Linear.Quaternion:$csinh from linear-1.19.1.3

Percentage Accurate: 99.9% → 99.4%

Time: 7.5s

Alternatives: 17

Speedup: 1.0×

• 49.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ t_1 := t\_0 \cdot \cosh x\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right) \cdot \cosh x\\ \mathbf{elif}\;t\_1 \leq 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), y \cdot y, 1\right) \cdot \cosh x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)) (t_1 (* t_0 (cosh x))))
   (if (<= t_1 (- INFINITY))
     (* (fma -0.16666666666666666 (* y y) 1.0) (cosh x))
     (if (<= t_1 1e-7)
       (*
        (fma
         (fma
          (fma 0.001388888888888889 (* x x) 0.041666666666666664)
          (* x x)
          0.5)
         (* x x)
         1.0)
        t_0)
       (*
        (fma
         (fma 0.008333333333333333 (* y y) -0.16666666666666666)
         (* y y)
         1.0)
        (cosh x))))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double t_1 = t_0 * cosh(x);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma(-0.16666666666666666, (y * y), 1.0) * cosh(x);
	} else if (t_1 <= 1e-7) {
		tmp = fma(fma(fma(0.001388888888888889, (x * x), 0.041666666666666664), (x * x), 0.5), (x * x), 1.0) * t_0;
	} else {
		tmp = fma(fma(0.008333333333333333, (y * y), -0.16666666666666666), (y * y), 1.0) * cosh(x);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(y) / y)
	t_1 = Float64(t_0 * cosh(x))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(fma(-0.16666666666666666, Float64(y * y), 1.0) * cosh(x));
	elseif (t_1 <= 1e-7)
		tmp = Float64(fma(fma(fma(0.001388888888888889, Float64(x * x), 0.041666666666666664), Float64(x * x), 0.5), Float64(x * x), 1.0) * t_0);
	else
		tmp = Float64(fma(fma(0.008333333333333333, Float64(y * y), -0.16666666666666666), Float64(y * y), 1.0) * cosh(x));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[Cosh[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * N[Cosh[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e-7], N[(N[(N[(N[(0.001388888888888889 * N[(x * x), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(N[(0.008333333333333333 * N[(y * y), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * N[Cosh[x], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -inf.0 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < 9.9999999999999995e-8

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 99.7

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

   5. Applied rewrites 99.7%

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

   if 9.9999999999999995e-8 < (*.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. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \cdot \cosh x \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right) \cdot \cosh x\\ \mathbf{elif}\;\frac{\sin y}{y} \cdot \cosh x \leq 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot \frac{\sin y}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), y \cdot y, 1\right) \cdot \cosh x\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ t_1 := t\_0 \cdot \cosh x\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right) \cdot \cosh x\\ \mathbf{elif}\;t\_1 \leq 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), y \cdot y, 1\right) \cdot \cosh x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)) (t_1 (* t_0 (cosh x))))
   (if (<= t_1 (- INFINITY))
     (* (fma -0.16666666666666666 (* y y) 1.0) (cosh x))
     (if (<= t_1 1e-7)
       (* (fma (fma 0.041666666666666664 (* x x) 0.5) (* x x) 1.0) t_0)
       (*
        (fma
         (fma 0.008333333333333333 (* y y) -0.16666666666666666)
         (* y y)
         1.0)
        (cosh x))))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double t_1 = t_0 * cosh(x);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma(-0.16666666666666666, (y * y), 1.0) * cosh(x);
	} else if (t_1 <= 1e-7) {
		tmp = fma(fma(0.041666666666666664, (x * x), 0.5), (x * x), 1.0) * t_0;
	} else {
		tmp = fma(fma(0.008333333333333333, (y * y), -0.16666666666666666), (y * y), 1.0) * cosh(x);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(y) / y)
	t_1 = Float64(t_0 * cosh(x))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(fma(-0.16666666666666666, Float64(y * y), 1.0) * cosh(x));
	elseif (t_1 <= 1e-7)
		tmp = Float64(fma(fma(0.041666666666666664, Float64(x * x), 0.5), Float64(x * x), 1.0) * t_0);
	else
		tmp = Float64(fma(fma(0.008333333333333333, Float64(y * y), -0.16666666666666666), Float64(y * y), 1.0) * cosh(x));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[Cosh[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * N[Cosh[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e-7], N[(N[(N[(0.041666666666666664 * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(N[(0.008333333333333333 * N[(y * y), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * N[Cosh[x], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -inf.0 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < 9.9999999999999995e-8

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 99.5

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

   5. Applied rewrites 99.5%

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

   if 9.9999999999999995e-8 < (*.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. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \cdot \cosh x \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right) \cdot \cosh x\\ \mathbf{elif}\;\frac{\sin y}{y} \cdot \cosh x \leq 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot \frac{\sin y}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), y \cdot y, 1\right) \cdot \cosh x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ t_1 := t\_0 \cdot \cosh x\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right) \cdot \cosh x\\ \mathbf{elif}\;t\_1 \leq 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, 0.5, 1\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), y \cdot y, 1\right) \cdot \cosh x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)) (t_1 (* t_0 (cosh x))))
   (if (<= t_1 (- INFINITY))
     (* (fma -0.16666666666666666 (* y y) 1.0) (cosh x))
     (if (<= t_1 1e-7)
       (* (fma (* x x) 0.5 1.0) t_0)
       (*
        (fma
         (fma 0.008333333333333333 (* y y) -0.16666666666666666)
         (* y y)
         1.0)
        (cosh x))))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double t_1 = t_0 * cosh(x);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma(-0.16666666666666666, (y * y), 1.0) * cosh(x);
	} else if (t_1 <= 1e-7) {
		tmp = fma((x * x), 0.5, 1.0) * t_0;
	} else {
		tmp = fma(fma(0.008333333333333333, (y * y), -0.16666666666666666), (y * y), 1.0) * cosh(x);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(y) / y)
	t_1 = Float64(t_0 * cosh(x))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(fma(-0.16666666666666666, Float64(y * y), 1.0) * cosh(x));
	elseif (t_1 <= 1e-7)
		tmp = Float64(fma(Float64(x * x), 0.5, 1.0) * t_0);
	else
		tmp = Float64(fma(fma(0.008333333333333333, Float64(y * y), -0.16666666666666666), Float64(y * y), 1.0) * cosh(x));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[Cosh[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * N[Cosh[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e-7], N[(N[(N[(x * x), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(N[(0.008333333333333333 * N[(y * y), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * N[Cosh[x], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -inf.0 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < 9.9999999999999995e-8

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 99.1

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

   5. Applied rewrites 99.1%

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

   if 9.9999999999999995e-8 < (*.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. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \cdot \cosh x \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right) \cdot \cosh x\\ \mathbf{elif}\;\frac{\sin y}{y} \cdot \cosh x \leq 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, 0.5, 1\right) \cdot \frac{\sin y}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), y \cdot y, 1\right) \cdot \cosh x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 93.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ t_1 := t\_0 \cdot \cosh x\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right) \cdot \cosh x\\ \mathbf{elif}\;t\_1 \leq 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, 0.5, 1\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x}{\mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)) (t_1 (* t_0 (cosh x))))
   (if (<= t_1 (- INFINITY))
     (* (fma -0.16666666666666666 (* y y) 1.0) (cosh x))
     (if (<= t_1 1e-7)
       (* (fma (* x x) 0.5 1.0) t_0)
       (/ (cosh x) (fma 0.16666666666666666 (* y y) 1.0))))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double t_1 = t_0 * cosh(x);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma(-0.16666666666666666, (y * y), 1.0) * cosh(x);
	} else if (t_1 <= 1e-7) {
		tmp = fma((x * x), 0.5, 1.0) * t_0;
	} else {
		tmp = cosh(x) / fma(0.16666666666666666, (y * y), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(y) / y)
	t_1 = Float64(t_0 * cosh(x))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(fma(-0.16666666666666666, Float64(y * y), 1.0) * cosh(x));
	elseif (t_1 <= 1e-7)
		tmp = Float64(fma(Float64(x * x), 0.5, 1.0) * t_0);
	else
		tmp = Float64(cosh(x) / fma(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[(t$95$0 * N[Cosh[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * N[Cosh[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e-7], N[(N[(N[(x * x), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[Cosh[x], $MachinePrecision] / N[(0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -inf.0 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < 9.9999999999999995e-8

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 99.1

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

   5. Applied rewrites 99.1%

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

   if 9.9999999999999995e-8 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y))

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

         \[\leadsto \frac{\cosh x}{\color{blue}{\frac{1}{6} \cdot {y}^{2} + 1}} \]

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 96.5

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

   7. Applied rewrites 96.5%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \cdot \cosh x \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right) \cdot \cosh x\\ \mathbf{elif}\;\frac{\sin y}{y} \cdot \cosh x \leq 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, 0.5, 1\right) \cdot \frac{\sin y}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x}{\mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 93.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ t_1 := t\_0 \cdot \cosh x\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right) \cdot \cosh x\\ \mathbf{elif}\;t\_1 \leq 10^{-7}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x}{\mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)) (t_1 (* t_0 (cosh x))))
   (if (<= t_1 (- INFINITY))
     (* (fma -0.16666666666666666 (* y y) 1.0) (cosh x))
     (if (<= t_1 1e-7)
       t_0
       (/ (cosh x) (fma 0.16666666666666666 (* y y) 1.0))))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double t_1 = t_0 * cosh(x);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma(-0.16666666666666666, (y * y), 1.0) * cosh(x);
	} else if (t_1 <= 1e-7) {
		tmp = t_0;
	} else {
		tmp = cosh(x) / fma(0.16666666666666666, (y * y), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(y) / y)
	t_1 = Float64(t_0 * cosh(x))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(fma(-0.16666666666666666, Float64(y * y), 1.0) * cosh(x));
	elseif (t_1 <= 1e-7)
		tmp = t_0;
	else
		tmp = Float64(cosh(x) / fma(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[(t$95$0 * N[Cosh[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * N[Cosh[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e-7], t$95$0, N[(N[Cosh[x], $MachinePrecision] / N[(0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -inf.0 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < 9.9999999999999995e-8

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 98.3

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

   5. Applied rewrites 98.3%

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

   if 9.9999999999999995e-8 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y))

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

         \[\leadsto \frac{\cosh x}{\color{blue}{\frac{1}{6} \cdot {y}^{2} + 1}} \]

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 96.5

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

   7. Applied rewrites 96.5%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \cdot \cosh x \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right) \cdot \cosh x\\ \mathbf{elif}\;\frac{\sin y}{y} \cdot \cosh x \leq 10^{-7}:\\ \;\;\;\;\frac{\sin y}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x}{\mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 92.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ t_1 := t\_0 \cdot \cosh x\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right) \cdot \cosh x\\ \mathbf{elif}\;t\_1 \leq 10^{-7}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), y \cdot y, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)) (t_1 (* t_0 (cosh x))))
   (if (<= t_1 (- INFINITY))
     (* (fma -0.16666666666666666 (* y y) 1.0) (cosh x))
     (if (<= t_1 1e-7)
       t_0
       (*
        (fma (fma 0.041666666666666664 (* x x) 0.5) (* x x) 1.0)
        (fma
         (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 = t_0 * cosh(x);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma(-0.16666666666666666, (y * y), 1.0) * cosh(x);
	} else if (t_1 <= 1e-7) {
		tmp = t_0;
	} else {
		tmp = fma(fma(0.041666666666666664, (x * x), 0.5), (x * x), 1.0) * fma(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(t_0 * cosh(x))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(fma(-0.16666666666666666, Float64(y * y), 1.0) * cosh(x));
	elseif (t_1 <= 1e-7)
		tmp = t_0;
	else
		tmp = Float64(fma(fma(0.041666666666666664, Float64(x * x), 0.5), Float64(x * x), 1.0) * fma(fma(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[(t$95$0 * N[Cosh[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * N[Cosh[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e-7], t$95$0, N[(N[(N[(0.041666666666666664 * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(0.008333333333333333 * N[(y * y), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -inf.0 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < 9.9999999999999995e-8

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 98.3

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

   5. Applied rewrites 98.3%

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

   if 9.9999999999999995e-8 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 91.0

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

   5. Applied rewrites 91.0%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 93.9

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

   8. Applied rewrites 93.9%

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

4. Final simplification 95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \cdot \cosh x \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right) \cdot \cosh x\\ \mathbf{elif}\;\frac{\sin y}{y} \cdot \cosh x \leq 10^{-7}:\\ \;\;\;\;\frac{\sin y}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), y \cdot y, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 92.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ t_1 := t\_0 \cdot \cosh x\\ t_2 := \mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\left(\left(t\_2 \cdot x\right) \cdot x\right) \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{elif}\;t\_1 \leq 10^{-7}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_2, x \cdot x, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), y \cdot y, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y))
        (t_1 (* t_0 (cosh x)))
        (t_2 (fma 0.041666666666666664 (* x x) 0.5)))
   (if (<= t_1 (- INFINITY))
     (* (* (* t_2 x) x) (fma -0.16666666666666666 (* y y) 1.0))
     (if (<= t_1 1e-7)
       t_0
       (*
        (fma t_2 (* x x) 1.0)
        (fma
         (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 = t_0 * cosh(x);
	double t_2 = fma(0.041666666666666664, (x * x), 0.5);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = ((t_2 * x) * x) * fma(-0.16666666666666666, (y * y), 1.0);
	} else if (t_1 <= 1e-7) {
		tmp = t_0;
	} else {
		tmp = fma(t_2, (x * x), 1.0) * fma(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(t_0 * cosh(x))
	t_2 = fma(0.041666666666666664, Float64(x * x), 0.5)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(t_2 * x) * x) * fma(-0.16666666666666666, Float64(y * y), 1.0));
	elseif (t_1 <= 1e-7)
		tmp = t_0;
	else
		tmp = Float64(fma(t_2, Float64(x * x), 1.0) * fma(fma(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[(t$95$0 * N[Cosh[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(0.041666666666666664 * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(t$95$2 * x), $MachinePrecision] * x), $MachinePrecision] * N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e-7], t$95$0, N[(N[(t$95$2 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(0.008333333333333333 * N[(y * y), $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 x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 61.9

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

   5. Applied rewrites 61.9%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 86.7

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

   8. Applied rewrites 86.7%

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

   9. Taylor expanded in x around inf

      \[\leadsto \left({x}^{4} \cdot \color{blue}{\left(\frac{1}{24} + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, y \cdot y, 1\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 86.7%

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

   if -inf.0 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < 9.9999999999999995e-8

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 98.3

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

   5. Applied rewrites 98.3%

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

   if 9.9999999999999995e-8 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 91.0

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

   5. Applied rewrites 91.0%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 93.9

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

   8. Applied rewrites 93.9%

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

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \cdot \cosh x \leq -\infty:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right) \cdot x\right) \cdot x\right) \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{elif}\;\frac{\sin y}{y} \cdot \cosh x \leq 10^{-7}:\\ \;\;\;\;\frac{\sin y}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), y \cdot y, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 68.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)\\ t_1 := \frac{\sin y}{y}\\ t_2 := \mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right)\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{-307}:\\ \;\;\;\;\left(\left(t\_2 \cdot x\right) \cdot x\right) \cdot t\_0\\ \mathbf{elif}\;t\_1 \leq 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(0.008333333333333333 \cdot \left(y \cdot y\right), y \cdot y, 1\right) \cdot \mathsf{fma}\left(x \cdot x, 0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_2, x \cdot x, 1\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma -0.16666666666666666 (* y y) 1.0))
        (t_1 (/ (sin y) y))
        (t_2 (fma 0.041666666666666664 (* x x) 0.5)))
   (if (<= t_1 -4e-307)
     (* (* (* t_2 x) x) t_0)
     (if (<= t_1 1e-7)
       (*
        (fma (* 0.008333333333333333 (* y y)) (* y y) 1.0)
        (fma (* x x) 0.5 1.0))
       (* (fma t_2 (* 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 t_2 = fma(0.041666666666666664, (x * x), 0.5);
	double tmp;
	if (t_1 <= -4e-307) {
		tmp = ((t_2 * x) * x) * t_0;
	} else if (t_1 <= 1e-7) {
		tmp = fma((0.008333333333333333 * (y * y)), (y * y), 1.0) * fma((x * x), 0.5, 1.0);
	} else {
		tmp = fma(t_2, (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)
	t_2 = fma(0.041666666666666664, Float64(x * x), 0.5)
	tmp = 0.0
	if (t_1 <= -4e-307)
		tmp = Float64(Float64(Float64(t_2 * x) * x) * t_0);
	elseif (t_1 <= 1e-7)
		tmp = Float64(fma(Float64(0.008333333333333333 * Float64(y * y)), Float64(y * y), 1.0) * fma(Float64(x * x), 0.5, 1.0));
	else
		tmp = Float64(fma(t_2, 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]}, Block[{t$95$2 = N[(0.041666666666666664 * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision]}, If[LessEqual[t$95$1, -4e-307], N[(N[(N[(t$95$2 * x), $MachinePrecision] * x), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[t$95$1, 1e-7], N[(N[(N[(0.008333333333333333 * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$2 * 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) < -3.99999999999999964e-307

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 81.3

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

   5. Applied rewrites 81.3%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 43.4

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

   8. Applied rewrites 43.4%

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

   9. Taylor expanded in x around inf

      \[\leadsto \left({x}^{4} \cdot \color{blue}{\left(\frac{1}{24} + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, y \cdot y, 1\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 43.6%

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

   if -3.99999999999999964e-307 < (/.f64 (sin.f64 y) y) < 9.9999999999999995e-8

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 79.0

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

   5. Applied rewrites 79.0%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 39.4

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

   8. Applied rewrites 39.4%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 39.4%

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

   if 9.9999999999999995e-8 < (/.f64 (sin.f64 y) y)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 94.8

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

   5. Applied rewrites 94.8%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 94.8

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

   8. Applied rewrites 94.8%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq -4 \cdot 10^{-307}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right) \cdot x\right) \cdot x\right) \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{elif}\;\frac{\sin y}{y} \leq 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(0.008333333333333333 \cdot \left(y \cdot y\right), y \cdot y, 1\right) \cdot \mathsf{fma}\left(x \cdot x, 0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 68.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right)\\ \mathbf{if}\;\frac{\sin y}{y} \cdot \cosh x \leq -2 \cdot 10^{-173}:\\ \;\;\;\;\left(\left(t\_0 \cdot x\right) \cdot x\right) \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, x \cdot x, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), y \cdot y, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma 0.041666666666666664 (* x x) 0.5)))
   (if (<= (* (/ (sin y) y) (cosh x)) -2e-173)
     (* (* (* t_0 x) x) (fma -0.16666666666666666 (* y y) 1.0))
     (*
      (fma t_0 (* x x) 1.0)
      (fma
       (fma 0.008333333333333333 (* y y) -0.16666666666666666)
       (* y y)
       1.0)))))

C

double code(double x, double y) {
	double t_0 = fma(0.041666666666666664, (x * x), 0.5);
	double tmp;
	if (((sin(y) / y) * cosh(x)) <= -2e-173) {
		tmp = ((t_0 * x) * x) * fma(-0.16666666666666666, (y * y), 1.0);
	} else {
		tmp = fma(t_0, (x * x), 1.0) * fma(fma(0.008333333333333333, (y * y), -0.16666666666666666), (y * y), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(0.041666666666666664, Float64(x * x), 0.5)
	tmp = 0.0
	if (Float64(Float64(sin(y) / y) * cosh(x)) <= -2e-173)
		tmp = Float64(Float64(Float64(t_0 * x) * x) * fma(-0.16666666666666666, Float64(y * y), 1.0));
	else
		tmp = Float64(fma(t_0, Float64(x * x), 1.0) * fma(fma(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[(0.041666666666666664 * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision] * N[Cosh[x], $MachinePrecision]), $MachinePrecision], -2e-173], N[(N[(N[(t$95$0 * x), $MachinePrecision] * x), $MachinePrecision] * N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(0.008333333333333333 * N[(y * y), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < -2.0000000000000001e-173

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 77.4

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

   5. Applied rewrites 77.4%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 52.1

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

   8. Applied rewrites 52.1%

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

   9. Taylor expanded in x around inf

      \[\leadsto \left({x}^{4} \cdot \color{blue}{\left(\frac{1}{24} + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, y \cdot y, 1\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 52.5%

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

   if -2.0000000000000001e-173 < (*.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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 92.6

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

   5. Applied rewrites 92.6%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 77.2

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

   8. Applied rewrites 77.2%

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

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \cdot \cosh x \leq -2 \cdot 10^{-173}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right) \cdot x\right) \cdot x\right) \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), y \cdot y, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 59.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{-307}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984 \cdot \left(y \cdot y\right), y \cdot y, -0.16666666666666666\right), y \cdot y, 1\right)\\ \mathbf{elif}\;t\_0 \leq 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, 0.5, 1\right) \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)))
   (if (<= t_0 -4e-307)
     (fma
      (fma (* -0.0001984126984126984 (* y y)) (* y y) -0.16666666666666666)
      (* y y)
      1.0)
     (if (<= t_0 1e-7)
       (fma
        (fma 0.008333333333333333 (* y y) -0.16666666666666666)
        (* y y)
        1.0)
       (* (fma (* x x) 0.5 1.0) (fma -0.16666666666666666 (* y y) 1.0))))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double tmp;
	if (t_0 <= -4e-307) {
		tmp = fma(fma((-0.0001984126984126984 * (y * y)), (y * y), -0.16666666666666666), (y * y), 1.0);
	} else if (t_0 <= 1e-7) {
		tmp = fma(fma(0.008333333333333333, (y * y), -0.16666666666666666), (y * y), 1.0);
	} else {
		tmp = fma((x * x), 0.5, 1.0) * fma(-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 <= -4e-307)
		tmp = fma(fma(Float64(-0.0001984126984126984 * Float64(y * y)), Float64(y * y), -0.16666666666666666), Float64(y * y), 1.0);
	elseif (t_0 <= 1e-7)
		tmp = fma(fma(0.008333333333333333, Float64(y * y), -0.16666666666666666), Float64(y * y), 1.0);
	else
		tmp = Float64(fma(Float64(x * x), 0.5, 1.0) * fma(-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[LessEqual[t$95$0, -4e-307], N[(N[(N[(-0.0001984126984126984 * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(y * y), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[t$95$0, 1e-7], N[(N[(0.008333333333333333 * N[(y * y), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(N[(x * x), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] * N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 y) y) < -3.99999999999999964e-307

   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 51.2

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

   5. Applied rewrites 51.2%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 42.9%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 42.9%

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

   if -3.99999999999999964e-307 < (/.f64 (sin.f64 y) y) < 9.9999999999999995e-8

   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 55.9

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

   5. Applied rewrites 55.9%

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

   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 26.3%

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

   if 9.9999999999999995e-8 < (/.f64 (sin.f64 y) y)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-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 81.9

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

   5. Applied rewrites 81.9%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 81.9

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

   8. Applied rewrites 81.9%

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

Alternative 11: 59.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ t_1 := \mathsf{fma}\left(x \cdot x, 0.5, 1\right) \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{-307}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y))
        (t_1 (* (fma (* x x) 0.5 1.0) (fma -0.16666666666666666 (* y y) 1.0))))
   (if (<= t_0 -4e-307)
     t_1
     (if (<= t_0 1e-7)
       (fma
        (fma 0.008333333333333333 (* y y) -0.16666666666666666)
        (* y y)
        1.0)
       t_1))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double t_1 = fma((x * x), 0.5, 1.0) * fma(-0.16666666666666666, (y * y), 1.0);
	double tmp;
	if (t_0 <= -4e-307) {
		tmp = t_1;
	} else if (t_0 <= 1e-7) {
		tmp = fma(fma(0.008333333333333333, (y * y), -0.16666666666666666), (y * y), 1.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(y) / y)
	t_1 = Float64(fma(Float64(x * x), 0.5, 1.0) * fma(-0.16666666666666666, Float64(y * y), 1.0))
	tmp = 0.0
	if (t_0 <= -4e-307)
		tmp = t_1;
	elseif (t_0 <= 1e-7)
		tmp = fma(fma(0.008333333333333333, Float64(y * y), -0.16666666666666666), Float64(y * y), 1.0);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(x * x), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] * N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -4e-307], t$95$1, If[LessEqual[t$95$0, 1e-7], N[(N[(0.008333333333333333 * N[(y * y), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 y) y) < -3.99999999999999964e-307 or 9.9999999999999995e-8 < (/.f64 (sin.f64 y) y)

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + \frac{1}{2} \cdot {x}^{2}\right)} \cdot \frac{\sin y}{y} \]
   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 78.8

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

   5. Applied rewrites 78.8%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 70.1

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

   8. Applied rewrites 70.1%

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

   if -3.99999999999999964e-307 < (/.f64 (sin.f64 y) y) < 9.9999999999999995e-8

   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 55.9

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

   5. Applied rewrites 55.9%

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

   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 26.3%

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

Alternative 12: 60.9% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (/ (sin y) y) (cosh x)) 2.0)
   (fma
    (fma (* -0.0001984126984126984 (* y y)) (* y y) -0.16666666666666666)
    (* y y)
    1.0)
   (*
    (* 0.5 (* x x))
    (fma
     (fma 0.008333333333333333 (* y y) -0.16666666666666666)
     (* y y)
     1.0))))

C

double code(double x, double y) {
	double tmp;
	if (((sin(y) / y) * cosh(x)) <= 2.0) {
		tmp = fma(fma((-0.0001984126984126984 * (y * y)), (y * y), -0.16666666666666666), (y * y), 1.0);
	} else {
		tmp = (0.5 * (x * x)) * fma(fma(0.008333333333333333, (y * y), -0.16666666666666666), (y * y), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(sin(y) / y) * cosh(x)) <= 2.0)
		tmp = fma(fma(Float64(-0.0001984126984126984 * Float64(y * y)), Float64(y * y), -0.16666666666666666), Float64(y * y), 1.0);
	else
		tmp = Float64(Float64(0.5 * Float64(x * x)) * fma(fma(0.008333333333333333, Float64(y * y), -0.16666666666666666), Float64(y * y), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision] * N[Cosh[x], $MachinePrecision]), $MachinePrecision], 2.0], N[(N[(N[(-0.0001984126984126984 * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(y * y), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(N[(0.008333333333333333 * N[(y * y), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.9%

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

   3. Taylor expanded in 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 81.7

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

   5. Applied rewrites 81.7%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 61.8%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 61.8%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-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 60.4

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

   5. Applied rewrites 60.4%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 67.8

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

   8. Applied rewrites 67.8%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 67.8%

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

4. Final simplification 64.0%

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

Alternative 13: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Final simplification 99.9%

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

Alternative 14: 42.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq -4 \cdot 10^{-307}:\\ \;\;\;\;\left(y \cdot y\right) \cdot -0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), y \cdot y, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (sin y) y) -4e-307)
   (* (* y y) -0.16666666666666666)
   (fma (fma 0.008333333333333333 (* y y) -0.16666666666666666) (* y y) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((sin(y) / y) <= -4e-307) {
		tmp = (y * y) * -0.16666666666666666;
	} else {
		tmp = fma(fma(0.008333333333333333, (y * y), -0.16666666666666666), (y * y), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(sin(y) / y) <= -4e-307)
		tmp = Float64(Float64(y * y) * -0.16666666666666666);
	else
		tmp = fma(fma(0.008333333333333333, Float64(y * y), -0.16666666666666666), Float64(y * y), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], -4e-307], N[(N[(y * y), $MachinePrecision] * -0.16666666666666666), $MachinePrecision], N[(N[(0.008333333333333333 * N[(y * y), $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) < -3.99999999999999964e-307

   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 51.2

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

   5. Applied rewrites 51.2%

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

   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 31.7%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 31.7%

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

   if -3.99999999999999964e-307 < (/.f64 (sin.f64 y) y)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 53.6

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

   5. Applied rewrites 53.6%

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

   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 46.0%

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

4. Final simplification 42.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq -4 \cdot 10^{-307}:\\ \;\;\;\;\left(y \cdot y\right) \cdot -0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), y \cdot y, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 46.6% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.14 \cdot 10^{+26}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984 \cdot \left(y \cdot y\right), y \cdot y, -0.16666666666666666\right), y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right) \cdot x\right) \cdot x\right) \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1.14e+26)
   (fma
    (fma (* -0.0001984126984126984 (* y y)) (* y y) -0.16666666666666666)
    (* y y)
    1.0)
   (*
    (* (* (fma 0.041666666666666664 (* x x) 0.5) x) x)
    (fma -0.16666666666666666 (* y y) 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.14e+26) {
		tmp = fma(fma((-0.0001984126984126984 * (y * y)), (y * y), -0.16666666666666666), (y * y), 1.0);
	} else {
		tmp = ((fma(0.041666666666666664, (x * x), 0.5) * x) * x) * fma(-0.16666666666666666, (y * y), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.14e+26)
		tmp = fma(fma(Float64(-0.0001984126984126984 * Float64(y * y)), Float64(y * y), -0.16666666666666666), Float64(y * y), 1.0);
	else
		tmp = Float64(Float64(Float64(fma(0.041666666666666664, Float64(x * x), 0.5) * x) * x) * fma(-0.16666666666666666, Float64(y * y), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.14e+26], N[(N[(N[(-0.0001984126984126984 * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(y * y), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(N[(N[(0.041666666666666664 * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.1399999999999999e26

   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 69.5

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

   5. Applied rewrites 69.5%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 47.6%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 47.6%

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

   if 1.1399999999999999e26 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 84.8

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

   5. Applied rewrites 84.8%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 70.1

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

   8. Applied rewrites 70.1%

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

   9. Taylor expanded in x around inf

      \[\leadsto \left({x}^{4} \cdot \color{blue}{\left(\frac{1}{24} + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)}\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, y \cdot y, 1\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 70.1%

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

Alternative 16: 33.0% 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 53.1

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

5. Applied rewrites 53.1%

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

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 37.7%

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

Alternative 17: 8.0% accurate, 19.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(N[(y * y), $MachinePrecision] * -0.16666666666666666), $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 53.1

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

5. Applied rewrites 53.1%

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

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 37.7%

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

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. Final simplification 8.5%

    \[\leadsto \left(y \cdot y\right) \cdot -0.16666666666666666 \]
13. 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 2024288 
(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)))