Linear.Quaternion:$ccosh from linear-1.19.1.3

Percentage Accurate: 89.8% → 99.9%

Time: 10.5s

Alternatives: 18

Speedup: 1.0×

• 51.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 89.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.8%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l/ N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 100.0

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

4. Applied rewrites 100.0%

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

5. Final simplification 100.0%

   \[\leadsto \sinh y \cdot \frac{\sin x}{x} \]
6. Add Preprocessing

Alternative 2: 84.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y \cdot \sin x}{x}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984 \cdot \left(y \cdot y\right), y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot y\right) \cdot \left(\mathsf{fma}\left(-0.16666666666666666, x \cdot x, 1\right) \cdot x\right)}{x}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-74}:\\ \;\;\;\;\left(\mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right) \cdot \sin x\right) \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \sinh y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sinh y) (sin x)) x)))
   (if (<= t_0 (- INFINITY))
     (/
      (*
       (*
        (fma
         (fma (* 0.0001984126984126984 (* y y)) (* y y) 0.16666666666666666)
         (* y y)
         1.0)
        y)
       (* (fma -0.16666666666666666 (* x x) 1.0) x))
      x)
     (if (<= t_0 2e-74)
       (* (* (fma (* y y) 0.16666666666666666 1.0) (sin x)) (/ y x))
       (* 1.0 (sinh y))))))

C

double code(double x, double y) {
	double t_0 = (sinh(y) * sin(x)) / x;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = ((fma(fma((0.0001984126984126984 * (y * y)), (y * y), 0.16666666666666666), (y * y), 1.0) * y) * (fma(-0.16666666666666666, (x * x), 1.0) * x)) / x;
	} else if (t_0 <= 2e-74) {
		tmp = (fma((y * y), 0.16666666666666666, 1.0) * sin(x)) * (y / x);
	} else {
		tmp = 1.0 * sinh(y);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(sinh(y) * sin(x)) / x)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(fma(fma(Float64(0.0001984126984126984 * Float64(y * y)), Float64(y * y), 0.16666666666666666), Float64(y * y), 1.0) * y) * Float64(fma(-0.16666666666666666, Float64(x * x), 1.0) * x)) / x);
	elseif (t_0 <= 2e-74)
		tmp = Float64(Float64(fma(Float64(y * y), 0.16666666666666666, 1.0) * sin(x)) * Float64(y / x));
	else
		tmp = Float64(1.0 * sinh(y));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sinh[y], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[(N[(N[(0.0001984126984126984 * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(y * y), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision] * N[(N[(-0.16666666666666666 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[t$95$0, 2e-74], N[(N[(N[(N[(y * y), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision], N[(1.0 * N[Sinh[y], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 86.4

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

   5. Applied rewrites 86.4%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 86.4%

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

   9. Taylor expanded in x around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. lower-fma.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 71.9

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

   11. Applied rewrites 71.9%

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

   if -inf.0 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 1.99999999999999992e-74

   1. Initial program 77.6%

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

   3. Taylor expanded in y around 0

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

      1. distribute-rgt-in N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. associate-*l/ N/A

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

      8. associate-/l* N/A

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

      9. distribute-rgt-out N/A

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

      10. +-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

   5. Applied rewrites 98.9%

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

   if 1.99999999999999992e-74 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 100.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 75.0%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -\infty:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984 \cdot \left(y \cdot y\right), y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot y\right) \cdot \left(\mathsf{fma}\left(-0.16666666666666666, x \cdot x, 1\right) \cdot x\right)}{x}\\ \mathbf{elif}\;\frac{\sinh y \cdot \sin x}{x} \leq 2 \cdot 10^{-74}:\\ \;\;\;\;\left(\mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right) \cdot \sin x\right) \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \sinh y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y \cdot \sin x}{x}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984 \cdot \left(y \cdot y\right), y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot y\right) \cdot \left(\mathsf{fma}\left(-0.16666666666666666, x \cdot x, 1\right) \cdot x\right)}{x}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-74}:\\ \;\;\;\;y \cdot \frac{\sin x}{x}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \sinh y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sinh y) (sin x)) x)))
   (if (<= t_0 (- INFINITY))
     (/
      (*
       (*
        (fma
         (fma (* 0.0001984126984126984 (* y y)) (* y y) 0.16666666666666666)
         (* y y)
         1.0)
        y)
       (* (fma -0.16666666666666666 (* x x) 1.0) x))
      x)
     (if (<= t_0 2e-74) (* y (/ (sin x) x)) (* 1.0 (sinh y))))))

C

double code(double x, double y) {
	double t_0 = (sinh(y) * sin(x)) / x;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = ((fma(fma((0.0001984126984126984 * (y * y)), (y * y), 0.16666666666666666), (y * y), 1.0) * y) * (fma(-0.16666666666666666, (x * x), 1.0) * x)) / x;
	} else if (t_0 <= 2e-74) {
		tmp = y * (sin(x) / x);
	} else {
		tmp = 1.0 * sinh(y);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(sinh(y) * sin(x)) / x)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(fma(fma(Float64(0.0001984126984126984 * Float64(y * y)), Float64(y * y), 0.16666666666666666), Float64(y * y), 1.0) * y) * Float64(fma(-0.16666666666666666, Float64(x * x), 1.0) * x)) / x);
	elseif (t_0 <= 2e-74)
		tmp = Float64(y * Float64(sin(x) / x));
	else
		tmp = Float64(1.0 * sinh(y));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sinh[y], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[(N[(N[(0.0001984126984126984 * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(y * y), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision] * N[(N[(-0.16666666666666666 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[t$95$0, 2e-74], N[(y * N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(1.0 * N[Sinh[y], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 86.4

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

   5. Applied rewrites 86.4%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 86.4%

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

   9. Taylor expanded in x around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. lower-fma.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 71.9

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

   11. Applied rewrites 71.9%

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

   if -inf.0 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 1.99999999999999992e-74

   1. Initial program 77.6%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sin.f64 98.7

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

   5. Applied rewrites 98.7%

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

   if 1.99999999999999992e-74 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 100.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 75.0%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -\infty:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984 \cdot \left(y \cdot y\right), y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot y\right) \cdot \left(\mathsf{fma}\left(-0.16666666666666666, x \cdot x, 1\right) \cdot x\right)}{x}\\ \mathbf{elif}\;\frac{\sinh y \cdot \sin x}{x} \leq 2 \cdot 10^{-74}:\\ \;\;\;\;y \cdot \frac{\sin x}{x}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \sinh y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 72.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y \cdot \sin x}{x}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-170}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984 \cdot \left(y \cdot y\right), y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot y\right) \cdot \left(\mathsf{fma}\left(-0.16666666666666666, x \cdot x, 1\right) \cdot x\right)}{x}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-216}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right) \cdot x}{\frac{x}{y}}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \sinh y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sinh y) (sin x)) x)))
   (if (<= t_0 -2e-170)
     (/
      (*
       (*
        (fma
         (fma (* 0.0001984126984126984 (* y y)) (* y y) 0.16666666666666666)
         (* y y)
         1.0)
        y)
       (* (fma -0.16666666666666666 (* x x) 1.0) x))
      x)
     (if (<= t_0 2e-216)
       (/ (* (fma 0.16666666666666666 (* y y) 1.0) x) (/ x y))
       (* 1.0 (sinh y))))))

C

double code(double x, double y) {
	double t_0 = (sinh(y) * sin(x)) / x;
	double tmp;
	if (t_0 <= -2e-170) {
		tmp = ((fma(fma((0.0001984126984126984 * (y * y)), (y * y), 0.16666666666666666), (y * y), 1.0) * y) * (fma(-0.16666666666666666, (x * x), 1.0) * x)) / x;
	} else if (t_0 <= 2e-216) {
		tmp = (fma(0.16666666666666666, (y * y), 1.0) * x) / (x / y);
	} else {
		tmp = 1.0 * sinh(y);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(sinh(y) * sin(x)) / x)
	tmp = 0.0
	if (t_0 <= -2e-170)
		tmp = Float64(Float64(Float64(fma(fma(Float64(0.0001984126984126984 * Float64(y * y)), Float64(y * y), 0.16666666666666666), Float64(y * y), 1.0) * y) * Float64(fma(-0.16666666666666666, Float64(x * x), 1.0) * x)) / x);
	elseif (t_0 <= 2e-216)
		tmp = Float64(Float64(fma(0.16666666666666666, Float64(y * y), 1.0) * x) / Float64(x / y));
	else
		tmp = Float64(1.0 * sinh(y));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sinh[y], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-170], N[(N[(N[(N[(N[(N[(0.0001984126984126984 * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(y * y), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision] * N[(N[(-0.16666666666666666 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[t$95$0, 2e-216], N[(N[(N[(0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision] / N[(x / y), $MachinePrecision]), $MachinePrecision], N[(1.0 * N[Sinh[y], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -1.99999999999999997e-170

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 90.0

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

   5. Applied rewrites 90.0%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 90.0%

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

   9. Taylor expanded in x around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. lower-fma.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 74.9

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

   11. Applied rewrites 74.9%

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

   if -1.99999999999999997e-170 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 2.0000000000000001e-216

   1. Initial program 65.5%

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

   3. Taylor expanded in y around 0

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

      1. distribute-rgt-in N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. associate-*l/ N/A

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

      8. associate-/l* N/A

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

      9. distribute-rgt-out N/A

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

      10. +-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

   5. Applied rewrites 98.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 77.6%

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

   10. Applied rewrites 79.4%

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

   if 2.0000000000000001e-216 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 100.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 70.9%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -2 \cdot 10^{-170}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984 \cdot \left(y \cdot y\right), y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot y\right) \cdot \left(\mathsf{fma}\left(-0.16666666666666666, x \cdot x, 1\right) \cdot x\right)}{x}\\ \mathbf{elif}\;\frac{\sinh y \cdot \sin x}{x} \leq 2 \cdot 10^{-216}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right) \cdot x}{\frac{x}{y}}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \sinh y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 71.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y \cdot \sin x}{x}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-170}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984 \cdot \left(y \cdot y\right), y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot y\right) \cdot \left(\mathsf{fma}\left(-0.16666666666666666, x \cdot x, 1\right) \cdot x\right)}{x}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-74}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right) \cdot x}{\frac{x}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984, y \cdot y, 0.008333333333333333\right), y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot y\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right), x \cdot x, 1\right) \cdot x\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sinh y) (sin x)) x)))
   (if (<= t_0 -2e-170)
     (/
      (*
       (*
        (fma
         (fma (* 0.0001984126984126984 (* y y)) (* y y) 0.16666666666666666)
         (* y y)
         1.0)
        y)
       (* (fma -0.16666666666666666 (* x x) 1.0) x))
      x)
     (if (<= t_0 2e-74)
       (/ (* (fma 0.16666666666666666 (* y y) 1.0) x) (/ x y))
       (/
        (*
         (*
          (fma
           (fma
            (fma 0.0001984126984126984 (* y y) 0.008333333333333333)
            (* y y)
            0.16666666666666666)
           (* y y)
           1.0)
          y)
         (*
          (fma
           (fma 0.008333333333333333 (* x x) -0.16666666666666666)
           (* x x)
           1.0)
          x))
        x)))))

C

double code(double x, double y) {
	double t_0 = (sinh(y) * sin(x)) / x;
	double tmp;
	if (t_0 <= -2e-170) {
		tmp = ((fma(fma((0.0001984126984126984 * (y * y)), (y * y), 0.16666666666666666), (y * y), 1.0) * y) * (fma(-0.16666666666666666, (x * x), 1.0) * x)) / x;
	} else if (t_0 <= 2e-74) {
		tmp = (fma(0.16666666666666666, (y * y), 1.0) * x) / (x / y);
	} else {
		tmp = ((fma(fma(fma(0.0001984126984126984, (y * y), 0.008333333333333333), (y * y), 0.16666666666666666), (y * y), 1.0) * y) * (fma(fma(0.008333333333333333, (x * x), -0.16666666666666666), (x * x), 1.0) * x)) / x;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(sinh(y) * sin(x)) / x)
	tmp = 0.0
	if (t_0 <= -2e-170)
		tmp = Float64(Float64(Float64(fma(fma(Float64(0.0001984126984126984 * Float64(y * y)), Float64(y * y), 0.16666666666666666), Float64(y * y), 1.0) * y) * Float64(fma(-0.16666666666666666, Float64(x * x), 1.0) * x)) / x);
	elseif (t_0 <= 2e-74)
		tmp = Float64(Float64(fma(0.16666666666666666, Float64(y * y), 1.0) * x) / Float64(x / y));
	else
		tmp = Float64(Float64(Float64(fma(fma(fma(0.0001984126984126984, Float64(y * y), 0.008333333333333333), Float64(y * y), 0.16666666666666666), Float64(y * y), 1.0) * y) * Float64(fma(fma(0.008333333333333333, Float64(x * x), -0.16666666666666666), Float64(x * x), 1.0) * x)) / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sinh[y], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-170], N[(N[(N[(N[(N[(N[(0.0001984126984126984 * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(y * y), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision] * N[(N[(-0.16666666666666666 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[t$95$0, 2e-74], N[(N[(N[(0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision] / N[(x / y), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(0.0001984126984126984 * N[(y * y), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] * N[(y * y), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision] * N[(N[(N[(0.008333333333333333 * N[(x * x), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -1.99999999999999997e-170

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 90.0

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

   5. Applied rewrites 90.0%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 90.0%

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

   9. Taylor expanded in x around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. lower-fma.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 74.9

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

   11. Applied rewrites 74.9%

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

   if -1.99999999999999997e-170 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 1.99999999999999992e-74

   1. Initial program 71.4%

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

   3. Taylor expanded in y around 0

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

      1. distribute-rgt-in N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. associate-*l/ N/A

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

      8. associate-/l* N/A

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

      9. distribute-rgt-out N/A

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

      10. +-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

   5. Applied rewrites 98.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 72.0%

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

   10. Applied rewrites 73.5%

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

   if 1.99999999999999992e-74 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 92.0

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

   5. Applied rewrites 92.0%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 68.1

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

   8. Applied rewrites 68.1%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -2 \cdot 10^{-170}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984 \cdot \left(y \cdot y\right), y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot y\right) \cdot \left(\mathsf{fma}\left(-0.16666666666666666, x \cdot x, 1\right) \cdot x\right)}{x}\\ \mathbf{elif}\;\frac{\sinh y \cdot \sin x}{x} \leq 2 \cdot 10^{-74}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right) \cdot x}{\frac{x}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984, y \cdot y, 0.008333333333333333\right), y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot y\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right), x \cdot x, 1\right) \cdot x\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 71.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y \cdot \sin x}{x}\\ t_1 := \mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984 \cdot \left(y \cdot y\right), y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot y\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-170}:\\ \;\;\;\;\frac{t\_1 \cdot \left(\mathsf{fma}\left(-0.16666666666666666, x \cdot x, 1\right) \cdot x\right)}{x}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-74}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right) \cdot x}{\frac{x}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right), x \cdot x, 1\right) \cdot x\right) \cdot t\_1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sinh y) (sin x)) x))
        (t_1
         (*
          (fma
           (fma (* 0.0001984126984126984 (* y y)) (* y y) 0.16666666666666666)
           (* y y)
           1.0)
          y)))
   (if (<= t_0 -2e-170)
     (/ (* t_1 (* (fma -0.16666666666666666 (* x x) 1.0) x)) x)
     (if (<= t_0 2e-74)
       (/ (* (fma 0.16666666666666666 (* y y) 1.0) x) (/ x y))
       (/
        (*
         (*
          (fma
           (fma 0.008333333333333333 (* x x) -0.16666666666666666)
           (* x x)
           1.0)
          x)
         t_1)
        x)))))

C

double code(double x, double y) {
	double t_0 = (sinh(y) * sin(x)) / x;
	double t_1 = fma(fma((0.0001984126984126984 * (y * y)), (y * y), 0.16666666666666666), (y * y), 1.0) * y;
	double tmp;
	if (t_0 <= -2e-170) {
		tmp = (t_1 * (fma(-0.16666666666666666, (x * x), 1.0) * x)) / x;
	} else if (t_0 <= 2e-74) {
		tmp = (fma(0.16666666666666666, (y * y), 1.0) * x) / (x / y);
	} else {
		tmp = ((fma(fma(0.008333333333333333, (x * x), -0.16666666666666666), (x * x), 1.0) * x) * t_1) / x;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(sinh(y) * sin(x)) / x)
	t_1 = Float64(fma(fma(Float64(0.0001984126984126984 * Float64(y * y)), Float64(y * y), 0.16666666666666666), Float64(y * y), 1.0) * y)
	tmp = 0.0
	if (t_0 <= -2e-170)
		tmp = Float64(Float64(t_1 * Float64(fma(-0.16666666666666666, Float64(x * x), 1.0) * x)) / x);
	elseif (t_0 <= 2e-74)
		tmp = Float64(Float64(fma(0.16666666666666666, Float64(y * y), 1.0) * x) / Float64(x / y));
	else
		tmp = Float64(Float64(Float64(fma(fma(0.008333333333333333, Float64(x * x), -0.16666666666666666), Float64(x * x), 1.0) * x) * t_1) / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sinh[y], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(0.0001984126984126984 * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(y * y), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-170], N[(N[(t$95$1 * N[(N[(-0.16666666666666666 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[t$95$0, 2e-74], N[(N[(N[(0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision] / N[(x / y), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(0.008333333333333333 * N[(x * x), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision] * t$95$1), $MachinePrecision] / x), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -1.99999999999999997e-170

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 90.0

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

   5. Applied rewrites 90.0%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 90.0%

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

   9. Taylor expanded in x around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. lower-fma.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 74.9

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

   11. Applied rewrites 74.9%

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

   if -1.99999999999999997e-170 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 1.99999999999999992e-74

   1. Initial program 71.4%

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

   3. Taylor expanded in y around 0

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

      1. distribute-rgt-in N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. associate-*l/ N/A

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

      8. associate-/l* N/A

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

      9. distribute-rgt-out N/A

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

      10. +-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

   5. Applied rewrites 98.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 72.0%

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

   10. Applied rewrites 73.5%

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

   if 1.99999999999999992e-74 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 92.0

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

   5. Applied rewrites 92.0%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 91.8%

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

   9. Taylor expanded in x around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. sub-neg N/A

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

       7. metadata-eval N/A

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

       8. lower-fma.f64 N/A

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

       9. unpow2 N/A

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

       10. lower-*.f64 N/A

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

       11. unpow2 N/A

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

       12. lower-*.f64 67.9

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

   11. Applied rewrites 67.9%

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

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -2 \cdot 10^{-170}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984 \cdot \left(y \cdot y\right), y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot y\right) \cdot \left(\mathsf{fma}\left(-0.16666666666666666, x \cdot x, 1\right) \cdot x\right)}{x}\\ \mathbf{elif}\;\frac{\sinh y \cdot \sin x}{x} \leq 2 \cdot 10^{-74}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right) \cdot x}{\frac{x}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right), x \cdot x, 1\right) \cdot x\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984 \cdot \left(y \cdot y\right), y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot y\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 70.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y \cdot \sin x}{x}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-170}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984 \cdot \left(y \cdot y\right), y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot y\right) \cdot \left(\mathsf{fma}\left(-0.16666666666666666, x \cdot x, 1\right) \cdot x\right)}{x}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-74}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right) \cdot x}{\frac{x}{y}}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right), x \cdot x, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right)\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sinh y) (sin x)) x)))
   (if (<= t_0 -2e-170)
     (/
      (*
       (*
        (fma
         (fma (* 0.0001984126984126984 (* y y)) (* y y) 0.16666666666666666)
         (* y y)
         1.0)
        y)
       (* (fma -0.16666666666666666 (* x x) 1.0) x))
      x)
     (if (<= t_0 2e-74)
       (/ (* (fma 0.16666666666666666 (* y y) 1.0) x) (/ x y))
       (*
        (*
         (fma
          (fma 0.008333333333333333 (* x x) -0.16666666666666666)
          (* x x)
          1.0)
         (fma
          (fma 0.008333333333333333 (* y y) 0.16666666666666666)
          (* y y)
          1.0))
        y)))))

C

double code(double x, double y) {
	double t_0 = (sinh(y) * sin(x)) / x;
	double tmp;
	if (t_0 <= -2e-170) {
		tmp = ((fma(fma((0.0001984126984126984 * (y * y)), (y * y), 0.16666666666666666), (y * y), 1.0) * y) * (fma(-0.16666666666666666, (x * x), 1.0) * x)) / x;
	} else if (t_0 <= 2e-74) {
		tmp = (fma(0.16666666666666666, (y * y), 1.0) * x) / (x / y);
	} else {
		tmp = (fma(fma(0.008333333333333333, (x * x), -0.16666666666666666), (x * x), 1.0) * fma(fma(0.008333333333333333, (y * y), 0.16666666666666666), (y * y), 1.0)) * y;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(sinh(y) * sin(x)) / x)
	tmp = 0.0
	if (t_0 <= -2e-170)
		tmp = Float64(Float64(Float64(fma(fma(Float64(0.0001984126984126984 * Float64(y * y)), Float64(y * y), 0.16666666666666666), Float64(y * y), 1.0) * y) * Float64(fma(-0.16666666666666666, Float64(x * x), 1.0) * x)) / x);
	elseif (t_0 <= 2e-74)
		tmp = Float64(Float64(fma(0.16666666666666666, Float64(y * y), 1.0) * x) / Float64(x / y));
	else
		tmp = Float64(Float64(fma(fma(0.008333333333333333, Float64(x * x), -0.16666666666666666), Float64(x * x), 1.0) * fma(fma(0.008333333333333333, Float64(y * y), 0.16666666666666666), Float64(y * y), 1.0)) * y);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sinh[y], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-170], N[(N[(N[(N[(N[(N[(0.0001984126984126984 * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(y * y), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision] * N[(N[(-0.16666666666666666 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[t$95$0, 2e-74], N[(N[(N[(0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision] / N[(x / y), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.008333333333333333 * N[(x * x), $MachinePrecision] + -0.16666666666666666), $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] * y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -1.99999999999999997e-170

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 90.0

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

   5. Applied rewrites 90.0%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 90.0%

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

   9. Taylor expanded in x around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. lower-fma.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 74.9

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

   11. Applied rewrites 74.9%

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

   if -1.99999999999999997e-170 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 1.99999999999999992e-74

   1. Initial program 71.4%

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

   3. Taylor expanded in y around 0

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

      1. distribute-rgt-in N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. associate-*l/ N/A

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

      8. associate-/l* N/A

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

      9. distribute-rgt-out N/A

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

      10. +-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

   5. Applied rewrites 98.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 72.0%

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

   10. Applied rewrites 73.5%

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

   if 1.99999999999999992e-74 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 80.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 62.6%

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

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -2 \cdot 10^{-170}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984 \cdot \left(y \cdot y\right), y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot y\right) \cdot \left(\mathsf{fma}\left(-0.16666666666666666, x \cdot x, 1\right) \cdot x\right)}{x}\\ \mathbf{elif}\;\frac{\sinh y \cdot \sin x}{x} \leq 2 \cdot 10^{-74}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right) \cdot x}{\frac{x}{y}}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right), x \cdot x, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right)\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 60.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y \cdot \sin x}{x}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-296}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.027777777777777776, x \cdot x, 0.16666666666666666\right) \cdot y\right) \cdot \left(y \cdot y\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-74}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right) \cdot x}{\frac{x}{y}}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right), x \cdot x, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right)\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sinh y) (sin x)) x)))
   (if (<= t_0 -2e-296)
     (* (* (fma -0.027777777777777776 (* x x) 0.16666666666666666) y) (* y y))
     (if (<= t_0 2e-74)
       (/ (* (fma 0.16666666666666666 (* y y) 1.0) x) (/ x y))
       (*
        (*
         (fma
          (fma 0.008333333333333333 (* x x) -0.16666666666666666)
          (* x x)
          1.0)
         (fma
          (fma 0.008333333333333333 (* y y) 0.16666666666666666)
          (* y y)
          1.0))
        y)))))

C

double code(double x, double y) {
	double t_0 = (sinh(y) * sin(x)) / x;
	double tmp;
	if (t_0 <= -2e-296) {
		tmp = (fma(-0.027777777777777776, (x * x), 0.16666666666666666) * y) * (y * y);
	} else if (t_0 <= 2e-74) {
		tmp = (fma(0.16666666666666666, (y * y), 1.0) * x) / (x / y);
	} else {
		tmp = (fma(fma(0.008333333333333333, (x * x), -0.16666666666666666), (x * x), 1.0) * fma(fma(0.008333333333333333, (y * y), 0.16666666666666666), (y * y), 1.0)) * y;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(sinh(y) * sin(x)) / x)
	tmp = 0.0
	if (t_0 <= -2e-296)
		tmp = Float64(Float64(fma(-0.027777777777777776, Float64(x * x), 0.16666666666666666) * y) * Float64(y * y));
	elseif (t_0 <= 2e-74)
		tmp = Float64(Float64(fma(0.16666666666666666, Float64(y * y), 1.0) * x) / Float64(x / y));
	else
		tmp = Float64(Float64(fma(fma(0.008333333333333333, Float64(x * x), -0.16666666666666666), Float64(x * x), 1.0) * fma(fma(0.008333333333333333, Float64(y * y), 0.16666666666666666), Float64(y * y), 1.0)) * y);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sinh[y], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-296], N[(N[(N[(-0.027777777777777776 * N[(x * x), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * y), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e-74], N[(N[(N[(0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision] / N[(x / y), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.008333333333333333 * N[(x * x), $MachinePrecision] + -0.16666666666666666), $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] * y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -2e-296

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. distribute-rgt-in N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. associate-*l/ N/A

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

      8. associate-/l* N/A

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

      9. distribute-rgt-out N/A

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

      10. +-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

   5. Applied rewrites 75.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 60.7%

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

   9. Taylor expanded in y around inf

      \[\leadsto \frac{1}{6} \cdot \left({y}^{3} \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {x}^{2}\right)}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 40.1%

       \[\leadsto \left(\mathsf{fma}\left(-0.027777777777777776, x \cdot x, 0.16666666666666666\right) \cdot y\right) \cdot \left(y \cdot \color{blue}{y}\right) \]

   if -2e-296 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 1.99999999999999992e-74

   1. Initial program 66.0%

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

   3. Taylor expanded in y around 0

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

      1. distribute-rgt-in N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. associate-*l/ N/A

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

      8. associate-/l* N/A

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

      9. distribute-rgt-out N/A

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

      10. +-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

   5. Applied rewrites 98.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 80.4%

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

   10. Applied rewrites 82.2%

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

   if 1.99999999999999992e-74 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 80.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 62.6%

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

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -2 \cdot 10^{-296}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.027777777777777776, x \cdot x, 0.16666666666666666\right) \cdot y\right) \cdot \left(y \cdot y\right)\\ \mathbf{elif}\;\frac{\sinh y \cdot \sin x}{x} \leq 2 \cdot 10^{-74}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right) \cdot x}{\frac{x}{y}}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right), x \cdot x, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right)\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 60.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y \cdot \sin x}{x}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-296}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.027777777777777776, x \cdot x, 0.16666666666666666\right) \cdot y\right) \cdot \left(y \cdot y\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-156}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right) \cdot x}{\frac{x}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot y\right) \cdot \left(1 \cdot x\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sinh y) (sin x)) x)))
   (if (<= t_0 -2e-296)
     (* (* (fma -0.027777777777777776 (* x x) 0.16666666666666666) y) (* y y))
     (if (<= t_0 2e-156)
       (/ (* (fma 0.16666666666666666 (* y y) 1.0) x) (/ x y))
       (/
        (*
         (*
          (fma
           (fma 0.008333333333333333 (* y y) 0.16666666666666666)
           (* y y)
           1.0)
          y)
         (* 1.0 x))
        x)))))

C

double code(double x, double y) {
	double t_0 = (sinh(y) * sin(x)) / x;
	double tmp;
	if (t_0 <= -2e-296) {
		tmp = (fma(-0.027777777777777776, (x * x), 0.16666666666666666) * y) * (y * y);
	} else if (t_0 <= 2e-156) {
		tmp = (fma(0.16666666666666666, (y * y), 1.0) * x) / (x / y);
	} else {
		tmp = ((fma(fma(0.008333333333333333, (y * y), 0.16666666666666666), (y * y), 1.0) * y) * (1.0 * x)) / x;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(sinh(y) * sin(x)) / x)
	tmp = 0.0
	if (t_0 <= -2e-296)
		tmp = Float64(Float64(fma(-0.027777777777777776, Float64(x * x), 0.16666666666666666) * y) * Float64(y * y));
	elseif (t_0 <= 2e-156)
		tmp = Float64(Float64(fma(0.16666666666666666, Float64(y * y), 1.0) * x) / Float64(x / y));
	else
		tmp = Float64(Float64(Float64(fma(fma(0.008333333333333333, Float64(y * y), 0.16666666666666666), Float64(y * y), 1.0) * y) * Float64(1.0 * x)) / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sinh[y], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-296], N[(N[(N[(-0.027777777777777776 * N[(x * x), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * y), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e-156], N[(N[(N[(0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision] / N[(x / y), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(0.008333333333333333 * N[(y * y), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision] * N[(1.0 * x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -2e-296

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. distribute-rgt-in N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. associate-*l/ N/A

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

      8. associate-/l* N/A

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

      9. distribute-rgt-out N/A

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

      10. +-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

   5. Applied rewrites 75.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 60.7%

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

   9. Taylor expanded in y around inf

      \[\leadsto \frac{1}{6} \cdot \left({y}^{3} \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {x}^{2}\right)}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 40.1%

       \[\leadsto \left(\mathsf{fma}\left(-0.027777777777777776, x \cdot x, 0.16666666666666666\right) \cdot y\right) \cdot \left(y \cdot \color{blue}{y}\right) \]

   if -2e-296 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 2.00000000000000008e-156

   1. Initial program 60.3%

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

   3. Taylor expanded in y around 0

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

      1. distribute-rgt-in N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. associate-*l/ N/A

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

      8. associate-/l* N/A

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

      9. distribute-rgt-out N/A

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

      10. +-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

   5. Applied rewrites 98.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 85.3%

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

   10. Applied rewrites 87.4%

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

   if 2.00000000000000008e-156 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 70.7

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

   5. Applied rewrites 70.7%

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

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 59.5

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

   8. Applied rewrites 59.5%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 60.5%

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

4. Final simplification 58.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -2 \cdot 10^{-296}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.027777777777777776, x \cdot x, 0.16666666666666666\right) \cdot y\right) \cdot \left(y \cdot y\right)\\ \mathbf{elif}\;\frac{\sinh y \cdot \sin x}{x} \leq 2 \cdot 10^{-156}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right) \cdot x}{\frac{x}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot y\right) \cdot \left(1 \cdot x\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 59.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y \cdot \sin x}{x}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-296}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.027777777777777776, x \cdot x, 0.16666666666666666\right) \cdot y\right) \cdot \left(y \cdot y\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-216}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right) \cdot x}{\frac{x}{y}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sinh y) (sin x)) x)))
   (if (<= t_0 -2e-296)
     (* (* (fma -0.027777777777777776 (* x x) 0.16666666666666666) y) (* y y))
     (if (<= t_0 2e-216)
       (/ (* (fma 0.16666666666666666 (* y y) 1.0) x) (/ x y))
       (*
        (fma
         (fma 0.008333333333333333 (* y y) 0.16666666666666666)
         (* y y)
         1.0)
        y)))))

C

double code(double x, double y) {
	double t_0 = (sinh(y) * sin(x)) / x;
	double tmp;
	if (t_0 <= -2e-296) {
		tmp = (fma(-0.027777777777777776, (x * x), 0.16666666666666666) * y) * (y * y);
	} else if (t_0 <= 2e-216) {
		tmp = (fma(0.16666666666666666, (y * y), 1.0) * x) / (x / y);
	} else {
		tmp = fma(fma(0.008333333333333333, (y * y), 0.16666666666666666), (y * y), 1.0) * y;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(sinh(y) * sin(x)) / x)
	tmp = 0.0
	if (t_0 <= -2e-296)
		tmp = Float64(Float64(fma(-0.027777777777777776, Float64(x * x), 0.16666666666666666) * y) * Float64(y * y));
	elseif (t_0 <= 2e-216)
		tmp = Float64(Float64(fma(0.16666666666666666, Float64(y * y), 1.0) * x) / Float64(x / y));
	else
		tmp = Float64(fma(fma(0.008333333333333333, Float64(y * y), 0.16666666666666666), Float64(y * y), 1.0) * y);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sinh[y], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-296], N[(N[(N[(-0.027777777777777776 * N[(x * x), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * y), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e-216], N[(N[(N[(0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision] / N[(x / y), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.008333333333333333 * N[(y * y), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -2e-296

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. distribute-rgt-in N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. associate-*l/ N/A

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

      8. associate-/l* N/A

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

      9. distribute-rgt-out N/A

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

      10. +-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

   5. Applied rewrites 75.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 60.7%

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

   9. Taylor expanded in y around inf

      \[\leadsto \frac{1}{6} \cdot \left({y}^{3} \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {x}^{2}\right)}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 40.1%

       \[\leadsto \left(\mathsf{fma}\left(-0.027777777777777776, x \cdot x, 0.16666666666666666\right) \cdot y\right) \cdot \left(y \cdot \color{blue}{y}\right) \]

   if -2e-296 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 2.0000000000000001e-216

   1. Initial program 57.4%

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

   3. Taylor expanded in y around 0

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

      1. distribute-rgt-in N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. associate-*l/ N/A

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

      8. associate-/l* N/A

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

      9. distribute-rgt-out N/A

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

      10. +-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

   5. Applied rewrites 98.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 89.5%

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

   10. Applied rewrites 91.8%

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

   if 2.0000000000000001e-216 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 82.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 58.4%

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

4. Final simplification 58.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -2 \cdot 10^{-296}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.027777777777777776, x \cdot x, 0.16666666666666666\right) \cdot y\right) \cdot \left(y \cdot y\right)\\ \mathbf{elif}\;\frac{\sinh y \cdot \sin x}{x} \leq 2 \cdot 10^{-216}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right) \cdot x}{\frac{x}{y}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 59.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y \cdot \sin x}{x}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-296}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.027777777777777776, x \cdot x, 0.16666666666666666\right) \cdot y\right) \cdot \left(y \cdot y\right)\\ \mathbf{elif}\;t\_0 \leq 10^{-164}:\\ \;\;\;\;\left(1 \cdot x\right) \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sinh y) (sin x)) x)))
   (if (<= t_0 -2e-296)
     (* (* (fma -0.027777777777777776 (* x x) 0.16666666666666666) y) (* y y))
     (if (<= t_0 1e-164)
       (* (* 1.0 x) (/ y x))
       (*
        (fma
         (fma 0.008333333333333333 (* y y) 0.16666666666666666)
         (* y y)
         1.0)
        y)))))

C

double code(double x, double y) {
	double t_0 = (sinh(y) * sin(x)) / x;
	double tmp;
	if (t_0 <= -2e-296) {
		tmp = (fma(-0.027777777777777776, (x * x), 0.16666666666666666) * y) * (y * y);
	} else if (t_0 <= 1e-164) {
		tmp = (1.0 * x) * (y / x);
	} else {
		tmp = fma(fma(0.008333333333333333, (y * y), 0.16666666666666666), (y * y), 1.0) * y;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(sinh(y) * sin(x)) / x)
	tmp = 0.0
	if (t_0 <= -2e-296)
		tmp = Float64(Float64(fma(-0.027777777777777776, Float64(x * x), 0.16666666666666666) * y) * Float64(y * y));
	elseif (t_0 <= 1e-164)
		tmp = Float64(Float64(1.0 * x) * Float64(y / x));
	else
		tmp = Float64(fma(fma(0.008333333333333333, Float64(y * y), 0.16666666666666666), Float64(y * y), 1.0) * y);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sinh[y], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-296], N[(N[(N[(-0.027777777777777776 * N[(x * x), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * y), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e-164], N[(N[(1.0 * x), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.008333333333333333 * N[(y * y), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -2e-296

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. distribute-rgt-in N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. associate-*l/ N/A

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

      8. associate-/l* N/A

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

      9. distribute-rgt-out N/A

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

      10. +-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

   5. Applied rewrites 75.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 60.7%

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

   9. Taylor expanded in y around inf

      \[\leadsto \frac{1}{6} \cdot \left({y}^{3} \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {x}^{2}\right)}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 40.1%

       \[\leadsto \left(\mathsf{fma}\left(-0.027777777777777776, x \cdot x, 0.16666666666666666\right) \cdot y\right) \cdot \left(y \cdot \color{blue}{y}\right) \]

   if -2e-296 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 9.99999999999999962e-165

   1. Initial program 59.6%

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

   3. Taylor expanded in y around 0

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

      1. distribute-rgt-in N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. associate-*l/ N/A

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

      8. associate-/l* N/A

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

      9. distribute-rgt-out N/A

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

      10. +-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

   5. Applied rewrites 98.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 86.7%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 86.7%

       \[\leadsto \frac{y}{x} \cdot \left(1 \cdot x\right) \]

   if 9.99999999999999962e-165 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 82.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 59.1%

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

4. Final simplification 58.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -2 \cdot 10^{-296}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.027777777777777776, x \cdot x, 0.16666666666666666\right) \cdot y\right) \cdot \left(y \cdot y\right)\\ \mathbf{elif}\;\frac{\sinh y \cdot \sin x}{x} \leq 10^{-164}:\\ \;\;\;\;\left(1 \cdot x\right) \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 56.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y \cdot \sin x}{x}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-296}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.027777777777777776, x \cdot x, 0.16666666666666666\right) \cdot y\right) \cdot \left(y \cdot y\right)\\ \mathbf{elif}\;t\_0 \leq 10^{-164}:\\ \;\;\;\;\left(1 \cdot x\right) \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.16666666666666666 \cdot y, y, 1\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sinh y) (sin x)) x)))
   (if (<= t_0 -2e-296)
     (* (* (fma -0.027777777777777776 (* x x) 0.16666666666666666) y) (* y y))
     (if (<= t_0 1e-164)
       (* (* 1.0 x) (/ y x))
       (* (fma (* 0.16666666666666666 y) y 1.0) y)))))

C

double code(double x, double y) {
	double t_0 = (sinh(y) * sin(x)) / x;
	double tmp;
	if (t_0 <= -2e-296) {
		tmp = (fma(-0.027777777777777776, (x * x), 0.16666666666666666) * y) * (y * y);
	} else if (t_0 <= 1e-164) {
		tmp = (1.0 * x) * (y / x);
	} else {
		tmp = fma((0.16666666666666666 * y), y, 1.0) * y;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(sinh(y) * sin(x)) / x)
	tmp = 0.0
	if (t_0 <= -2e-296)
		tmp = Float64(Float64(fma(-0.027777777777777776, Float64(x * x), 0.16666666666666666) * y) * Float64(y * y));
	elseif (t_0 <= 1e-164)
		tmp = Float64(Float64(1.0 * x) * Float64(y / x));
	else
		tmp = Float64(fma(Float64(0.16666666666666666 * y), y, 1.0) * y);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sinh[y], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-296], N[(N[(N[(-0.027777777777777776 * N[(x * x), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * y), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e-164], N[(N[(1.0 * x), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.16666666666666666 * y), $MachinePrecision] * y + 1.0), $MachinePrecision] * y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -2e-296

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. distribute-rgt-in N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. associate-*l/ N/A

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

      8. associate-/l* N/A

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

      9. distribute-rgt-out N/A

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

      10. +-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

   5. Applied rewrites 75.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 60.7%

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

   9. Taylor expanded in y around inf

      \[\leadsto \frac{1}{6} \cdot \left({y}^{3} \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {x}^{2}\right)}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 40.1%

       \[\leadsto \left(\mathsf{fma}\left(-0.027777777777777776, x \cdot x, 0.16666666666666666\right) \cdot y\right) \cdot \left(y \cdot \color{blue}{y}\right) \]

   if -2e-296 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 9.99999999999999962e-165

   1. Initial program 59.6%

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

   3. Taylor expanded in y around 0

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

      1. distribute-rgt-in N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. associate-*l/ N/A

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

      8. associate-/l* N/A

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

      9. distribute-rgt-out N/A

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

      10. +-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

   5. Applied rewrites 98.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 86.7%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 86.7%

       \[\leadsto \frac{y}{x} \cdot \left(1 \cdot x\right) \]

   if 9.99999999999999962e-165 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. distribute-rgt-in N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. associate-*l/ N/A

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

      8. associate-/l* N/A

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

      9. distribute-rgt-out N/A

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

      10. +-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

   5. Applied rewrites 63.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 53.0%

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

   10. Applied rewrites 53.0%

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

4. Final simplification 55.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -2 \cdot 10^{-296}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.027777777777777776, x \cdot x, 0.16666666666666666\right) \cdot y\right) \cdot \left(y \cdot y\right)\\ \mathbf{elif}\;\frac{\sinh y \cdot \sin x}{x} \leq 10^{-164}:\\ \;\;\;\;\left(1 \cdot x\right) \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.16666666666666666 \cdot y, y, 1\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 53.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y \cdot \sin x}{x}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-170}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, x \cdot x, 1\right) \cdot y\\ \mathbf{elif}\;t\_0 \leq 10^{-164}:\\ \;\;\;\;\left(1 \cdot x\right) \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.16666666666666666 \cdot y, y, 1\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sinh y) (sin x)) x)))
   (if (<= t_0 -2e-170)
     (* (fma -0.16666666666666666 (* x x) 1.0) y)
     (if (<= t_0 1e-164)
       (* (* 1.0 x) (/ y x))
       (* (fma (* 0.16666666666666666 y) y 1.0) y)))))

C

double code(double x, double y) {
	double t_0 = (sinh(y) * sin(x)) / x;
	double tmp;
	if (t_0 <= -2e-170) {
		tmp = fma(-0.16666666666666666, (x * x), 1.0) * y;
	} else if (t_0 <= 1e-164) {
		tmp = (1.0 * x) * (y / x);
	} else {
		tmp = fma((0.16666666666666666 * y), y, 1.0) * y;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(sinh(y) * sin(x)) / x)
	tmp = 0.0
	if (t_0 <= -2e-170)
		tmp = Float64(fma(-0.16666666666666666, Float64(x * x), 1.0) * y);
	elseif (t_0 <= 1e-164)
		tmp = Float64(Float64(1.0 * x) * Float64(y / x));
	else
		tmp = Float64(fma(Float64(0.16666666666666666 * y), y, 1.0) * y);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sinh[y], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-170], N[(N[(-0.16666666666666666 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t$95$0, 1e-164], N[(N[(1.0 * x), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.16666666666666666 * y), $MachinePrecision] * y + 1.0), $MachinePrecision] * y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -1.99999999999999997e-170

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sin.f64 30.2

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

   5. Applied rewrites 30.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 39.9%

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

   if -1.99999999999999997e-170 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 9.99999999999999962e-165

   1. Initial program 67.0%

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

   3. Taylor expanded in y around 0

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

      1. distribute-rgt-in N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. associate-*l/ N/A

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

      8. associate-/l* N/A

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

      9. distribute-rgt-out N/A

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

      10. +-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

   5. Applied rewrites 98.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 75.8%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 75.8%

       \[\leadsto \frac{y}{x} \cdot \left(1 \cdot x\right) \]

   if 9.99999999999999962e-165 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. distribute-rgt-in N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. associate-*l/ N/A

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

      8. associate-/l* N/A

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

      9. distribute-rgt-out N/A

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

      10. +-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

   5. Applied rewrites 63.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 53.0%

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

   10. Applied rewrites 53.0%

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

4. Final simplification 54.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -2 \cdot 10^{-170}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, x \cdot x, 1\right) \cdot y\\ \mathbf{elif}\;\frac{\sinh y \cdot \sin x}{x} \leq 10^{-164}:\\ \;\;\;\;\left(1 \cdot x\right) \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.16666666666666666 \cdot y, y, 1\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 87.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq 2 \cdot 10^{-74}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot \frac{\sin x}{x}\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \sinh y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (sinh y) (sin x)) x) 2e-74)
   (*
    (*
     (fma (fma 0.008333333333333333 (* y y) 0.16666666666666666) (* y y) 1.0)
     (/ (sin x) x))
    y)
   (* 1.0 (sinh y))))

C

double code(double x, double y) {
	double tmp;
	if (((sinh(y) * sin(x)) / x) <= 2e-74) {
		tmp = (fma(fma(0.008333333333333333, (y * y), 0.16666666666666666), (y * y), 1.0) * (sin(x) / x)) * y;
	} else {
		tmp = 1.0 * sinh(y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(sinh(y) * sin(x)) / x) <= 2e-74)
		tmp = Float64(Float64(fma(fma(0.008333333333333333, Float64(y * y), 0.16666666666666666), Float64(y * y), 1.0) * Float64(sin(x) / x)) * y);
	else
		tmp = Float64(1.0 * sinh(y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[Sinh[y], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], 2e-74], N[(N[(N[(N[(0.008333333333333333 * N[(y * y), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], N[(1.0 * N[Sinh[y], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 1.99999999999999992e-74

   1. Initial program 86.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 91.5%

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

   if 1.99999999999999992e-74 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 100.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 75.0%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq 2 \cdot 10^{-74}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot \frac{\sin x}{x}\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \sinh y\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 43.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -2 \cdot 10^{-170}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, x \cdot x, 1\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.16666666666666666 \cdot y, y, 1\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (sinh y) (sin x)) x) -2e-170)
   (* (fma -0.16666666666666666 (* x x) 1.0) y)
   (* (fma (* 0.16666666666666666 y) y 1.0) y)))

C

double code(double x, double y) {
	double tmp;
	if (((sinh(y) * sin(x)) / x) <= -2e-170) {
		tmp = fma(-0.16666666666666666, (x * x), 1.0) * y;
	} else {
		tmp = fma((0.16666666666666666 * y), y, 1.0) * y;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(sinh(y) * sin(x)) / x) <= -2e-170)
		tmp = Float64(fma(-0.16666666666666666, Float64(x * x), 1.0) * y);
	else
		tmp = Float64(fma(Float64(0.16666666666666666 * y), y, 1.0) * y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[Sinh[y], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], -2e-170], N[(N[(-0.16666666666666666 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision], N[(N[(N[(0.16666666666666666 * y), $MachinePrecision] * y + 1.0), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -1.99999999999999997e-170

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sin.f64 30.2

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

   5. Applied rewrites 30.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 39.9%

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

   if -1.99999999999999997e-170 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 86.2%

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

   3. Taylor expanded in y around 0

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

      1. distribute-rgt-in N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. associate-*l/ N/A

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

      8. associate-/l* N/A

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

      9. distribute-rgt-out N/A

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

      10. +-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

   5. Applied rewrites 78.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 49.3%

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

   10. Applied rewrites 49.3%

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

4. Final simplification 46.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -2 \cdot 10^{-170}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, x \cdot x, 1\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.16666666666666666 \cdot y, y, 1\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 39.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq 1:\\ \;\;\;\;1 \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y \cdot y\right) \cdot y\right) \cdot 0.16666666666666666\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (sinh y) (sin x)) x) 1.0)
   (* 1.0 y)
   (* (* (* y y) y) 0.16666666666666666)))

C

double code(double x, double y) {
	double tmp;
	if (((sinh(y) * sin(x)) / x) <= 1.0) {
		tmp = 1.0 * y;
	} else {
		tmp = ((y * y) * y) * 0.16666666666666666;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (((sinh(y) * sin(x)) / x) <= 1.0d0) then
        tmp = 1.0d0 * y
    else
        tmp = ((y * y) * y) * 0.16666666666666666d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (((Math.sinh(y) * Math.sin(x)) / x) <= 1.0) {
		tmp = 1.0 * y;
	} else {
		tmp = ((y * y) * y) * 0.16666666666666666;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((math.sinh(y) * math.sin(x)) / x) <= 1.0:
		tmp = 1.0 * y
	else:
		tmp = ((y * y) * y) * 0.16666666666666666
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(sinh(y) * sin(x)) / x) <= 1.0)
		tmp = Float64(1.0 * y);
	else
		tmp = Float64(Float64(Float64(y * y) * y) * 0.16666666666666666);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((sinh(y) * sin(x)) / x) <= 1.0)
		tmp = 1.0 * y;
	else
		tmp = ((y * y) * y) * 0.16666666666666666;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[Sinh[y], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], 1.0], N[(1.0 * y), $MachinePrecision], N[(N[(N[(y * y), $MachinePrecision] * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 1

   1. Initial program 86.6%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sin.f64 64.7

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

   5. Applied rewrites 64.7%

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

   6. Taylor expanded in x around 0

      \[\leadsto 1 \cdot y \]
   7. Step-by-step derivation

   8. Applied rewrites 37.3%

      \[\leadsto 1 \cdot y \]

   if 1 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(\frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x}\right) \cdot y + \frac{\sin x}{x} \cdot y} \]

      2. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{6} \cdot \left({y}^{2} \cdot \sin x\right)}{x}} \cdot y + \frac{\sin x}{x} \cdot y \]

      3. *-commutative N/A

         \[\leadsto \frac{\frac{1}{6} \cdot \color{blue}{\left(\sin x \cdot {y}^{2}\right)}}{x} \cdot y + \frac{\sin x}{x} \cdot y \]

      4. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}}}{x} \cdot y + \frac{\sin x}{x} \cdot y \]

      5. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\left(\left(\frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}\right) \cdot y}{x}} + \frac{\sin x}{x} \cdot y \]

      6. associate-/l* N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}\right) \cdot \frac{y}{x}} + \frac{\sin x}{x} \cdot y \]

      7. associate-*l/ N/A

         \[\leadsto \left(\left(\frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}\right) \cdot \frac{y}{x} + \color{blue}{\frac{\sin x \cdot y}{x}} \]

      8. associate-/l* N/A

         \[\leadsto \left(\left(\frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}\right) \cdot \frac{y}{x} + \color{blue}{\sin x \cdot \frac{y}{x}} \]

      9. distribute-rgt-out N/A

         \[\leadsto \color{blue}{\frac{y}{x} \cdot \left(\left(\frac{1}{6} \cdot \sin x\right) \cdot {y}^{2} + \sin x\right)} \]

      10. +-commutative N/A

          \[\leadsto \frac{y}{x} \cdot \color{blue}{\left(\sin x + \left(\frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{y}{x} \cdot \left(\sin x + \left(\frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}\right)} \]

      12. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{y}{x}} \cdot \left(\sin x + \left(\frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}\right) \]

      13. associate-*r* N/A

          \[\leadsto \frac{y}{x} \cdot \left(\sin x + \color{blue}{\frac{1}{6} \cdot \left(\sin x \cdot {y}^{2}\right)}\right) \]

      14. *-commutative N/A

          \[\leadsto \frac{y}{x} \cdot \left(\sin x + \frac{1}{6} \cdot \color{blue}{\left({y}^{2} \cdot \sin x\right)}\right) \]

      15. associate-*r* N/A

          \[\leadsto \frac{y}{x} \cdot \left(\sin x + \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \sin x}\right) \]

   5. Applied rewrites 55.8%

      \[\leadsto \color{blue}{\frac{y}{x} \cdot \left(\mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right) \cdot \sin x\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto y \cdot \color{blue}{\left(1 + \frac{1}{6} \cdot {y}^{2}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 50.0%

      \[\leadsto \mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right) \cdot \color{blue}{y} \]

   9. Taylor expanded in y around inf

      \[\leadsto \frac{1}{6} \cdot {y}^{\color{blue}{3}} \]
   10. Step-by-step derivation

   11. Applied rewrites 50.0%

       \[\leadsto \left(\left(y \cdot y\right) \cdot y\right) \cdot 0.16666666666666666 \]
3. Recombined 2 regimes into one program.

4. Final simplification 41.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq 1:\\ \;\;\;\;1 \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y \cdot y\right) \cdot y\right) \cdot 0.16666666666666666\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 55.7% accurate, 9.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.9 \cdot 10^{+28}:\\ \;\;\;\;\mathsf{fma}\left(0.16666666666666666 \cdot y, y, 1\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y \cdot y\right) \cdot y\right) \cdot 0.16666666666666666\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 2.9e+28)
   (* (fma (* 0.16666666666666666 y) y 1.0) y)
   (* (* (* y y) y) 0.16666666666666666)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 2.9e+28) {
		tmp = fma((0.16666666666666666 * y), y, 1.0) * y;
	} else {
		tmp = ((y * y) * y) * 0.16666666666666666;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 2.9e+28)
		tmp = Float64(fma(Float64(0.16666666666666666 * y), y, 1.0) * y);
	else
		tmp = Float64(Float64(Float64(y * y) * y) * 0.16666666666666666);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 2.9e+28], N[(N[(N[(0.16666666666666666 * y), $MachinePrecision] * y + 1.0), $MachinePrecision] * y), $MachinePrecision], N[(N[(N[(y * y), $MachinePrecision] * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.9000000000000001e28

   1. Initial program 87.7%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left(\frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{\sin x}{x}\right)} \]
   4. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(\frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x}\right) \cdot y + \frac{\sin x}{x} \cdot y} \]

      2. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{6} \cdot \left({y}^{2} \cdot \sin x\right)}{x}} \cdot y + \frac{\sin x}{x} \cdot y \]

      3. *-commutative N/A

         \[\leadsto \frac{\frac{1}{6} \cdot \color{blue}{\left(\sin x \cdot {y}^{2}\right)}}{x} \cdot y + \frac{\sin x}{x} \cdot y \]

      4. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}}}{x} \cdot y + \frac{\sin x}{x} \cdot y \]

      5. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\left(\left(\frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}\right) \cdot y}{x}} + \frac{\sin x}{x} \cdot y \]

      6. associate-/l* N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}\right) \cdot \frac{y}{x}} + \frac{\sin x}{x} \cdot y \]

      7. associate-*l/ N/A

         \[\leadsto \left(\left(\frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}\right) \cdot \frac{y}{x} + \color{blue}{\frac{\sin x \cdot y}{x}} \]

      8. associate-/l* N/A

         \[\leadsto \left(\left(\frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}\right) \cdot \frac{y}{x} + \color{blue}{\sin x \cdot \frac{y}{x}} \]

      9. distribute-rgt-out N/A

         \[\leadsto \color{blue}{\frac{y}{x} \cdot \left(\left(\frac{1}{6} \cdot \sin x\right) \cdot {y}^{2} + \sin x\right)} \]

      10. +-commutative N/A

          \[\leadsto \frac{y}{x} \cdot \color{blue}{\left(\sin x + \left(\frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{y}{x} \cdot \left(\sin x + \left(\frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}\right)} \]

      12. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{y}{x}} \cdot \left(\sin x + \left(\frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}\right) \]

      13. associate-*r* N/A

          \[\leadsto \frac{y}{x} \cdot \left(\sin x + \color{blue}{\frac{1}{6} \cdot \left(\sin x \cdot {y}^{2}\right)}\right) \]

      14. *-commutative N/A

          \[\leadsto \frac{y}{x} \cdot \left(\sin x + \frac{1}{6} \cdot \color{blue}{\left({y}^{2} \cdot \sin x\right)}\right) \]

      15. associate-*r* N/A

          \[\leadsto \frac{y}{x} \cdot \left(\sin x + \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \sin x}\right) \]

   5. Applied rewrites 78.9%

      \[\leadsto \color{blue}{\frac{y}{x} \cdot \left(\mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right) \cdot \sin x\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto y \cdot \color{blue}{\left(1 + \frac{1}{6} \cdot {y}^{2}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 62.3%

      \[\leadsto \mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right) \cdot \color{blue}{y} \]
   9. Step-by-step derivation

   10. Applied rewrites 62.3%

       \[\leadsto \mathsf{fma}\left(0.16666666666666666 \cdot y, y, 1\right) \cdot y \]

   if 2.9000000000000001e28 < x

   1. Initial program 99.9%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left(\frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{\sin x}{x}\right)} \]
   4. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(\frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x}\right) \cdot y + \frac{\sin x}{x} \cdot y} \]

      2. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{6} \cdot \left({y}^{2} \cdot \sin x\right)}{x}} \cdot y + \frac{\sin x}{x} \cdot y \]

      3. *-commutative N/A

         \[\leadsto \frac{\frac{1}{6} \cdot \color{blue}{\left(\sin x \cdot {y}^{2}\right)}}{x} \cdot y + \frac{\sin x}{x} \cdot y \]

      4. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}}}{x} \cdot y + \frac{\sin x}{x} \cdot y \]

      5. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\left(\left(\frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}\right) \cdot y}{x}} + \frac{\sin x}{x} \cdot y \]

      6. associate-/l* N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}\right) \cdot \frac{y}{x}} + \frac{\sin x}{x} \cdot y \]

      7. associate-*l/ N/A

         \[\leadsto \left(\left(\frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}\right) \cdot \frac{y}{x} + \color{blue}{\frac{\sin x \cdot y}{x}} \]

      8. associate-/l* N/A

         \[\leadsto \left(\left(\frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}\right) \cdot \frac{y}{x} + \color{blue}{\sin x \cdot \frac{y}{x}} \]

      9. distribute-rgt-out N/A

         \[\leadsto \color{blue}{\frac{y}{x} \cdot \left(\left(\frac{1}{6} \cdot \sin x\right) \cdot {y}^{2} + \sin x\right)} \]

      10. +-commutative N/A

          \[\leadsto \frac{y}{x} \cdot \color{blue}{\left(\sin x + \left(\frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{y}{x} \cdot \left(\sin x + \left(\frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}\right)} \]

      12. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{y}{x}} \cdot \left(\sin x + \left(\frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}\right) \]

      13. associate-*r* N/A

          \[\leadsto \frac{y}{x} \cdot \left(\sin x + \color{blue}{\frac{1}{6} \cdot \left(\sin x \cdot {y}^{2}\right)}\right) \]

      14. *-commutative N/A

          \[\leadsto \frac{y}{x} \cdot \left(\sin x + \frac{1}{6} \cdot \color{blue}{\left({y}^{2} \cdot \sin x\right)}\right) \]

      15. associate-*r* N/A

          \[\leadsto \frac{y}{x} \cdot \left(\sin x + \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \sin x}\right) \]

   5. Applied rewrites 68.0%

      \[\leadsto \color{blue}{\frac{y}{x} \cdot \left(\mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right) \cdot \sin x\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto y \cdot \color{blue}{\left(1 + \frac{1}{6} \cdot {y}^{2}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 24.8%

      \[\leadsto \mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right) \cdot \color{blue}{y} \]

   9. Taylor expanded in y around inf

      \[\leadsto \frac{1}{6} \cdot {y}^{\color{blue}{3}} \]
   10. Step-by-step derivation

   11. Applied rewrites 35.2%

       \[\leadsto \left(\left(y \cdot y\right) \cdot y\right) \cdot 0.16666666666666666 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 27.6% accurate, 36.2× speedup

Math

\[\begin{array}{l} \\ 1 \cdot y \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* 1.0 y))

C

double code(double x, double y) {
	return 1.0 * y;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 * y
end function

Java

public static double code(double x, double y) {
	return 1.0 * y;
}

Python

def code(x, y):
	return 1.0 * y

Julia

function code(x, y)
	return Float64(1.0 * y)
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0 * y;
end

Wolfram

code[x_, y_] := N[(1.0 * y), $MachinePrecision]

Derivation

1. Initial program 90.8%

   \[\frac{\sin x \cdot \sinh y}{x} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\frac{y \cdot \sin x}{x}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\sin x \cdot y}}{x} \]

   2. associate-*l/ N/A

      \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot y} \]

   3. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot y} \]

   4. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\sin x}{x}} \cdot y \]

   5. lower-sin.f64 45.7

      \[\leadsto \frac{\color{blue}{\sin x}}{x} \cdot y \]

5. Applied rewrites 45.7%

   \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot y} \]

6. Taylor expanded in x around 0

   \[\leadsto 1 \cdot y \]
7. Step-by-step derivation

8. Applied rewrites 26.8%

   \[\leadsto 1 \cdot y \]
9. Add Preprocessing

Developer Target 1: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sin x \cdot \frac{\sinh y}{x} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* (sin x) (/ (sinh y) x)))

C

double code(double x, double y) {
	return sin(x) * (sinh(y) / x);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = sin(x) * (sinh(y) / x)
end function

Java

public static double code(double x, double y) {
	return Math.sin(x) * (Math.sinh(y) / x);
}

Python

def code(x, y):
	return math.sin(x) * (math.sinh(y) / x)

Julia

function code(x, y)
	return Float64(sin(x) * Float64(sinh(y) / x))
end

MATLAB

function tmp = code(x, y)
	tmp = sin(x) * (sinh(y) / x);
end

Wolfram

code[x_, y_] := N[(N[Sin[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024235 
(FPCore (x y)
  :name "Linear.Quaternion:$ccosh from linear-1.19.1.3"
  :precision binary64

  :alt
  (! :herbie-platform default (* (sin x) (/ (sinh y) x)))

  (/ (* (sin x) (sinh y)) x))