041666666666666664, (x * x), 0.5), (x * x), 1.0) * t_0;
	} else if (t_1 <= 0.9999995) {
		tmp = fma((x * x), 0.5, 1.0) * fma(((0.008333333333333333 * (y * y)) - 0.16666666666666666), (y * y), 1.0);
	} else {
		tmp = fma(fma((fma((x * x), 0.001388888888888889, 0.041666666666666664) * x), x, 0.5), (x * x), 1.0) * t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(-0.16666666666666666, Float64(y * y), 1.0)
	t_1 = Float64(sin(y) / y)
	tmp = 0.0
	if (t_1 <= -1e-293)
		tmp = Float64(fma(fma(0.041666666666666664, Float64(x * x), 0.5), Float64(x * x), 1.0) * t_0);
	elseif (t_1 <= 0.9999995)
		tmp = Float64(fma(Float64(x * x), 0.5, 1.0) * fma(Float64(Float64(0.008333333333333333 * Float64(y * y)) - 0.16666666666666666), Float64(y * y), 1.0));
	else
		tmp = Float64(fma(fma(Float64(fma(Float64(x * x), 0.001388888888888889, 0.041666666666666664) * x), x, 0.5), Float64(x * x), 1.0) * t_0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-293], N[(N[(N[(0.041666666666666664 * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[t$95$1, 0.9999995], N[(N[(N[(x * x), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] * N[(N[(N[(0.008333333333333333 * N[(y * y), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(x * x), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision] * x), $MachinePrecision] * x + 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 y) y) < -1.0000000000000001e-293

   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. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 61.3

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

   5. Applied rewrites 61.3%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 61.3

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

   8. Applied rewrites 61.3%

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

   if -1.0000000000000001e-293 < (/.f64 (sin.f64 y) y) < 0.999999500000000041

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 47.5

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

   5. Applied rewrites 47.5%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 47.5

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

   8. Applied rewrites 47.5%

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

   if 0.999999500000000041 < (/.f64 (sin.f64 y) y)

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 93.5

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

   8. Applied rewrites 93.5%

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

   10. Applied rewrites 93.5%

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

Alternative 3: 69.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)\\ t_1 := \frac{\sin y}{y}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-293}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot t\_0\\ \mathbf{elif}\;t\_1 \leq 0.9999995:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, 0.5, 1\right) \cdot \mathsf{fma}\left(0.008333333333333333 \cdot \left(y \cdot y\right) - 0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889 \cdot \left(x \cdot x\right), x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma -0.16666666666666666 (* y y) 1.0)) (t_1 (/ (sin y) y)))
   (if (<= t_1 -1e-293)
     (* (fma (fma 0.041666666666666664 (* x x) 0.5) (* x x) 1.0) t_0)
     (if (<= t_1 0.9999995)
       (*
        (fma (* x x) 0.5 1.0)
        (fma
         (- (* 0.008333333333333333 (* y y)) 0.16666666666666666)
         (* y y)
         1.0))
       (*
        (fma (fma (* 0.001388888888888889 (* x x)) (* x x) 0.5) (* x x) 1.0)
        t_0)))))

C

double code(double x, double y) {
	double t_0 = fma(-0.16666666666666666, (y * y), 1.0);
	double t_1 = sin(y) / y;
	double tmp;
	if (t_1 <= -1e-293) {
		tmp = fma(fma(0.041666666666666664, (x * x), 0.5), (x * x), 1.0) * t_0;
	} else if (t_1 <= 0.9999995) {
		tmp = fma((x * x), 0.5, 1.0) * fma(((0.008333333333333333 * (y * y)) - 0.16666666666666666), (y * y), 1.0);
	} else {
		tmp = fma(fma((0.001388888888888889 * (x * x)), (x * x), 0.5), (x * x), 1.0) * t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(-0.16666666666666666, Float64(y * y), 1.0)
	t_1 = Float64(sin(y) / y)
	tmp = 0.0
	if (t_1 <= -1e-293)
		tmp = Float64(fma(fma(0.041666666666666664, Float64(x * x), 0.5), Float64(x * x), 1.0) * t_0);
	elseif (t_1 <= 0.9999995)
		tmp = Float64(fma(Float64(x * x), 0.5, 1.0) * fma(Float64(Float64(0.008333333333333333 * Float64(y * y)) - 0.16666666666666666), Float64(y * y), 1.0));
	else
		tmp = Float64(fma(fma(Float64(0.001388888888888889 * Float64(x * x)), Float64(x * x), 0.5), Float64(x * x), 1.0) * t_0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-293], N[(N[(N[(0.041666666666666664 * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[t$95$1, 0.9999995], N[(N[(N[(x * x), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] * N[(N[(N[(0.008333333333333333 * N[(y * y), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.001388888888888889 * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 y) y) < -1.0000000000000001e-293

   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. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 61.3

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

   5. Applied rewrites 61.3%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 61.3

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

   8. Applied rewrites 61.3%

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

   if -1.0000000000000001e-293 < (/.f64 (sin.f64 y) y) < 0.999999500000000041

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 47.5

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

   5. Applied rewrites 47.5%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 47.5

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

   8. Applied rewrites 47.5%

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

   if 0.999999500000000041 < (/.f64 (sin.f64 y) y)

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 93.5

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

   8. Applied rewrites 93.5%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 93.5%

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

Alternative 4: 67.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-293} \lor \neg \left(t\_0 \leq 0.9999995\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, 0.5, 1\right) \cdot \mathsf{fma}\left(0.008333333333333333 \cdot \left(y \cdot y\right) - 0.16666666666666666, y \cdot y, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)))
   (if (or (<= t_0 -1e-293) (not (<= t_0 0.9999995)))
     (*
      (fma (fma 0.041666666666666664 (* x x) 0.5) (* x x) 1.0)
      (fma -0.16666666666666666 (* y y) 1.0))
     (*
      (fma (* x x) 0.5 1.0)
      (fma
       (- (* 0.008333333333333333 (* y y)) 0.16666666666666666)
       (* y y)
       1.0)))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double tmp;
	if ((t_0 <= -1e-293) || !(t_0 <= 0.9999995)) {
		tmp = fma(fma(0.041666666666666664, (x * x), 0.5), (x * x), 1.0) * fma(-0.16666666666666666, (y * y), 1.0);
	} else {
		tmp = fma((x * x), 0.5, 1.0) * fma(((0.008333333333333333 * (y * y)) - 0.16666666666666666), (y * y), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(y) / y)
	tmp = 0.0
	if ((t_0 <= -1e-293) || !(t_0 <= 0.9999995))
		tmp = Float64(fma(fma(0.041666666666666664, Float64(x * x), 0.5), Float64(x * x), 1.0) * fma(-0.16666666666666666, Float64(y * y), 1.0));
	else
		tmp = Float64(fma(Float64(x * x), 0.5, 1.0) * fma(Float64(Float64(0.008333333333333333 * Float64(y * y)) - 0.16666666666666666), Float64(y * y), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -1e-293], N[Not[LessEqual[t$95$0, 0.9999995]], $MachinePrecision]], N[(N[(N[(0.041666666666666664 * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * x), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] * N[(N[(N[(0.008333333333333333 * N[(y * y), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 y) y) < -1.0000000000000001e-293 or 0.999999500000000041 < (/.f64 (sin.f64 y) y)

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 90.0

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

   5. Applied rewrites 90.0%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 81.6

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

   8. Applied rewrites 81.6%

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

   if -1.0000000000000001e-293 < (/.f64 (sin.f64 y) y) < 0.999999500000000041

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 47.5

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

   5. Applied rewrites 47.5%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 47.5

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

   8. Applied rewrites 47.5%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq -1 \cdot 10^{-293} \lor \neg \left(\frac{\sin y}{y} \leq 0.9999995\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, 0.5, 1\right) \cdot \mathsf{fma}\left(0.008333333333333333 \cdot \left(y \cdot y\right) - 0.16666666666666666, y \cdot y, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ \mathbf{if}\;t\_0 \leq 10^{-300} \lor \neg \left(t\_0 \leq 5 \cdot 10^{-18}\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot 0.5\right) \cdot \mathsf{fma}\left(0.008333333333333333 \cdot \left(y \cdot y\right) - 0.16666666666666666, y \cdot y, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)))
   (if (or (<= t_0 1e-300) (not (<= t_0 5e-18)))
     (*
      (fma (fma 0.041666666666666664 (* x x) 0.5) (* x x) 1.0)
      (fma -0.16666666666666666 (* y y) 1.0))
     (*
      (* (* x x) 0.5)
      (fma
       (- (* 0.008333333333333333 (* y y)) 0.16666666666666666)
       (* y y)
       1.0)))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double tmp;
	if ((t_0 <= 1e-300) || !(t_0 <= 5e-18)) {
		tmp = fma(fma(0.041666666666666664, (x * x), 0.5), (x * x), 1.0) * fma(-0.16666666666666666, (y * y), 1.0);
	} else {
		tmp = ((x * x) * 0.5) * fma(((0.008333333333333333 * (y * y)) - 0.16666666666666666), (y * y), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(y) / y)
	tmp = 0.0
	if ((t_0 <= 1e-300) || !(t_0 <= 5e-18))
		tmp = Float64(fma(fma(0.041666666666666664, Float64(x * x), 0.5), Float64(x * x), 1.0) * fma(-0.16666666666666666, Float64(y * y), 1.0));
	else
		tmp = Float64(Float64(Float64(x * x) * 0.5) * fma(Float64(Float64(0.008333333333333333 * Float64(y * y)) - 0.16666666666666666), Float64(y * y), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, If[Or[LessEqual[t$95$0, 1e-300], N[Not[LessEqual[t$95$0, 5e-18]], $MachinePrecision]], N[(N[(N[(0.041666666666666664 * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[(N[(0.008333333333333333 * N[(y * y), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 y) y) < 1.00000000000000003e-300 or 5.00000000000000036e-18 < (/.f64 (sin.f64 y) y)

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 88.5

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

   5. Applied rewrites 88.5%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 80.4

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

   8. Applied rewrites 80.4%

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

   if 1.00000000000000003e-300 < (/.f64 (sin.f64 y) y) < 5.00000000000000036e-18

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 48.0

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

   5. Applied rewrites 48.0%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 48.0

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

   8. Applied rewrites 48.0%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 47.7%

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 10^{-300} \lor \neg \left(\frac{\sin y}{y} \leq 5 \cdot 10^{-18}\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot 0.5\right) \cdot \mathsf{fma}\left(0.008333333333333333 \cdot \left(y \cdot y\right) - 0.16666666666666666, y \cdot y, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 65.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-293} \lor \neg \left(t\_0 \leq 0.9999995\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.008333333333333333 \cdot \left(y \cdot y\right) - 0.16666666666666666, y \cdot y, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)))
   (if (or (<= t_0 -1e-293) (not (<= t_0 0.9999995)))
     (*
      (fma (fma 0.041666666666666664 (* x x) 0.5) (* x x) 1.0)
      (fma -0.16666666666666666 (* y y) 1.0))
     (fma
      (- (* 0.008333333333333333 (* y y)) 0.16666666666666666)
      (* y y)
      1.0))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double tmp;
	if ((t_0 <= -1e-293) || !(t_0 <= 0.9999995)) {
		tmp = fma(fma(0.041666666666666664, (x * x), 0.5), (x * x), 1.0) * fma(-0.16666666666666666, (y * y), 1.0);
	} else {
		tmp = fma(((0.008333333333333333 * (y * y)) - 0.16666666666666666), (y * y), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(y) / y)
	tmp = 0.0
	if ((t_0 <= -1e-293) || !(t_0 <= 0.9999995))
		tmp = Float64(fma(fma(0.041666666666666664, Float64(x * x), 0.5), Float64(x * x), 1.0) * fma(-0.16666666666666666, Float64(y * y), 1.0));
	else
		tmp = fma(Float64(Float64(0.008333333333333333 * Float64(y * y)) - 0.16666666666666666), Float64(y * y), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -1e-293], N[Not[LessEqual[t$95$0, 0.9999995]], $MachinePrecision]], N[(N[(N[(0.041666666666666664 * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.008333333333333333 * N[(y * y), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 y) y) < -1.0000000000000001e-293 or 0.999999500000000041 < (/.f64 (sin.f64 y) y)

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 90.0

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

   5. Applied rewrites 90.0%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 81.6

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

   8. Applied rewrites 81.6%

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

   if -1.0000000000000001e-293 < (/.f64 (sin.f64 y) y) < 0.999999500000000041

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 57.9

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

   5. Applied rewrites 57.9%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 41.5%

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

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq -1 \cdot 10^{-293} \lor \neg \left(\frac{\sin y}{y} \leq 0.9999995\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.008333333333333333 \cdot \left(y \cdot y\right) - 0.16666666666666666, y \cdot y, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 57.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-293} \lor \neg \left(t\_0 \leq 0.9999995\right):\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, 0.5, 1\right) \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.008333333333333333 \cdot \left(y \cdot y\right) - 0.16666666666666666, y \cdot y, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)))
   (if (or (<= t_0 -1e-293) (not (<= t_0 0.9999995)))
     (* (fma (* x x) 0.5 1.0) (fma -0.16666666666666666 (* y y) 1.0))
     (fma
      (- (* 0.008333333333333333 (* y y)) 0.16666666666666666)
      (* y y)
      1.0))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double tmp;
	if ((t_0 <= -1e-293) || !(t_0 <= 0.9999995)) {
		tmp = fma((x * x), 0.5, 1.0) * fma(-0.16666666666666666, (y * y), 1.0);
	} else {
		tmp = fma(((0.008333333333333333 * (y * y)) - 0.16666666666666666), (y * y), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(y) / y)
	tmp = 0.0
	if ((t_0 <= -1e-293) || !(t_0 <= 0.9999995))
		tmp = Float64(fma(Float64(x * x), 0.5, 1.0) * fma(-0.16666666666666666, Float64(y * y), 1.0));
	else
		tmp = fma(Float64(Float64(0.008333333333333333 * Float64(y * y)) - 0.16666666666666666), Float64(y * y), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -1e-293], N[Not[LessEqual[t$95$0, 0.9999995]], $MachinePrecision]], N[(N[(N[(x * x), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] * N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.008333333333333333 * N[(y * y), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 y) y) < -1.0000000000000001e-293 or 0.999999500000000041 < (/.f64 (sin.f64 y) y)

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 90.0

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

   5. Applied rewrites 90.0%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 73.3

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

   8. Applied rewrites 73.3%

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

   if -1.0000000000000001e-293 < (/.f64 (sin.f64 y) y) < 0.999999500000000041

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 57.9

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

   5. Applied rewrites 57.9%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 41.5%

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

4. Final simplification 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq -1 \cdot 10^{-293} \lor \neg \left(\frac{\sin y}{y} \leq 0.9999995\right):\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, 0.5, 1\right) \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.008333333333333333 \cdot \left(y \cdot y\right) - 0.16666666666666666, y \cdot y, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 41.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -1 \cdot 10^{-278}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.008333333333333333 \cdot \left(y \cdot y\right) - 0.16666666666666666, y \cdot y, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cosh x) (/ (sin y) y)) -1e-278)
   (* -0.16666666666666666 (* y y))
   (fma (- (* 0.008333333333333333 (* y y)) 0.16666666666666666) (* y y) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((cosh(x) * (sin(y) / y)) <= -1e-278) {
		tmp = -0.16666666666666666 * (y * y);
	} else {
		tmp = fma(((0.008333333333333333 * (y * y)) - 0.16666666666666666), (y * y), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(cosh(x) * Float64(sin(y) / y)) <= -1e-278)
		tmp = Float64(-0.16666666666666666 * Float64(y * y));
	else
		tmp = fma(Float64(Float64(0.008333333333333333 * Float64(y * y)) - 0.16666666666666666), Float64(y * y), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Cosh[x], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], -1e-278], N[(-0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.008333333333333333 * N[(y * y), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < -9.99999999999999938e-279

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 41.1

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

   5. Applied rewrites 41.1%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 36.8%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 36.8%

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

   if -9.99999999999999938e-279 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y))

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 51.4

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

   5. Applied rewrites 51.4%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 46.4%

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

Alternative 9: 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 10: 70.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ \mathbf{if}\;x \leq 2.9 \cdot 10^{-24}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.62 \cdot 10^{+153}:\\ \;\;\;\;\cosh x \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, 0.5, 1\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)))
   (if (<= x 2.9e-24)
     t_0
     (if (<= x 1.62e+153)
       (* (cosh x) (fma -0.16666666666666666 (* y y) 1.0))
       (* (fma (* x x) 0.5 1.0) t_0)))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double tmp;
	if (x <= 2.9e-24) {
		tmp = t_0;
	} else if (x <= 1.62e+153) {
		tmp = cosh(x) * fma(-0.16666666666666666, (y * y), 1.0);
	} else {
		tmp = fma((x * x), 0.5, 1.0) * t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(y) / y)
	tmp = 0.0
	if (x <= 2.9e-24)
		tmp = t_0;
	elseif (x <= 1.62e+153)
		tmp = Float64(cosh(x) * fma(-0.16666666666666666, Float64(y * y), 1.0));
	else
		tmp = Float64(fma(Float64(x * x), 0.5, 1.0) * t_0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[x, 2.9e-24], t$95$0, If[LessEqual[x, 1.62e+153], N[(N[Cosh[x], $MachinePrecision] * N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * x), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < 2.8999999999999999e-24

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 63.2

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

   5. Applied rewrites 63.2%

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

   if 2.8999999999999999e-24 < x < 1.62e153

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 92.9

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

   5. Applied rewrites 92.9%

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

   if 1.62e153 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

Alternative 11: 67.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.9 \cdot 10^{-24}:\\ \;\;\;\;\frac{\sin y}{y}\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{+214}:\\ \;\;\;\;\cosh x \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot 0.5\right) \cdot \mathsf{fma}\left(0.008333333333333333 \cdot \left(y \cdot y\right) - 0.16666666666666666, y \cdot y, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 2.9e-24)
   (/ (sin y) y)
   (if (<= x 2.5e+214)
     (* (cosh x) (fma -0.16666666666666666 (* y y) 1.0))
     (*
      (* (* x x) 0.5)
      (fma
       (- (* 0.008333333333333333 (* y y)) 0.16666666666666666)
       (* y y)
       1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 2.9e-24) {
		tmp = sin(y) / y;
	} else if (x <= 2.5e+214) {
		tmp = cosh(x) * fma(-0.16666666666666666, (y * y), 1.0);
	} else {
		tmp = ((x * x) * 0.5) * fma(((0.008333333333333333 * (y * y)) - 0.16666666666666666), (y * y), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 2.9e-24)
		tmp = Float64(sin(y) / y);
	elseif (x <= 2.5e+214)
		tmp = Float64(cosh(x) * fma(-0.16666666666666666, Float64(y * y), 1.0));
	else
		tmp = Float64(Float64(Float64(x * x) * 0.5) * fma(Float64(Float64(0.008333333333333333 * Float64(y * y)) - 0.16666666666666666), Float64(y * y), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 2.9e-24], N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], If[LessEqual[x, 2.5e+214], N[(N[Cosh[x], $MachinePrecision] * N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[(N[(0.008333333333333333 * N[(y * y), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 2.8999999999999999e-24

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 63.2

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

   5. Applied rewrites 63.2%

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

   if 2.8999999999999999e-24 < x < 2.49999999999999977e214

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 90.9

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

   5. Applied rewrites 90.9%

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

   if 2.49999999999999977e214 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 85.7

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

   5. Applied rewrites 85.7%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 85.7

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

   8. Applied rewrites 85.7%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 85.7%

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

Alternative 12: 65.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.9 \cdot 10^{-24}:\\ \;\;\;\;\frac{\sin y}{y}\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{+214}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right) \cdot x, x, 0.5\right), x \cdot x, 1\right) \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot 0.5\right) \cdot \mathsf{fma}\left(0.008333333333333333 \cdot \left(y \cdot y\right) - 0.16666666666666666, y \cdot y, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 2.9e-24)
   (/ (sin y) y)
   (if (<= x 2.5e+214)
     (*
      (fma
       (fma
        (* (fma (* x x) 0.001388888888888889 0.041666666666666664) x)
        x
        0.5)
       (* x x)
       1.0)
      (fma -0.16666666666666666 (* y y) 1.0))
     (*
      (* (* x x) 0.5)
      (fma
       (- (* 0.008333333333333333 (* y y)) 0.16666666666666666)
       (* y y)
       1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 2.9e-24) {
		tmp = sin(y) / y;
	} else if (x <= 2.5e+214) {
		tmp = fma(fma((fma((x * x), 0.001388888888888889, 0.041666666666666664) * x), x, 0.5), (x * x), 1.0) * fma(-0.16666666666666666, (y * y), 1.0);
	} else {
		tmp = ((x * x) * 0.5) * fma(((0.008333333333333333 * (y * y)) - 0.16666666666666666), (y * y), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 2.9e-24)
		tmp = Float64(sin(y) / y);
	elseif (x <= 2.5e+214)
		tmp = Float64(fma(fma(Float64(fma(Float64(x * x), 0.001388888888888889, 0.041666666666666664) * x), x, 0.5), Float64(x * x), 1.0) * fma(-0.16666666666666666, Float64(y * y), 1.0));
	else
		tmp = Float64(Float64(Float64(x * x) * 0.5) * fma(Float64(Float64(0.008333333333333333 * Float64(y * y)) - 0.16666666666666666), Float64(y * y), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 2.9e-24], N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], If[LessEqual[x, 2.5e+214], N[(N[(N[(N[(N[(N[(x * x), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision] * x), $MachinePrecision] * x + 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[(N[(0.008333333333333333 * N[(y * y), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 2.8999999999999999e-24

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 63.2

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

   5. Applied rewrites 63.2%

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

   if 2.8999999999999999e-24 < x < 2.49999999999999977e214

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 90.9

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

   5. Applied rewrites 90.9%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 80.3

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

   8. Applied rewrites 80.3%

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

   10. Applied rewrites 80.3%

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

   if 2.49999999999999977e214 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 85.7

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

   5. Applied rewrites 85.7%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 85.7

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

   8. Applied rewrites 85.7%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 85.7%

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

Alternative 13: 41.6% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.2 \cdot 10^{+40}:\\ \;\;\;\;\mathsf{fma}\left(0.008333333333333333 \cdot \left(y \cdot y\right) - 0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot 0.5\right) \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 4.2e+40)
   (fma (- (* 0.008333333333333333 (* y y)) 0.16666666666666666) (* y y) 1.0)
   (* (* (* x x) 0.5) (fma -0.16666666666666666 (* y y) 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 4.2e+40) {
		tmp = fma(((0.008333333333333333 * (y * y)) - 0.16666666666666666), (y * y), 1.0);
	} else {
		tmp = ((x * x) * 0.5) * fma(-0.16666666666666666, (y * y), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 4.2e+40)
		tmp = fma(Float64(Float64(0.008333333333333333 * Float64(y * y)) - 0.16666666666666666), Float64(y * y), 1.0);
	else
		tmp = Float64(Float64(Float64(x * x) * 0.5) * fma(-0.16666666666666666, Float64(y * y), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 4.2e+40], N[(N[(N[(0.008333333333333333 * N[(y * y), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision] * N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.2000000000000002e40

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 62.7

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

   5. Applied rewrites 62.7%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 43.6%

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

   if 4.2000000000000002e40 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 84.2

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

   5. Applied rewrites 84.2%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 62.7

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

   8. Applied rewrites 62.7%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 62.7%

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

Alternative 14: 31.6% accurate, 18.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in x around 0

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

   1. lower-/.f64 N/A

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

   2. lower-sin.f64 49.4

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

5. Applied rewrites 49.4%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 35.1%

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

Alternative 15: 8.1% accurate, 19.7× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (-0.16666666666666666d0) * (y * y)
end function

Java

public static double code(double x, double y) {
	return -0.16666666666666666 * (y * y);
}

Python

def code(x, y):
	return -0.16666666666666666 * (y * y)

Julia

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

MATLAB

function tmp = code(x, y)
	tmp = -0.16666666666666666 * (y * y);
end

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in x around 0

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

   1. lower-/.f64 N/A

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

   2. lower-sin.f64 49.4

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

5. Applied rewrites 49.4%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 35.1%

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

9. Taylor expanded in y around inf

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

11. Applied rewrites 8.5%

    \[\leadsto -0.16666666666666666 \cdot \left(y \cdot \color{blue}{y}\right) \]
12. Add Preprocessing

Developer Target 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024326 
(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)))