Linear.Quaternion:$ccosh from linear-1.19.1.3

Percentage Accurate: 88.9% → 99.9%

Time: 12.2s

Alternatives: 23

Speedup: 1.0×

• 49.6% 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: 88.9% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.8%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. lift-*.f64 N/A

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

   4. associate-/r* N/A

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

   5. clear-num N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 99.9

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

4. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\frac{\sinh y}{\frac{x}{\sin x}}} \]
5. Add Preprocessing

Alternative 2: 84.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y \cdot \sin x}{x}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot -0.16666666666666666, x\right) \cdot \mathsf{fma}\left(y \cdot y, y \cdot \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), y\right)}{x}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.16666666666666666, y \cdot \left(y \cdot y\right), y\right)}{\frac{1}{\frac{\sin x}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sinh y}{x} \cdot \mathsf{fma}\left(x \cdot x, x \cdot \mathsf{fma}\left(x, x \cdot 0.008333333333333333, -0.16666666666666666\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sinh y) (sin x)) x)))
   (if (<= t_0 (- INFINITY))
     (/
      (*
       (fma x (* (* x x) -0.16666666666666666) x)
       (fma
        (* y y)
        (* y (fma y (* y 0.008333333333333333) 0.16666666666666666))
        y))
      x)
     (if (<= t_0 2e-6)
       (/ (fma 0.16666666666666666 (* y (* y y)) y) (/ 1.0 (/ (sin x) x)))
       (*
        (/ (sinh y) x)
        (fma
         (* x x)
         (* x (fma x (* x 0.008333333333333333) -0.16666666666666666))
         x))))))

C

double code(double x, double y) {
	double t_0 = (sinh(y) * sin(x)) / x;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (fma(x, ((x * x) * -0.16666666666666666), x) * fma((y * y), (y * fma(y, (y * 0.008333333333333333), 0.16666666666666666)), y)) / x;
	} else if (t_0 <= 2e-6) {
		tmp = fma(0.16666666666666666, (y * (y * y)), y) / (1.0 / (sin(x) / x));
	} else {
		tmp = (sinh(y) / x) * fma((x * x), (x * fma(x, (x * 0.008333333333333333), -0.16666666666666666)), x);
	}
	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(fma(x, Float64(Float64(x * x) * -0.16666666666666666), x) * fma(Float64(y * y), Float64(y * fma(y, Float64(y * 0.008333333333333333), 0.16666666666666666)), y)) / x);
	elseif (t_0 <= 2e-6)
		tmp = Float64(fma(0.16666666666666666, Float64(y * Float64(y * y)), y) / Float64(1.0 / Float64(sin(x) / x)));
	else
		tmp = Float64(Float64(sinh(y) / x) * fma(Float64(x * x), Float64(x * fma(x, Float64(x * 0.008333333333333333), -0.16666666666666666)), 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, (-Infinity)], N[(N[(N[(x * N[(N[(x * x), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + x), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * N[(y * 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[t$95$0, 2e-6], N[(N[(0.16666666666666666 * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] / N[(1.0 / N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sinh[y], $MachinePrecision] / x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(x * 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + x), $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 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

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 84.2

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

   5. Applied rewrites 84.2%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-*l* N/A

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

      4. *-lft-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 66.1

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

   8. Applied rewrites 66.1%

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

   if -inf.0 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 1.99999999999999991e-6

   1. Initial program 75.9%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. clear-num N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. unpow2 N/A

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

      6. unpow3 N/A

         \[\leadsto \frac{\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot \color{blue}{{y}^{3}} + 1 \cdot y}{\frac{x}{\sin x}} \]

      7. *-lft-identity N/A

         \[\leadsto \frac{\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot {y}^{3} + \color{blue}{y}}{\frac{x}{\sin x}} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right), {y}^{3}, y\right)}}{\frac{x}{\sin x}} \]

   7. Applied rewrites 99.3%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)}}{\frac{x}{\sin x}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lift-sin.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lower-/.f64 99.2

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

   9. Applied rewrites 99.2%

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

   10. Taylor expanded in y around 0

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

   12. Applied rewrites 99.2%

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

   if 1.99999999999999991e-6 < (/.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

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. sub-neg N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 83.0

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

   5. Applied rewrites 83.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.008333333333333333, -0.16666666666666666\right), x \cdot \left(x \cdot x\right), x\right)} \cdot \sinh y}{x} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 83.1

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

   7. Applied rewrites 83.1%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -\infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot -0.16666666666666666, x\right) \cdot \mathsf{fma}\left(y \cdot y, y \cdot \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), y\right)}{x}\\ \mathbf{elif}\;\frac{\sinh y \cdot \sin x}{x} \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.16666666666666666, y \cdot \left(y \cdot y\right), y\right)}{\frac{1}{\frac{\sin x}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sinh y}{x} \cdot \mathsf{fma}\left(x \cdot x, x \cdot \mathsf{fma}\left(x, x \cdot 0.008333333333333333, -0.16666666666666666\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y \cdot \sin x}{x}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot -0.16666666666666666, x\right) \cdot \mathsf{fma}\left(y \cdot y, y \cdot \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), y\right)}{x}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\frac{\sin x}{x} \cdot \mathsf{fma}\left(y \cdot y, y \cdot 0.16666666666666666, y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sinh y}{x} \cdot \mathsf{fma}\left(x \cdot x, x \cdot \mathsf{fma}\left(x, x \cdot 0.008333333333333333, -0.16666666666666666\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sinh y) (sin x)) x)))
   (if (<= t_0 (- INFINITY))
     (/
      (*
       (fma x (* (* x x) -0.16666666666666666) x)
       (fma
        (* y y)
        (* y (fma y (* y 0.008333333333333333) 0.16666666666666666))
        y))
      x)
     (if (<= t_0 2e-6)
       (* (/ (sin x) x) (fma (* y y) (* y 0.16666666666666666) y))
       (*
        (/ (sinh y) x)
        (fma
         (* x x)
         (* x (fma x (* x 0.008333333333333333) -0.16666666666666666))
         x))))))

C

double code(double x, double y) {
	double t_0 = (sinh(y) * sin(x)) / x;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (fma(x, ((x * x) * -0.16666666666666666), x) * fma((y * y), (y * fma(y, (y * 0.008333333333333333), 0.16666666666666666)), y)) / x;
	} else if (t_0 <= 2e-6) {
		tmp = (sin(x) / x) * fma((y * y), (y * 0.16666666666666666), y);
	} else {
		tmp = (sinh(y) / x) * fma((x * x), (x * fma(x, (x * 0.008333333333333333), -0.16666666666666666)), x);
	}
	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(fma(x, Float64(Float64(x * x) * -0.16666666666666666), x) * fma(Float64(y * y), Float64(y * fma(y, Float64(y * 0.008333333333333333), 0.16666666666666666)), y)) / x);
	elseif (t_0 <= 2e-6)
		tmp = Float64(Float64(sin(x) / x) * fma(Float64(y * y), Float64(y * 0.16666666666666666), y));
	else
		tmp = Float64(Float64(sinh(y) / x) * fma(Float64(x * x), Float64(x * fma(x, Float64(x * 0.008333333333333333), -0.16666666666666666)), 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, (-Infinity)], N[(N[(N[(x * N[(N[(x * x), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + x), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * N[(y * 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[t$95$0, 2e-6], N[(N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(y * 0.16666666666666666), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sinh[y], $MachinePrecision] / x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(x * 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + x), $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 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

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 84.2

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

   5. Applied rewrites 84.2%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-*l* N/A

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

      4. *-lft-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 66.1

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

   8. Applied rewrites 66.1%

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

   if -inf.0 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 1.99999999999999991e-6

   1. Initial program 75.9%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*r/ N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. associate-*r* N/A

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

      11. associate-/l* N/A

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

      12. distribute-rgt-out N/A

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

      13. *-rgt-identity N/A

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

      14. distribute-lft-in N/A

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

   5. Applied rewrites 99.2%

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

   if 1.99999999999999991e-6 < (/.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

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. sub-neg N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 83.0

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

   5. Applied rewrites 83.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.008333333333333333, -0.16666666666666666\right), x \cdot \left(x \cdot x\right), x\right)} \cdot \sinh y}{x} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 83.1

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

   7. Applied rewrites 83.1%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -\infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot -0.16666666666666666, x\right) \cdot \mathsf{fma}\left(y \cdot y, y \cdot \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), y\right)}{x}\\ \mathbf{elif}\;\frac{\sinh y \cdot \sin x}{x} \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\frac{\sin x}{x} \cdot \mathsf{fma}\left(y \cdot y, y \cdot 0.16666666666666666, y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sinh y}{x} \cdot \mathsf{fma}\left(x \cdot x, x \cdot \mathsf{fma}\left(x, x \cdot 0.008333333333333333, -0.16666666666666666\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y \cdot \sin x}{x}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot -0.16666666666666666, x\right) \cdot \mathsf{fma}\left(y \cdot y, y \cdot \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), y\right)}{x}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\frac{\sin x}{x} \cdot \mathsf{fma}\left(y \cdot y, y \cdot 0.16666666666666666, y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sinh y}{1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sinh y) (sin x)) x)))
   (if (<= t_0 (- INFINITY))
     (/
      (*
       (fma x (* (* x x) -0.16666666666666666) x)
       (fma
        (* y y)
        (* y (fma y (* y 0.008333333333333333) 0.16666666666666666))
        y))
      x)
     (if (<= t_0 2e-6)
       (* (/ (sin x) x) (fma (* y y) (* y 0.16666666666666666) y))
       (/ (sinh y) 1.0)))))

C

double code(double x, double y) {
	double t_0 = (sinh(y) * sin(x)) / x;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (fma(x, ((x * x) * -0.16666666666666666), x) * fma((y * y), (y * fma(y, (y * 0.008333333333333333), 0.16666666666666666)), y)) / x;
	} else if (t_0 <= 2e-6) {
		tmp = (sin(x) / x) * fma((y * y), (y * 0.16666666666666666), y);
	} else {
		tmp = sinh(y) / 1.0;
	}
	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(fma(x, Float64(Float64(x * x) * -0.16666666666666666), x) * fma(Float64(y * y), Float64(y * fma(y, Float64(y * 0.008333333333333333), 0.16666666666666666)), y)) / x);
	elseif (t_0 <= 2e-6)
		tmp = Float64(Float64(sin(x) / x) * fma(Float64(y * y), Float64(y * 0.16666666666666666), y));
	else
		tmp = Float64(sinh(y) / 1.0);
	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[(x * N[(N[(x * x), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + x), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * N[(y * 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[t$95$0, 2e-6], N[(N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(y * 0.16666666666666666), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision], N[(N[Sinh[y], $MachinePrecision] / 1.0), $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 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

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 84.2

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

   5. Applied rewrites 84.2%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-*l* N/A

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

      4. *-lft-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 66.1

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

   8. Applied rewrites 66.1%

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

   if -inf.0 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 1.99999999999999991e-6

   1. Initial program 75.9%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*r/ N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. associate-*r* N/A

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

      11. associate-/l* N/A

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

      12. distribute-rgt-out N/A

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

      13. *-rgt-identity N/A

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

      14. distribute-lft-in N/A

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

   5. Applied rewrites 99.2%

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

   if 1.99999999999999991e-6 < (/.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. clear-num N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. clear-num N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 82.3%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -\infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot -0.16666666666666666, x\right) \cdot \mathsf{fma}\left(y \cdot y, y \cdot \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), y\right)}{x}\\ \mathbf{elif}\;\frac{\sinh y \cdot \sin x}{x} \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\frac{\sin x}{x} \cdot \mathsf{fma}\left(y \cdot y, y \cdot 0.16666666666666666, y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sinh y}{1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y \cdot \sin x}{x}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot -0.16666666666666666, x\right) \cdot \mathsf{fma}\left(y \cdot y, y \cdot \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), y\right)}{x}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-23}:\\ \;\;\;\;\frac{y}{\frac{x}{\sin x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sinh y}{1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sinh y) (sin x)) x)))
   (if (<= t_0 (- INFINITY))
     (/
      (*
       (fma x (* (* x x) -0.16666666666666666) x)
       (fma
        (* y y)
        (* y (fma y (* y 0.008333333333333333) 0.16666666666666666))
        y))
      x)
     (if (<= t_0 5e-23) (/ y (/ x (sin x))) (/ (sinh y) 1.0)))))

C

double code(double x, double y) {
	double t_0 = (sinh(y) * sin(x)) / x;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (fma(x, ((x * x) * -0.16666666666666666), x) * fma((y * y), (y * fma(y, (y * 0.008333333333333333), 0.16666666666666666)), y)) / x;
	} else if (t_0 <= 5e-23) {
		tmp = y / (x / sin(x));
	} else {
		tmp = sinh(y) / 1.0;
	}
	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(fma(x, Float64(Float64(x * x) * -0.16666666666666666), x) * fma(Float64(y * y), Float64(y * fma(y, Float64(y * 0.008333333333333333), 0.16666666666666666)), y)) / x);
	elseif (t_0 <= 5e-23)
		tmp = Float64(y / Float64(x / sin(x)));
	else
		tmp = Float64(sinh(y) / 1.0);
	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[(x * N[(N[(x * x), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + x), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * N[(y * 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[t$95$0, 5e-23], N[(y / N[(x / N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sinh[y], $MachinePrecision] / 1.0), $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 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

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 84.2

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

   5. Applied rewrites 84.2%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-*l* N/A

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

      4. *-lft-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 66.1

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

   8. Applied rewrites 66.1%

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

   if -inf.0 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 5.0000000000000002e-23

   1. Initial program 75.6%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sin.f64 98.7

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

   5. Applied rewrites 98.7%

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

   7. Applied rewrites 98.7%

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

   if 5.0000000000000002e-23 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 99.9%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. clear-num N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 81.2%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -\infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot -0.16666666666666666, x\right) \cdot \mathsf{fma}\left(y \cdot y, y \cdot \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), y\right)}{x}\\ \mathbf{elif}\;\frac{\sinh y \cdot \sin x}{x} \leq 5 \cdot 10^{-23}:\\ \;\;\;\;\frac{y}{\frac{x}{\sin x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sinh y}{1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 83.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y \cdot \sin x}{x}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot -0.16666666666666666, x\right) \cdot \mathsf{fma}\left(y \cdot y, y \cdot \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), y\right)}{x}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-23}:\\ \;\;\;\;y \cdot \frac{\sin x}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sinh y}{1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sinh y) (sin x)) x)))
   (if (<= t_0 (- INFINITY))
     (/
      (*
       (fma x (* (* x x) -0.16666666666666666) x)
       (fma
        (* y y)
        (* y (fma y (* y 0.008333333333333333) 0.16666666666666666))
        y))
      x)
     (if (<= t_0 5e-23) (* y (/ (sin x) x)) (/ (sinh y) 1.0)))))

C

double code(double x, double y) {
	double t_0 = (sinh(y) * sin(x)) / x;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (fma(x, ((x * x) * -0.16666666666666666), x) * fma((y * y), (y * fma(y, (y * 0.008333333333333333), 0.16666666666666666)), y)) / x;
	} else if (t_0 <= 5e-23) {
		tmp = y * (sin(x) / x);
	} else {
		tmp = sinh(y) / 1.0;
	}
	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(fma(x, Float64(Float64(x * x) * -0.16666666666666666), x) * fma(Float64(y * y), Float64(y * fma(y, Float64(y * 0.008333333333333333), 0.16666666666666666)), y)) / x);
	elseif (t_0 <= 5e-23)
		tmp = Float64(y * Float64(sin(x) / x));
	else
		tmp = Float64(sinh(y) / 1.0);
	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[(x * N[(N[(x * x), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + x), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * N[(y * 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[t$95$0, 5e-23], N[(y * N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[Sinh[y], $MachinePrecision] / 1.0), $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 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

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 84.2

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

   5. Applied rewrites 84.2%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-*l* N/A

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

      4. *-lft-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 66.1

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

   8. Applied rewrites 66.1%

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

   if -inf.0 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 5.0000000000000002e-23

   1. Initial program 75.6%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sin.f64 98.7

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

   5. Applied rewrites 98.7%

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

   if 5.0000000000000002e-23 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 99.9%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. clear-num N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 81.2%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -\infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot -0.16666666666666666, x\right) \cdot \mathsf{fma}\left(y \cdot y, y \cdot \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), y\right)}{x}\\ \mathbf{elif}\;\frac{\sinh y \cdot \sin x}{x} \leq 5 \cdot 10^{-23}:\\ \;\;\;\;y \cdot \frac{\sin x}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sinh y}{1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 72.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y \cdot \sin x}{x}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-92}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot -0.16666666666666666, x\right) \cdot \mathsf{fma}\left(y \cdot y, y \cdot \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), y\right)}{x}\\ \mathbf{elif}\;t\_0 \leq 1.5 \cdot 10^{-146}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.16666666666666666, y \cdot \left(y \cdot y\right), y\right)}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.019444444444444445, 0.16666666666666666\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sinh y}{1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sinh y) (sin x)) x)))
   (if (<= t_0 -2e-92)
     (/
      (*
       (fma x (* (* x x) -0.16666666666666666) x)
       (fma
        (* y y)
        (* y (fma y (* y 0.008333333333333333) 0.16666666666666666))
        y))
      x)
     (if (<= t_0 1.5e-146)
       (/
        (fma 0.16666666666666666 (* y (* y y)) y)
        (fma
         x
         (* x (fma (* x x) 0.019444444444444445 0.16666666666666666))
         1.0))
       (/ (sinh y) 1.0)))))

C

double code(double x, double y) {
	double t_0 = (sinh(y) * sin(x)) / x;
	double tmp;
	if (t_0 <= -2e-92) {
		tmp = (fma(x, ((x * x) * -0.16666666666666666), x) * fma((y * y), (y * fma(y, (y * 0.008333333333333333), 0.16666666666666666)), y)) / x;
	} else if (t_0 <= 1.5e-146) {
		tmp = fma(0.16666666666666666, (y * (y * y)), y) / fma(x, (x * fma((x * x), 0.019444444444444445, 0.16666666666666666)), 1.0);
	} else {
		tmp = sinh(y) / 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(sinh(y) * sin(x)) / x)
	tmp = 0.0
	if (t_0 <= -2e-92)
		tmp = Float64(Float64(fma(x, Float64(Float64(x * x) * -0.16666666666666666), x) * fma(Float64(y * y), Float64(y * fma(y, Float64(y * 0.008333333333333333), 0.16666666666666666)), y)) / x);
	elseif (t_0 <= 1.5e-146)
		tmp = Float64(fma(0.16666666666666666, Float64(y * Float64(y * y)), y) / fma(x, Float64(x * fma(Float64(x * x), 0.019444444444444445, 0.16666666666666666)), 1.0));
	else
		tmp = Float64(sinh(y) / 1.0);
	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-92], N[(N[(N[(x * N[(N[(x * x), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + x), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * N[(y * 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[t$95$0, 1.5e-146], N[(N[(0.16666666666666666 * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] / N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 0.019444444444444445 + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Sinh[y], $MachinePrecision] / 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -1.99999999999999998e-92

   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

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 82.3

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

   5. Applied rewrites 82.3%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-*l* N/A

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

      4. *-lft-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 66.9

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

   8. Applied rewrites 66.9%

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

   if -1.99999999999999998e-92 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 1.50000000000000009e-146

   1. Initial program 69.1%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. clear-num N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. unpow2 N/A

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

      6. unpow3 N/A

         \[\leadsto \frac{\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot \color{blue}{{y}^{3}} + 1 \cdot y}{\frac{x}{\sin x}} \]

      7. *-lft-identity N/A

         \[\leadsto \frac{\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot {y}^{3} + \color{blue}{y}}{\frac{x}{\sin x}} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right), {y}^{3}, y\right)}}{\frac{x}{\sin x}} \]

   7. Applied rewrites 99.1%

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

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), y \cdot \left(y \cdot y\right), y\right)}{\color{blue}{1 + {x}^{2} \cdot \left(\frac{1}{6} + \frac{7}{360} \cdot {x}^{2}\right)}} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), y \cdot \left(y \cdot y\right), y\right)}{\color{blue}{{x}^{2} \cdot \left(\frac{1}{6} + \frac{7}{360} \cdot {x}^{2}\right) + 1}} \]

      2. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), y \cdot \left(y \cdot y\right), y\right)}{\color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{6} + \frac{7}{360} \cdot {x}^{2}\right) + 1} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), y \cdot \left(y \cdot y\right), y\right)}{\color{blue}{x \cdot \left(x \cdot \left(\frac{1}{6} + \frac{7}{360} \cdot {x}^{2}\right)\right)} + 1} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), y \cdot \left(y \cdot y\right), y\right)}{\color{blue}{\mathsf{fma}\left(x, x \cdot \left(\frac{1}{6} + \frac{7}{360} \cdot {x}^{2}\right), 1\right)}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), y \cdot \left(y \cdot y\right), y\right)}{\mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{6} + \frac{7}{360} \cdot {x}^{2}\right)}, 1\right)} \]

      6. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), y \cdot \left(y \cdot y\right), y\right)}{\mathsf{fma}\left(x, x \cdot \color{blue}{\left(\frac{7}{360} \cdot {x}^{2} + \frac{1}{6}\right)}, 1\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), y \cdot \left(y \cdot y\right), y\right)}{\mathsf{fma}\left(x, x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{7}{360}} + \frac{1}{6}\right), 1\right)} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), y \cdot \left(y \cdot y\right), y\right)}{\mathsf{fma}\left(x, x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{7}{360}, \frac{1}{6}\right)}, 1\right)} \]

      9. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), y \cdot \left(y \cdot y\right), y\right)}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{7}{360}, \frac{1}{6}\right), 1\right)} \]

      10. lower-*.f64 77.1

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.019444444444444445, 0.16666666666666666\right), 1\right)} \]

   10. Applied rewrites 77.1%

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)}{\color{blue}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.019444444444444445, 0.16666666666666666\right), 1\right)}} \]

   11. Taylor expanded in y around 0

       \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{6}, \color{blue}{y} \cdot \left(y \cdot y\right), y\right)}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \frac{7}{360}, \frac{1}{6}\right), 1\right)} \]
   12. Step-by-step derivation

   13. Applied rewrites 77.1%

       \[\leadsto \frac{\mathsf{fma}\left(0.16666666666666666, \color{blue}{y} \cdot \left(y \cdot y\right), y\right)}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.019444444444444445, 0.16666666666666666\right), 1\right)} \]

   if 1.50000000000000009e-146 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 99.9%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. clear-num N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 77.3%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -2 \cdot 10^{-92}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot -0.16666666666666666, x\right) \cdot \mathsf{fma}\left(y \cdot y, y \cdot \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), y\right)}{x}\\ \mathbf{elif}\;\frac{\sinh y \cdot \sin x}{x} \leq 1.5 \cdot 10^{-146}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.16666666666666666, y \cdot \left(y \cdot y\right), y\right)}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.019444444444444445, 0.16666666666666666\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sinh y}{1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 71.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y \cdot \sin x}{x}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-92}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot -0.16666666666666666, x\right) \cdot \mathsf{fma}\left(y \cdot y, y \cdot \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), y\right)}{x}\\ \mathbf{elif}\;t\_0 \leq 4:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.019444444444444445, 0.16666666666666666\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y\right)}{x} \cdot \mathsf{fma}\left(x \cdot x, x \cdot \mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sinh y) (sin x)) x)))
   (if (<= t_0 -2e-92)
     (/
      (*
       (fma x (* (* x x) -0.16666666666666666) x)
       (fma
        (* y y)
        (* y (fma y (* y 0.008333333333333333) 0.16666666666666666))
        y))
      x)
     (if (<= t_0 4.0)
       (/
        (fma
         (fma
          y
          (* y (fma (* y y) 0.0001984126984126984 0.008333333333333333))
          0.16666666666666666)
         (* y (* y y))
         y)
        (fma
         x
         (* x (fma (* x x) 0.019444444444444445 0.16666666666666666))
         1.0))
       (*
        (/
         (fma
          (* y y)
          (*
           y
           (fma
            (* y y)
            (fma y (* y 0.0001984126984126984) 0.008333333333333333)
            0.16666666666666666))
          y)
         x)
        (fma
         (* x x)
         (* x (fma 0.008333333333333333 (* x x) -0.16666666666666666))
         x))))))

C

double code(double x, double y) {
	double t_0 = (sinh(y) * sin(x)) / x;
	double tmp;
	if (t_0 <= -2e-92) {
		tmp = (fma(x, ((x * x) * -0.16666666666666666), x) * fma((y * y), (y * fma(y, (y * 0.008333333333333333), 0.16666666666666666)), y)) / x;
	} else if (t_0 <= 4.0) {
		tmp = fma(fma(y, (y * fma((y * y), 0.0001984126984126984, 0.008333333333333333)), 0.16666666666666666), (y * (y * y)), y) / fma(x, (x * fma((x * x), 0.019444444444444445, 0.16666666666666666)), 1.0);
	} else {
		tmp = (fma((y * y), (y * fma((y * y), fma(y, (y * 0.0001984126984126984), 0.008333333333333333), 0.16666666666666666)), y) / x) * fma((x * x), (x * fma(0.008333333333333333, (x * x), -0.16666666666666666)), x);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(sinh(y) * sin(x)) / x)
	tmp = 0.0
	if (t_0 <= -2e-92)
		tmp = Float64(Float64(fma(x, Float64(Float64(x * x) * -0.16666666666666666), x) * fma(Float64(y * y), Float64(y * fma(y, Float64(y * 0.008333333333333333), 0.16666666666666666)), y)) / x);
	elseif (t_0 <= 4.0)
		tmp = Float64(fma(fma(y, Float64(y * fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333)), 0.16666666666666666), Float64(y * Float64(y * y)), y) / fma(x, Float64(x * fma(Float64(x * x), 0.019444444444444445, 0.16666666666666666)), 1.0));
	else
		tmp = Float64(Float64(fma(Float64(y * y), Float64(y * fma(Float64(y * y), fma(y, Float64(y * 0.0001984126984126984), 0.008333333333333333), 0.16666666666666666)), y) / x) * fma(Float64(x * x), Float64(x * fma(0.008333333333333333, Float64(x * x), -0.16666666666666666)), 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-92], N[(N[(N[(x * N[(N[(x * x), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + x), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * N[(y * 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[t$95$0, 4.0], N[(N[(N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] / N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 0.019444444444444445 + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(y * y), $MachinePrecision] * N[(y * N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * 0.0001984126984126984), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] / x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * N[(0.008333333333333333 * N[(x * x), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -1.99999999999999998e-92

   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

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 82.3

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

   5. Applied rewrites 82.3%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-*l* N/A

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

      4. *-lft-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 66.9

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

   8. Applied rewrites 66.9%

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

   if -1.99999999999999998e-92 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 4

   1. Initial program 74.3%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. clear-num N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. unpow2 N/A

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

      6. unpow3 N/A

         \[\leadsto \frac{\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot \color{blue}{{y}^{3}} + 1 \cdot y}{\frac{x}{\sin x}} \]

      7. *-lft-identity N/A

         \[\leadsto \frac{\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot {y}^{3} + \color{blue}{y}}{\frac{x}{\sin x}} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right), {y}^{3}, y\right)}}{\frac{x}{\sin x}} \]

   7. Applied rewrites 98.8%

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

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), y \cdot \left(y \cdot y\right), y\right)}{\color{blue}{1 + {x}^{2} \cdot \left(\frac{1}{6} + \frac{7}{360} \cdot {x}^{2}\right)}} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), y \cdot \left(y \cdot y\right), y\right)}{\color{blue}{{x}^{2} \cdot \left(\frac{1}{6} + \frac{7}{360} \cdot {x}^{2}\right) + 1}} \]

      2. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), y \cdot \left(y \cdot y\right), y\right)}{\color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{6} + \frac{7}{360} \cdot {x}^{2}\right) + 1} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), y \cdot \left(y \cdot y\right), y\right)}{\color{blue}{x \cdot \left(x \cdot \left(\frac{1}{6} + \frac{7}{360} \cdot {x}^{2}\right)\right)} + 1} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), y \cdot \left(y \cdot y\right), y\right)}{\color{blue}{\mathsf{fma}\left(x, x \cdot \left(\frac{1}{6} + \frac{7}{360} \cdot {x}^{2}\right), 1\right)}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), y \cdot \left(y \cdot y\right), y\right)}{\mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{6} + \frac{7}{360} \cdot {x}^{2}\right)}, 1\right)} \]

      6. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), y \cdot \left(y \cdot y\right), y\right)}{\mathsf{fma}\left(x, x \cdot \color{blue}{\left(\frac{7}{360} \cdot {x}^{2} + \frac{1}{6}\right)}, 1\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), y \cdot \left(y \cdot y\right), y\right)}{\mathsf{fma}\left(x, x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{7}{360}} + \frac{1}{6}\right), 1\right)} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), y \cdot \left(y \cdot y\right), y\right)}{\mathsf{fma}\left(x, x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{7}{360}, \frac{1}{6}\right)}, 1\right)} \]

      9. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), y \cdot \left(y \cdot y\right), y\right)}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{7}{360}, \frac{1}{6}\right), 1\right)} \]

      10. lower-*.f64 74.6

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.019444444444444445, 0.16666666666666666\right), 1\right)} \]

   10. Applied rewrites 74.6%

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)}{\color{blue}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.019444444444444445, 0.16666666666666666\right), 1\right)}} \]

   if 4 < (/.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

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. sub-neg N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 82.8

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

   5. Applied rewrites 82.8%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. unpow2 N/A

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

      6. unpow3 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{1}{120}, \frac{-1}{6}\right), x \cdot \left(x \cdot x\right), x\right) \cdot \left(\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot \color{blue}{{y}^{3}} + 1 \cdot y\right)}{x} \]

      7. *-lft-identity N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{1}{120}, \frac{-1}{6}\right), x \cdot \left(x \cdot x\right), x\right) \cdot \left(\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot {y}^{3} + \color{blue}{y}\right)}{x} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{1}{120}, \frac{-1}{6}\right), x \cdot \left(x \cdot x\right), x\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right), {y}^{3}, y\right)}}{x} \]

   8. Applied rewrites 74.5%

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

   10. Applied rewrites 77.8%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -2 \cdot 10^{-92}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot -0.16666666666666666, x\right) \cdot \mathsf{fma}\left(y \cdot y, y \cdot \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), y\right)}{x}\\ \mathbf{elif}\;\frac{\sinh y \cdot \sin x}{x} \leq 4:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.019444444444444445, 0.16666666666666666\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y\right)}{x} \cdot \mathsf{fma}\left(x \cdot x, x \cdot \mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 70.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y \cdot \sin x}{x}\\ t_1 := y \cdot \left(y \cdot y\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-92}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot -0.16666666666666666, x\right) \cdot \mathsf{fma}\left(y \cdot y, y \cdot \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), y\right)}{x}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-23}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.16666666666666666, t\_1, y\right)}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.019444444444444445, 0.16666666666666666\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), t\_1, y\right) \cdot \mathsf{fma}\left(0.008333333333333333 \cdot \left(x \cdot x\right), x \cdot \left(x \cdot x\right), x\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sinh y) (sin x)) x)) (t_1 (* y (* y y))))
   (if (<= t_0 -2e-92)
     (/
      (*
       (fma x (* (* x x) -0.16666666666666666) x)
       (fma
        (* y y)
        (* y (fma y (* y 0.008333333333333333) 0.16666666666666666))
        y))
      x)
     (if (<= t_0 5e-23)
       (/
        (fma 0.16666666666666666 t_1 y)
        (fma
         x
         (* x (fma (* x x) 0.019444444444444445 0.16666666666666666))
         1.0))
       (/
        (*
         (fma
          (fma
           y
           (* y (fma (* y y) 0.0001984126984126984 0.008333333333333333))
           0.16666666666666666)
          t_1
          y)
         (fma (* 0.008333333333333333 (* x x)) (* x (* x x)) x))
        x)))))

C

double code(double x, double y) {
	double t_0 = (sinh(y) * sin(x)) / x;
	double t_1 = y * (y * y);
	double tmp;
	if (t_0 <= -2e-92) {
		tmp = (fma(x, ((x * x) * -0.16666666666666666), x) * fma((y * y), (y * fma(y, (y * 0.008333333333333333), 0.16666666666666666)), y)) / x;
	} else if (t_0 <= 5e-23) {
		tmp = fma(0.16666666666666666, t_1, y) / fma(x, (x * fma((x * x), 0.019444444444444445, 0.16666666666666666)), 1.0);
	} else {
		tmp = (fma(fma(y, (y * fma((y * y), 0.0001984126984126984, 0.008333333333333333)), 0.16666666666666666), t_1, y) * fma((0.008333333333333333 * (x * x)), (x * (x * x)), x)) / x;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(sinh(y) * sin(x)) / x)
	t_1 = Float64(y * Float64(y * y))
	tmp = 0.0
	if (t_0 <= -2e-92)
		tmp = Float64(Float64(fma(x, Float64(Float64(x * x) * -0.16666666666666666), x) * fma(Float64(y * y), Float64(y * fma(y, Float64(y * 0.008333333333333333), 0.16666666666666666)), y)) / x);
	elseif (t_0 <= 5e-23)
		tmp = Float64(fma(0.16666666666666666, t_1, y) / fma(x, Float64(x * fma(Float64(x * x), 0.019444444444444445, 0.16666666666666666)), 1.0));
	else
		tmp = Float64(Float64(fma(fma(y, Float64(y * fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333)), 0.16666666666666666), t_1, y) * fma(Float64(0.008333333333333333 * Float64(x * x)), Float64(x * Float64(x * x)), 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]}, Block[{t$95$1 = N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-92], N[(N[(N[(x * N[(N[(x * x), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + x), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * N[(y * 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[t$95$0, 5e-23], N[(N[(0.16666666666666666 * t$95$1 + y), $MachinePrecision] / N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 0.019444444444444445 + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * t$95$1 + y), $MachinePrecision] * N[(N[(0.008333333333333333 * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -1.99999999999999998e-92

   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

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 82.3

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

   5. Applied rewrites 82.3%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-*l* N/A

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

      4. *-lft-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 66.9

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

   8. Applied rewrites 66.9%

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

   if -1.99999999999999998e-92 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 5.0000000000000002e-23

   1. Initial program 73.7%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. clear-num N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. unpow2 N/A

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

      6. unpow3 N/A

         \[\leadsto \frac{\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot \color{blue}{{y}^{3}} + 1 \cdot y}{\frac{x}{\sin x}} \]

      7. *-lft-identity N/A

         \[\leadsto \frac{\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot {y}^{3} + \color{blue}{y}}{\frac{x}{\sin x}} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right), {y}^{3}, y\right)}}{\frac{x}{\sin x}} \]

   7. Applied rewrites 99.2%

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

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), y \cdot \left(y \cdot y\right), y\right)}{\color{blue}{1 + {x}^{2} \cdot \left(\frac{1}{6} + \frac{7}{360} \cdot {x}^{2}\right)}} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), y \cdot \left(y \cdot y\right), y\right)}{\color{blue}{{x}^{2} \cdot \left(\frac{1}{6} + \frac{7}{360} \cdot {x}^{2}\right) + 1}} \]

      2. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), y \cdot \left(y \cdot y\right), y\right)}{\color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{6} + \frac{7}{360} \cdot {x}^{2}\right) + 1} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), y \cdot \left(y \cdot y\right), y\right)}{\color{blue}{x \cdot \left(x \cdot \left(\frac{1}{6} + \frac{7}{360} \cdot {x}^{2}\right)\right)} + 1} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), y \cdot \left(y \cdot y\right), y\right)}{\color{blue}{\mathsf{fma}\left(x, x \cdot \left(\frac{1}{6} + \frac{7}{360} \cdot {x}^{2}\right), 1\right)}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), y \cdot \left(y \cdot y\right), y\right)}{\mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{6} + \frac{7}{360} \cdot {x}^{2}\right)}, 1\right)} \]

      6. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), y \cdot \left(y \cdot y\right), y\right)}{\mathsf{fma}\left(x, x \cdot \color{blue}{\left(\frac{7}{360} \cdot {x}^{2} + \frac{1}{6}\right)}, 1\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), y \cdot \left(y \cdot y\right), y\right)}{\mathsf{fma}\left(x, x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{7}{360}} + \frac{1}{6}\right), 1\right)} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), y \cdot \left(y \cdot y\right), y\right)}{\mathsf{fma}\left(x, x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{7}{360}, \frac{1}{6}\right)}, 1\right)} \]

      9. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), y \cdot \left(y \cdot y\right), y\right)}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{7}{360}, \frac{1}{6}\right), 1\right)} \]

      10. lower-*.f64 75.2

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.019444444444444445, 0.16666666666666666\right), 1\right)} \]

   10. Applied rewrites 75.2%

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)}{\color{blue}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.019444444444444445, 0.16666666666666666\right), 1\right)}} \]

   11. Taylor expanded in y around 0

       \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{6}, \color{blue}{y} \cdot \left(y \cdot y\right), y\right)}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \frac{7}{360}, \frac{1}{6}\right), 1\right)} \]
   12. Step-by-step derivation

   13. Applied rewrites 75.2%

       \[\leadsto \frac{\mathsf{fma}\left(0.16666666666666666, \color{blue}{y} \cdot \left(y \cdot y\right), y\right)}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.019444444444444445, 0.16666666666666666\right), 1\right)} \]

   if 5.0000000000000002e-23 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. sub-neg N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 81.9

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

   5. Applied rewrites 81.9%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. unpow2 N/A

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

      6. unpow3 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{1}{120}, \frac{-1}{6}\right), x \cdot \left(x \cdot x\right), x\right) \cdot \left(\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot \color{blue}{{y}^{3}} + 1 \cdot y\right)}{x} \]

      7. *-lft-identity N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{1}{120}, \frac{-1}{6}\right), x \cdot \left(x \cdot x\right), x\right) \cdot \left(\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot {y}^{3} + \color{blue}{y}\right)}{x} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{1}{120}, \frac{-1}{6}\right), x \cdot \left(x \cdot x\right), x\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right), {y}^{3}, y\right)}}{x} \]

   8. Applied rewrites 73.2%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 73.2%

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

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -2 \cdot 10^{-92}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot -0.16666666666666666, x\right) \cdot \mathsf{fma}\left(y \cdot y, y \cdot \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), y\right)}{x}\\ \mathbf{elif}\;\frac{\sinh y \cdot \sin x}{x} \leq 5 \cdot 10^{-23}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.16666666666666666, y \cdot \left(y \cdot y\right), y\right)}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.019444444444444445, 0.16666666666666666\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right) \cdot \mathsf{fma}\left(0.008333333333333333 \cdot \left(x \cdot x\right), x \cdot \left(x \cdot x\right), x\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 70.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot y\right)\\ t_1 := \frac{\sinh y \cdot \sin x}{x}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-92}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot -0.16666666666666666, x\right) \cdot \mathsf{fma}\left(y \cdot y, y \cdot \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), y\right)}{x}\\ \mathbf{elif}\;t\_1 \leq 4:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), t\_0, y\right)}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.019444444444444445, 0.16666666666666666\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.008333333333333333, -0.16666666666666666\right), x \cdot \left(x \cdot x\right), x\right) \cdot \mathsf{fma}\left(y \cdot \left(0.0001984126984126984 \cdot t\_0\right), t\_0, y\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y y))) (t_1 (/ (* (sinh y) (sin x)) x)))
   (if (<= t_1 -2e-92)
     (/
      (*
       (fma x (* (* x x) -0.16666666666666666) x)
       (fma
        (* y y)
        (* y (fma y (* y 0.008333333333333333) 0.16666666666666666))
        y))
      x)
     (if (<= t_1 4.0)
       (/
        (fma
         (fma
          y
          (* y (fma (* y y) 0.0001984126984126984 0.008333333333333333))
          0.16666666666666666)
         t_0
         y)
        (fma
         x
         (* x (fma (* x x) 0.019444444444444445 0.16666666666666666))
         1.0))
       (/
        (*
         (fma
          (fma (* x x) 0.008333333333333333 -0.16666666666666666)
          (* x (* x x))
          x)
         (fma (* y (* 0.0001984126984126984 t_0)) t_0 y))
        x)))))

C

double code(double x, double y) {
	double t_0 = y * (y * y);
	double t_1 = (sinh(y) * sin(x)) / x;
	double tmp;
	if (t_1 <= -2e-92) {
		tmp = (fma(x, ((x * x) * -0.16666666666666666), x) * fma((y * y), (y * fma(y, (y * 0.008333333333333333), 0.16666666666666666)), y)) / x;
	} else if (t_1 <= 4.0) {
		tmp = fma(fma(y, (y * fma((y * y), 0.0001984126984126984, 0.008333333333333333)), 0.16666666666666666), t_0, y) / fma(x, (x * fma((x * x), 0.019444444444444445, 0.16666666666666666)), 1.0);
	} else {
		tmp = (fma(fma((x * x), 0.008333333333333333, -0.16666666666666666), (x * (x * x)), x) * fma((y * (0.0001984126984126984 * t_0)), t_0, y)) / x;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(y * Float64(y * y))
	t_1 = Float64(Float64(sinh(y) * sin(x)) / x)
	tmp = 0.0
	if (t_1 <= -2e-92)
		tmp = Float64(Float64(fma(x, Float64(Float64(x * x) * -0.16666666666666666), x) * fma(Float64(y * y), Float64(y * fma(y, Float64(y * 0.008333333333333333), 0.16666666666666666)), y)) / x);
	elseif (t_1 <= 4.0)
		tmp = Float64(fma(fma(y, Float64(y * fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333)), 0.16666666666666666), t_0, y) / fma(x, Float64(x * fma(Float64(x * x), 0.019444444444444445, 0.16666666666666666)), 1.0));
	else
		tmp = Float64(Float64(fma(fma(Float64(x * x), 0.008333333333333333, -0.16666666666666666), Float64(x * Float64(x * x)), x) * fma(Float64(y * Float64(0.0001984126984126984 * t_0)), t_0, y)) / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Sinh[y], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-92], N[(N[(N[(x * N[(N[(x * x), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + x), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * N[(y * 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[t$95$1, 4.0], N[(N[(N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * t$95$0 + y), $MachinePrecision] / N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 0.019444444444444445 + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(x * x), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision] * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] * N[(N[(y * N[(0.0001984126984126984 * t$95$0), $MachinePrecision]), $MachinePrecision] * t$95$0 + y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -1.99999999999999998e-92

   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

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 82.3

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

   5. Applied rewrites 82.3%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-*l* N/A

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

      4. *-lft-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 66.9

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

   8. Applied rewrites 66.9%

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

   if -1.99999999999999998e-92 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 4

   1. Initial program 74.3%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. clear-num N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. unpow2 N/A

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

      6. unpow3 N/A

         \[\leadsto \frac{\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot \color{blue}{{y}^{3}} + 1 \cdot y}{\frac{x}{\sin x}} \]

      7. *-lft-identity N/A

         \[\leadsto \frac{\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot {y}^{3} + \color{blue}{y}}{\frac{x}{\sin x}} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right), {y}^{3}, y\right)}}{\frac{x}{\sin x}} \]

   7. Applied rewrites 98.8%

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

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), y \cdot \left(y \cdot y\right), y\right)}{\color{blue}{1 + {x}^{2} \cdot \left(\frac{1}{6} + \frac{7}{360} \cdot {x}^{2}\right)}} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), y \cdot \left(y \cdot y\right), y\right)}{\color{blue}{{x}^{2} \cdot \left(\frac{1}{6} + \frac{7}{360} \cdot {x}^{2}\right) + 1}} \]

      2. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), y \cdot \left(y \cdot y\right), y\right)}{\color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{6} + \frac{7}{360} \cdot {x}^{2}\right) + 1} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), y \cdot \left(y \cdot y\right), y\right)}{\color{blue}{x \cdot \left(x \cdot \left(\frac{1}{6} + \frac{7}{360} \cdot {x}^{2}\right)\right)} + 1} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), y \cdot \left(y \cdot y\right), y\right)}{\color{blue}{\mathsf{fma}\left(x, x \cdot \left(\frac{1}{6} + \frac{7}{360} \cdot {x}^{2}\right), 1\right)}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), y \cdot \left(y \cdot y\right), y\right)}{\mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{6} + \frac{7}{360} \cdot {x}^{2}\right)}, 1\right)} \]

      6. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), y \cdot \left(y \cdot y\right), y\right)}{\mathsf{fma}\left(x, x \cdot \color{blue}{\left(\frac{7}{360} \cdot {x}^{2} + \frac{1}{6}\right)}, 1\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), y \cdot \left(y \cdot y\right), y\right)}{\mathsf{fma}\left(x, x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{7}{360}} + \frac{1}{6}\right), 1\right)} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), y \cdot \left(y \cdot y\right), y\right)}{\mathsf{fma}\left(x, x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{7}{360}, \frac{1}{6}\right)}, 1\right)} \]

      9. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), y \cdot \left(y \cdot y\right), y\right)}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{7}{360}, \frac{1}{6}\right), 1\right)} \]

      10. lower-*.f64 74.6

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.019444444444444445, 0.16666666666666666\right), 1\right)} \]

   10. Applied rewrites 74.6%

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)}{\color{blue}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.019444444444444445, 0.16666666666666666\right), 1\right)}} \]

   if 4 < (/.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

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. sub-neg N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 82.8

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

   5. Applied rewrites 82.8%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. unpow2 N/A

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

      6. unpow3 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{1}{120}, \frac{-1}{6}\right), x \cdot \left(x \cdot x\right), x\right) \cdot \left(\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot \color{blue}{{y}^{3}} + 1 \cdot y\right)}{x} \]

      7. *-lft-identity N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{1}{120}, \frac{-1}{6}\right), x \cdot \left(x \cdot x\right), x\right) \cdot \left(\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot {y}^{3} + \color{blue}{y}\right)}{x} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{1}{120}, \frac{-1}{6}\right), x \cdot \left(x \cdot x\right), x\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right), {y}^{3}, y\right)}}{x} \]

   8. Applied rewrites 74.5%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 74.5%

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

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -2 \cdot 10^{-92}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot -0.16666666666666666, x\right) \cdot \mathsf{fma}\left(y \cdot y, y \cdot \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), y\right)}{x}\\ \mathbf{elif}\;\frac{\sinh y \cdot \sin x}{x} \leq 4:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.019444444444444445, 0.16666666666666666\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.008333333333333333, -0.16666666666666666\right), x \cdot \left(x \cdot x\right), x\right) \cdot \mathsf{fma}\left(y \cdot \left(0.0001984126984126984 \cdot \left(y \cdot \left(y \cdot y\right)\right)\right), y \cdot \left(y \cdot y\right), y\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 69.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y \cdot \sin x}{x}\\ t_1 := y \cdot \left(y \cdot y\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-92}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot -0.16666666666666666, x\right) \cdot \mathsf{fma}\left(y \cdot y, y \cdot \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), y\right)}{x}\\ \mathbf{elif}\;t\_0 \leq 1.5 \cdot 10^{-146}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.16666666666666666, t\_1, y\right)}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.019444444444444445, 0.16666666666666666\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), t\_1, y\right)}{1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sinh y) (sin x)) x)) (t_1 (* y (* y y))))
   (if (<= t_0 -2e-92)
     (/
      (*
       (fma x (* (* x x) -0.16666666666666666) x)
       (fma
        (* y y)
        (* y (fma y (* y 0.008333333333333333) 0.16666666666666666))
        y))
      x)
     (if (<= t_0 1.5e-146)
       (/
        (fma 0.16666666666666666 t_1 y)
        (fma
         x
         (* x (fma (* x x) 0.019444444444444445 0.16666666666666666))
         1.0))
       (/
        (fma
         (fma
          y
          (* y (fma (* y y) 0.0001984126984126984 0.008333333333333333))
          0.16666666666666666)
         t_1
         y)
        1.0)))))

C

double code(double x, double y) {
	double t_0 = (sinh(y) * sin(x)) / x;
	double t_1 = y * (y * y);
	double tmp;
	if (t_0 <= -2e-92) {
		tmp = (fma(x, ((x * x) * -0.16666666666666666), x) * fma((y * y), (y * fma(y, (y * 0.008333333333333333), 0.16666666666666666)), y)) / x;
	} else if (t_0 <= 1.5e-146) {
		tmp = fma(0.16666666666666666, t_1, y) / fma(x, (x * fma((x * x), 0.019444444444444445, 0.16666666666666666)), 1.0);
	} else {
		tmp = fma(fma(y, (y * fma((y * y), 0.0001984126984126984, 0.008333333333333333)), 0.16666666666666666), t_1, y) / 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(sinh(y) * sin(x)) / x)
	t_1 = Float64(y * Float64(y * y))
	tmp = 0.0
	if (t_0 <= -2e-92)
		tmp = Float64(Float64(fma(x, Float64(Float64(x * x) * -0.16666666666666666), x) * fma(Float64(y * y), Float64(y * fma(y, Float64(y * 0.008333333333333333), 0.16666666666666666)), y)) / x);
	elseif (t_0 <= 1.5e-146)
		tmp = Float64(fma(0.16666666666666666, t_1, y) / fma(x, Float64(x * fma(Float64(x * x), 0.019444444444444445, 0.16666666666666666)), 1.0));
	else
		tmp = Float64(fma(fma(y, Float64(y * fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333)), 0.16666666666666666), t_1, y) / 1.0);
	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[(y * N[(y * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-92], N[(N[(N[(x * N[(N[(x * x), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + x), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * N[(y * 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[t$95$0, 1.5e-146], N[(N[(0.16666666666666666 * t$95$1 + y), $MachinePrecision] / N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 0.019444444444444445 + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * t$95$1 + y), $MachinePrecision] / 1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -1.99999999999999998e-92

   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

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 82.3

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

   5. Applied rewrites 82.3%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-*l* N/A

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

      4. *-lft-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 66.9

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

   8. Applied rewrites 66.9%

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

   if -1.99999999999999998e-92 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 1.50000000000000009e-146

   1. Initial program 69.1%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. clear-num N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. unpow2 N/A

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

      6. unpow3 N/A

         \[\leadsto \frac{\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot \color{blue}{{y}^{3}} + 1 \cdot y}{\frac{x}{\sin x}} \]

      7. *-lft-identity N/A

         \[\leadsto \frac{\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot {y}^{3} + \color{blue}{y}}{\frac{x}{\sin x}} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right), {y}^{3}, y\right)}}{\frac{x}{\sin x}} \]

   7. Applied rewrites 99.1%

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

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), y \cdot \left(y \cdot y\right), y\right)}{\color{blue}{1 + {x}^{2} \cdot \left(\frac{1}{6} + \frac{7}{360} \cdot {x}^{2}\right)}} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), y \cdot \left(y \cdot y\right), y\right)}{\color{blue}{{x}^{2} \cdot \left(\frac{1}{6} + \frac{7}{360} \cdot {x}^{2}\right) + 1}} \]

      2. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), y \cdot \left(y \cdot y\right), y\right)}{\color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{6} + \frac{7}{360} \cdot {x}^{2}\right) + 1} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), y \cdot \left(y \cdot y\right), y\right)}{\color{blue}{x \cdot \left(x \cdot \left(\frac{1}{6} + \frac{7}{360} \cdot {x}^{2}\right)\right)} + 1} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), y \cdot \left(y \cdot y\right), y\right)}{\color{blue}{\mathsf{fma}\left(x, x \cdot \left(\frac{1}{6} + \frac{7}{360} \cdot {x}^{2}\right), 1\right)}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), y \cdot \left(y \cdot y\right), y\right)}{\mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{6} + \frac{7}{360} \cdot {x}^{2}\right)}, 1\right)} \]

      6. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), y \cdot \left(y \cdot y\right), y\right)}{\mathsf{fma}\left(x, x \cdot \color{blue}{\left(\frac{7}{360} \cdot {x}^{2} + \frac{1}{6}\right)}, 1\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), y \cdot \left(y \cdot y\right), y\right)}{\mathsf{fma}\left(x, x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{7}{360}} + \frac{1}{6}\right), 1\right)} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), y \cdot \left(y \cdot y\right), y\right)}{\mathsf{fma}\left(x, x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{7}{360}, \frac{1}{6}\right)}, 1\right)} \]

      9. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), y \cdot \left(y \cdot y\right), y\right)}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{7}{360}, \frac{1}{6}\right), 1\right)} \]

      10. lower-*.f64 77.1

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.019444444444444445, 0.16666666666666666\right), 1\right)} \]

   10. Applied rewrites 77.1%

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)}{\color{blue}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.019444444444444445, 0.16666666666666666\right), 1\right)}} \]

   11. Taylor expanded in y around 0

       \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{6}, \color{blue}{y} \cdot \left(y \cdot y\right), y\right)}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \frac{7}{360}, \frac{1}{6}\right), 1\right)} \]
   12. Step-by-step derivation

   13. Applied rewrites 77.1%

       \[\leadsto \frac{\mathsf{fma}\left(0.16666666666666666, \color{blue}{y} \cdot \left(y \cdot y\right), y\right)}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.019444444444444445, 0.16666666666666666\right), 1\right)} \]

   if 1.50000000000000009e-146 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 99.9%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. clear-num N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. unpow2 N/A

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

      6. unpow3 N/A

         \[\leadsto \frac{\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot \color{blue}{{y}^{3}} + 1 \cdot y}{\frac{x}{\sin x}} \]

      7. *-lft-identity N/A

         \[\leadsto \frac{\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot {y}^{3} + \color{blue}{y}}{\frac{x}{\sin x}} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right), {y}^{3}, y\right)}}{\frac{x}{\sin x}} \]

   7. Applied rewrites 92.1%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 71.2%

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

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -2 \cdot 10^{-92}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot -0.16666666666666666, x\right) \cdot \mathsf{fma}\left(y \cdot y, y \cdot \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), y\right)}{x}\\ \mathbf{elif}\;\frac{\sinh y \cdot \sin x}{x} \leq 1.5 \cdot 10^{-146}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.16666666666666666, y \cdot \left(y \cdot y\right), y\right)}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.019444444444444445, 0.16666666666666666\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)}{1}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 68.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y \cdot \sin x}{x}\\ t_1 := y \cdot \left(y \cdot y\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-92}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot -0.16666666666666666, x\right) \cdot \mathsf{fma}\left(y, \left(y \cdot y\right) \cdot 0.16666666666666666, y\right)}{x}\\ \mathbf{elif}\;t\_0 \leq 1.5 \cdot 10^{-146}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.16666666666666666, t\_1, y\right)}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.019444444444444445, 0.16666666666666666\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), t\_1, y\right)}{1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sinh y) (sin x)) x)) (t_1 (* y (* y y))))
   (if (<= t_0 -2e-92)
     (/
      (*
       (fma x (* (* x x) -0.16666666666666666) x)
       (fma y (* (* y y) 0.16666666666666666) y))
      x)
     (if (<= t_0 1.5e-146)
       (/
        (fma 0.16666666666666666 t_1 y)
        (fma
         x
         (* x (fma (* x x) 0.019444444444444445 0.16666666666666666))
         1.0))
       (/
        (fma
         (fma
          y
          (* y (fma (* y y) 0.0001984126984126984 0.008333333333333333))
          0.16666666666666666)
         t_1
         y)
        1.0)))))

C

double code(double x, double y) {
	double t_0 = (sinh(y) * sin(x)) / x;
	double t_1 = y * (y * y);
	double tmp;
	if (t_0 <= -2e-92) {
		tmp = (fma(x, ((x * x) * -0.16666666666666666), x) * fma(y, ((y * y) * 0.16666666666666666), y)) / x;
	} else if (t_0 <= 1.5e-146) {
		tmp = fma(0.16666666666666666, t_1, y) / fma(x, (x * fma((x * x), 0.019444444444444445, 0.16666666666666666)), 1.0);
	} else {
		tmp = fma(fma(y, (y * fma((y * y), 0.0001984126984126984, 0.008333333333333333)), 0.16666666666666666), t_1, y) / 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(sinh(y) * sin(x)) / x)
	t_1 = Float64(y * Float64(y * y))
	tmp = 0.0
	if (t_0 <= -2e-92)
		tmp = Float64(Float64(fma(x, Float64(Float64(x * x) * -0.16666666666666666), x) * fma(y, Float64(Float64(y * y) * 0.16666666666666666), y)) / x);
	elseif (t_0 <= 1.5e-146)
		tmp = Float64(fma(0.16666666666666666, t_1, y) / fma(x, Float64(x * fma(Float64(x * x), 0.019444444444444445, 0.16666666666666666)), 1.0));
	else
		tmp = Float64(fma(fma(y, Float64(y * fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333)), 0.16666666666666666), t_1, y) / 1.0);
	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[(y * N[(y * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-92], N[(N[(N[(x * N[(N[(x * x), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + x), $MachinePrecision] * N[(y * N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[t$95$0, 1.5e-146], N[(N[(0.16666666666666666 * t$95$1 + y), $MachinePrecision] / N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 0.019444444444444445 + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * t$95$1 + y), $MachinePrecision] / 1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -1.99999999999999998e-92

   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

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 82.3

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

   5. Applied rewrites 82.3%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 62.4

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

   8. Applied rewrites 62.4%

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

   if -1.99999999999999998e-92 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 1.50000000000000009e-146

   1. Initial program 69.1%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. clear-num N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. unpow2 N/A

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

      6. unpow3 N/A

         \[\leadsto \frac{\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot \color{blue}{{y}^{3}} + 1 \cdot y}{\frac{x}{\sin x}} \]

      7. *-lft-identity N/A

         \[\leadsto \frac{\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot {y}^{3} + \color{blue}{y}}{\frac{x}{\sin x}} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right), {y}^{3}, y\right)}}{\frac{x}{\sin x}} \]

   7. Applied rewrites 99.1%

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

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), y \cdot \left(y \cdot y\right), y\right)}{\color{blue}{1 + {x}^{2} \cdot \left(\frac{1}{6} + \frac{7}{360} \cdot {x}^{2}\right)}} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), y \cdot \left(y \cdot y\right), y\right)}{\color{blue}{{x}^{2} \cdot \left(\frac{1}{6} + \frac{7}{360} \cdot {x}^{2}\right) + 1}} \]

      2. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), y \cdot \left(y \cdot y\right), y\right)}{\color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{6} + \frac{7}{360} \cdot {x}^{2}\right) + 1} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), y \cdot \left(y \cdot y\right), y\right)}{\color{blue}{x \cdot \left(x \cdot \left(\frac{1}{6} + \frac{7}{360} \cdot {x}^{2}\right)\right)} + 1} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), y \cdot \left(y \cdot y\right), y\right)}{\color{blue}{\mathsf{fma}\left(x, x \cdot \left(\frac{1}{6} + \frac{7}{360} \cdot {x}^{2}\right), 1\right)}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), y \cdot \left(y \cdot y\right), y\right)}{\mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{6} + \frac{7}{360} \cdot {x}^{2}\right)}, 1\right)} \]

      6. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), y \cdot \left(y \cdot y\right), y\right)}{\mathsf{fma}\left(x, x \cdot \color{blue}{\left(\frac{7}{360} \cdot {x}^{2} + \frac{1}{6}\right)}, 1\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), y \cdot \left(y \cdot y\right), y\right)}{\mathsf{fma}\left(x, x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{7}{360}} + \frac{1}{6}\right), 1\right)} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), y \cdot \left(y \cdot y\right), y\right)}{\mathsf{fma}\left(x, x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{7}{360}, \frac{1}{6}\right)}, 1\right)} \]

      9. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), y \cdot \left(y \cdot y\right), y\right)}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{7}{360}, \frac{1}{6}\right), 1\right)} \]

      10. lower-*.f64 77.1

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.019444444444444445, 0.16666666666666666\right), 1\right)} \]

   10. Applied rewrites 77.1%

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)}{\color{blue}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.019444444444444445, 0.16666666666666666\right), 1\right)}} \]

   11. Taylor expanded in y around 0

       \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{6}, \color{blue}{y} \cdot \left(y \cdot y\right), y\right)}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \frac{7}{360}, \frac{1}{6}\right), 1\right)} \]
   12. Step-by-step derivation

   13. Applied rewrites 77.1%

       \[\leadsto \frac{\mathsf{fma}\left(0.16666666666666666, \color{blue}{y} \cdot \left(y \cdot y\right), y\right)}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.019444444444444445, 0.16666666666666666\right), 1\right)} \]

   if 1.50000000000000009e-146 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 99.9%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. clear-num N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. unpow2 N/A

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

      6. unpow3 N/A

         \[\leadsto \frac{\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot \color{blue}{{y}^{3}} + 1 \cdot y}{\frac{x}{\sin x}} \]

      7. *-lft-identity N/A

         \[\leadsto \frac{\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot {y}^{3} + \color{blue}{y}}{\frac{x}{\sin x}} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right), {y}^{3}, y\right)}}{\frac{x}{\sin x}} \]

   7. Applied rewrites 92.1%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 71.2%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -2 \cdot 10^{-92}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot -0.16666666666666666, x\right) \cdot \mathsf{fma}\left(y, \left(y \cdot y\right) \cdot 0.16666666666666666, y\right)}{x}\\ \mathbf{elif}\;\frac{\sinh y \cdot \sin x}{x} \leq 1.5 \cdot 10^{-146}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.16666666666666666, y \cdot \left(y \cdot y\right), y\right)}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.019444444444444445, 0.16666666666666666\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)}{1}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 66.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y \cdot \sin x}{x}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-92}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot -0.16666666666666666, x\right) \cdot \mathsf{fma}\left(y, \left(y \cdot y\right) \cdot 0.16666666666666666, y\right)}{x}\\ \mathbf{elif}\;t\_0 \leq 4:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.16666666666666666, y \cdot \left(y \cdot y\right), y\right)}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.019444444444444445, 0.16666666666666666\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, y \cdot 0.16666666666666666, y\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.008333333333333333, -0.16666666666666666\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sinh y) (sin x)) x)))
   (if (<= t_0 -2e-92)
     (/
      (*
       (fma x (* (* x x) -0.16666666666666666) x)
       (fma y (* (* y y) 0.16666666666666666) y))
      x)
     (if (<= t_0 4.0)
       (/
        (fma 0.16666666666666666 (* y (* y y)) y)
        (fma
         x
         (* x (fma (* x x) 0.019444444444444445 0.16666666666666666))
         1.0))
       (*
        (fma (* y y) (* y 0.16666666666666666) y)
        (fma
         (* x x)
         (fma x (* x 0.008333333333333333) -0.16666666666666666)
         1.0))))))

C

double code(double x, double y) {
	double t_0 = (sinh(y) * sin(x)) / x;
	double tmp;
	if (t_0 <= -2e-92) {
		tmp = (fma(x, ((x * x) * -0.16666666666666666), x) * fma(y, ((y * y) * 0.16666666666666666), y)) / x;
	} else if (t_0 <= 4.0) {
		tmp = fma(0.16666666666666666, (y * (y * y)), y) / fma(x, (x * fma((x * x), 0.019444444444444445, 0.16666666666666666)), 1.0);
	} else {
		tmp = fma((y * y), (y * 0.16666666666666666), y) * fma((x * x), fma(x, (x * 0.008333333333333333), -0.16666666666666666), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(sinh(y) * sin(x)) / x)
	tmp = 0.0
	if (t_0 <= -2e-92)
		tmp = Float64(Float64(fma(x, Float64(Float64(x * x) * -0.16666666666666666), x) * fma(y, Float64(Float64(y * y) * 0.16666666666666666), y)) / x);
	elseif (t_0 <= 4.0)
		tmp = Float64(fma(0.16666666666666666, Float64(y * Float64(y * y)), y) / fma(x, Float64(x * fma(Float64(x * x), 0.019444444444444445, 0.16666666666666666)), 1.0));
	else
		tmp = Float64(fma(Float64(y * y), Float64(y * 0.16666666666666666), y) * fma(Float64(x * x), fma(x, Float64(x * 0.008333333333333333), -0.16666666666666666), 1.0));
	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-92], N[(N[(N[(x * N[(N[(x * x), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + x), $MachinePrecision] * N[(y * N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[t$95$0, 4.0], N[(N[(0.16666666666666666 * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] / N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 0.019444444444444445 + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y * y), $MachinePrecision] * N[(y * 0.16666666666666666), $MachinePrecision] + y), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -1.99999999999999998e-92

   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

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 82.3

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

   5. Applied rewrites 82.3%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 62.4

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

   8. Applied rewrites 62.4%

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

   if -1.99999999999999998e-92 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 4

   1. Initial program 74.3%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. clear-num N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. unpow2 N/A

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

      6. unpow3 N/A

         \[\leadsto \frac{\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot \color{blue}{{y}^{3}} + 1 \cdot y}{\frac{x}{\sin x}} \]

      7. *-lft-identity N/A

         \[\leadsto \frac{\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot {y}^{3} + \color{blue}{y}}{\frac{x}{\sin x}} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right), {y}^{3}, y\right)}}{\frac{x}{\sin x}} \]

   7. Applied rewrites 98.8%

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

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), y \cdot \left(y \cdot y\right), y\right)}{\color{blue}{1 + {x}^{2} \cdot \left(\frac{1}{6} + \frac{7}{360} \cdot {x}^{2}\right)}} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), y \cdot \left(y \cdot y\right), y\right)}{\color{blue}{{x}^{2} \cdot \left(\frac{1}{6} + \frac{7}{360} \cdot {x}^{2}\right) + 1}} \]

      2. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), y \cdot \left(y \cdot y\right), y\right)}{\color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{6} + \frac{7}{360} \cdot {x}^{2}\right) + 1} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), y \cdot \left(y \cdot y\right), y\right)}{\color{blue}{x \cdot \left(x \cdot \left(\frac{1}{6} + \frac{7}{360} \cdot {x}^{2}\right)\right)} + 1} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), y \cdot \left(y \cdot y\right), y\right)}{\color{blue}{\mathsf{fma}\left(x, x \cdot \left(\frac{1}{6} + \frac{7}{360} \cdot {x}^{2}\right), 1\right)}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), y \cdot \left(y \cdot y\right), y\right)}{\mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{6} + \frac{7}{360} \cdot {x}^{2}\right)}, 1\right)} \]

      6. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), y \cdot \left(y \cdot y\right), y\right)}{\mathsf{fma}\left(x, x \cdot \color{blue}{\left(\frac{7}{360} \cdot {x}^{2} + \frac{1}{6}\right)}, 1\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), y \cdot \left(y \cdot y\right), y\right)}{\mathsf{fma}\left(x, x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{7}{360}} + \frac{1}{6}\right), 1\right)} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), y \cdot \left(y \cdot y\right), y\right)}{\mathsf{fma}\left(x, x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{7}{360}, \frac{1}{6}\right)}, 1\right)} \]

      9. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), y \cdot \left(y \cdot y\right), y\right)}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{7}{360}, \frac{1}{6}\right), 1\right)} \]

      10. lower-*.f64 74.6

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.019444444444444445, 0.16666666666666666\right), 1\right)} \]

   10. Applied rewrites 74.6%

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)}{\color{blue}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.019444444444444445, 0.16666666666666666\right), 1\right)}} \]

   11. Taylor expanded in y around 0

       \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{6}, \color{blue}{y} \cdot \left(y \cdot y\right), y\right)}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \frac{7}{360}, \frac{1}{6}\right), 1\right)} \]
   12. Step-by-step derivation

   13. Applied rewrites 74.5%

       \[\leadsto \frac{\mathsf{fma}\left(0.16666666666666666, \color{blue}{y} \cdot \left(y \cdot y\right), y\right)}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.019444444444444445, 0.16666666666666666\right), 1\right)} \]

   if 4 < (/.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. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*r/ N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. associate-*r* N/A

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

      11. associate-/l* N/A

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

      12. distribute-rgt-out N/A

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

      13. *-rgt-identity N/A

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

      14. distribute-lft-in N/A

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

   5. Applied rewrites 81.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 71.3%

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

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -2 \cdot 10^{-92}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot -0.16666666666666666, x\right) \cdot \mathsf{fma}\left(y, \left(y \cdot y\right) \cdot 0.16666666666666666, y\right)}{x}\\ \mathbf{elif}\;\frac{\sinh y \cdot \sin x}{x} \leq 4:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.16666666666666666, y \cdot \left(y \cdot y\right), y\right)}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.019444444444444445, 0.16666666666666666\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, y \cdot 0.16666666666666666, y\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.008333333333333333, -0.16666666666666666\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 90.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)}{\frac{x}{\sin x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sinh y}{x} \cdot \mathsf{fma}\left(x \cdot x, x \cdot \mathsf{fma}\left(x, x \cdot 0.008333333333333333, -0.16666666666666666\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (sinh y) (sin x)) x) 2e-6)
   (/
    (fma
     (fma
      y
      (* y (fma (* y y) 0.0001984126984126984 0.008333333333333333))
      0.16666666666666666)
     (* y (* y y))
     y)
    (/ x (sin x)))
   (*
    (/ (sinh y) x)
    (fma
     (* x x)
     (* x (fma x (* x 0.008333333333333333) -0.16666666666666666))
     x))))

C

double code(double x, double y) {
	double tmp;
	if (((sinh(y) * sin(x)) / x) <= 2e-6) {
		tmp = fma(fma(y, (y * fma((y * y), 0.0001984126984126984, 0.008333333333333333)), 0.16666666666666666), (y * (y * y)), y) / (x / sin(x));
	} else {
		tmp = (sinh(y) / x) * fma((x * x), (x * fma(x, (x * 0.008333333333333333), -0.16666666666666666)), x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(sinh(y) * sin(x)) / x) <= 2e-6)
		tmp = Float64(fma(fma(y, Float64(y * fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333)), 0.16666666666666666), Float64(y * Float64(y * y)), y) / Float64(x / sin(x)));
	else
		tmp = Float64(Float64(sinh(y) / x) * fma(Float64(x * x), Float64(x * fma(x, Float64(x * 0.008333333333333333), -0.16666666666666666)), x));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[Sinh[y], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], 2e-6], N[(N[(N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] / N[(x / N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sinh[y], $MachinePrecision] / x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(x * 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 1.99999999999999991e-6

   1. Initial program 82.9%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. clear-num N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. unpow2 N/A

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

      6. unpow3 N/A

         \[\leadsto \frac{\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot \color{blue}{{y}^{3}} + 1 \cdot y}{\frac{x}{\sin x}} \]

      7. *-lft-identity N/A

         \[\leadsto \frac{\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot {y}^{3} + \color{blue}{y}}{\frac{x}{\sin x}} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right), {y}^{3}, y\right)}}{\frac{x}{\sin x}} \]

   7. Applied rewrites 94.7%

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

   if 1.99999999999999991e-6 < (/.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

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. sub-neg N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 83.0

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

   5. Applied rewrites 83.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.008333333333333333, -0.16666666666666666\right), x \cdot \left(x \cdot x\right), x\right)} \cdot \sinh y}{x} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 83.1

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

   7. Applied rewrites 83.1%

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

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)}{\frac{x}{\sin x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sinh y}{x} \cdot \mathsf{fma}\left(x \cdot x, x \cdot \mathsf{fma}\left(x, x \cdot 0.008333333333333333, -0.16666666666666666\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 90.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\frac{\sin x}{x} \cdot \mathsf{fma}\left(y \cdot y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sinh y}{x} \cdot \mathsf{fma}\left(x \cdot x, x \cdot \mathsf{fma}\left(x, x \cdot 0.008333333333333333, -0.16666666666666666\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (sinh y) (sin x)) x) 2e-6)
   (*
    (/ (sin x) x)
    (fma
     (* y y)
     (*
      y
      (fma
       y
       (* y (fma y (* y 0.0001984126984126984) 0.008333333333333333))
       0.16666666666666666))
     y))
   (*
    (/ (sinh y) x)
    (fma
     (* x x)
     (* x (fma x (* x 0.008333333333333333) -0.16666666666666666))
     x))))

C

double code(double x, double y) {
	double tmp;
	if (((sinh(y) * sin(x)) / x) <= 2e-6) {
		tmp = (sin(x) / x) * fma((y * y), (y * fma(y, (y * fma(y, (y * 0.0001984126984126984), 0.008333333333333333)), 0.16666666666666666)), y);
	} else {
		tmp = (sinh(y) / x) * fma((x * x), (x * fma(x, (x * 0.008333333333333333), -0.16666666666666666)), x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(sinh(y) * sin(x)) / x) <= 2e-6)
		tmp = Float64(Float64(sin(x) / x) * fma(Float64(y * y), Float64(y * fma(y, Float64(y * fma(y, Float64(y * 0.0001984126984126984), 0.008333333333333333)), 0.16666666666666666)), y));
	else
		tmp = Float64(Float64(sinh(y) / x) * fma(Float64(x * x), Float64(x * fma(x, Float64(x * 0.008333333333333333), -0.16666666666666666)), x));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[Sinh[y], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], 2e-6], N[(N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * N[(y * N[(y * N[(y * 0.0001984126984126984), $MachinePrecision] + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sinh[y], $MachinePrecision] / x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(x * 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 1.99999999999999991e-6

   1. Initial program 82.9%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. clear-num N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. unpow2 N/A

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

      6. unpow3 N/A

         \[\leadsto \frac{\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot \color{blue}{{y}^{3}} + 1 \cdot y}{\frac{x}{\sin x}} \]

      7. *-lft-identity N/A

         \[\leadsto \frac{\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot {y}^{3} + \color{blue}{y}}{\frac{x}{\sin x}} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right), {y}^{3}, y\right)}}{\frac{x}{\sin x}} \]

   7. Applied rewrites 94.7%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)}}{\frac{x}{\sin x}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lift-sin.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lower-/.f64 94.7

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

   9. Applied rewrites 94.7%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)}{\color{blue}{\frac{1}{\frac{\sin x}{x}}}} \]
   10. Step-by-step derivation

       1. lift-/.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-/r/ N/A

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

       4. /-rgt-identity N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 94.7

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

   11. Applied rewrites 94.7%

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

   if 1.99999999999999991e-6 < (/.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

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. sub-neg N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{1}{120}, \frac{-1}{6}\right), x \cdot \color{blue}{\left(x \cdot x\right)}, x\right) \cdot \sinh y}{x} \]

      15. lower-*.f64 83.0

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.008333333333333333, -0.16666666666666666\right), x \cdot \color{blue}{\left(x \cdot x\right)}, x\right) \cdot \sinh y}{x} \]

   5. Applied rewrites 83.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.008333333333333333, -0.16666666666666666\right), x \cdot \left(x \cdot x\right), x\right)} \cdot \sinh y}{x} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{1}{120}, \frac{-1}{6}\right), x \cdot \left(x \cdot x\right), x\right) \cdot \sinh y}{x}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{1}{120}, \frac{-1}{6}\right), x \cdot \left(x \cdot x\right), x\right) \cdot \sinh y}}{x} \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{1}{120}, \frac{-1}{6}\right), x \cdot \left(x \cdot x\right), x\right) \cdot \frac{\sinh y}{x}} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\sinh y}{x} \cdot \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{1}{120}, \frac{-1}{6}\right), x \cdot \left(x \cdot x\right), x\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sinh y}{x} \cdot \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{1}{120}, \frac{-1}{6}\right), x \cdot \left(x \cdot x\right), x\right)} \]

      6. lower-/.f64 83.1

         \[\leadsto \color{blue}{\frac{\sinh y}{x}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.008333333333333333, -0.16666666666666666\right), x \cdot \left(x \cdot x\right), x\right) \]

   7. Applied rewrites 83.1%

      \[\leadsto \color{blue}{\frac{\sinh y}{x} \cdot \mathsf{fma}\left(x \cdot x, x \cdot \mathsf{fma}\left(x, x \cdot 0.008333333333333333, -0.16666666666666666\right), x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\frac{\sin x}{x} \cdot \mathsf{fma}\left(y \cdot y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sinh y}{x} \cdot \mathsf{fma}\left(x \cdot x, x \cdot \mathsf{fma}\left(x, x \cdot 0.008333333333333333, -0.16666666666666666\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 42.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq 4:\\ \;\;\;\;\mathsf{fma}\left(y, x \cdot \left(x \cdot -0.16666666666666666\right), y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(y \cdot y\right) \cdot 0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (sinh y) (sin x)) x) 4.0)
   (fma y (* x (* x -0.16666666666666666)) y)
   (* y (* (* y y) 0.16666666666666666))))

C

double code(double x, double y) {
	double tmp;
	if (((sinh(y) * sin(x)) / x) <= 4.0) {
		tmp = fma(y, (x * (x * -0.16666666666666666)), 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) <= 4.0)
		tmp = fma(y, Float64(x * Float64(x * -0.16666666666666666)), y);
	else
		tmp = Float64(y * Float64(Float64(y * y) * 0.16666666666666666));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[Sinh[y], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], 4.0], N[(y * N[(x * N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision], N[(y * N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 4

   1. Initial program 83.0%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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. associate-/l* N/A

         \[\leadsto \color{blue}{y \cdot \frac{\sin x}{x}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{y \cdot \frac{\sin x}{x}} \]

      3. lower-/.f64 N/A

         \[\leadsto y \cdot \color{blue}{\frac{\sin x}{x}} \]

      4. lower-sin.f64 71.0

         \[\leadsto y \cdot \frac{\color{blue}{\sin x}}{x} \]

   5. Applied rewrites 71.0%

      \[\leadsto \color{blue}{y \cdot \frac{\sin x}{x}} \]

   6. Taylor expanded in x around 0

      \[\leadsto y + \color{blue}{\frac{-1}{6} \cdot \left({x}^{2} \cdot y\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 44.5%

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{x \cdot \left(x \cdot -0.16666666666666666\right)}, y\right) \]

   if 4 < (/.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. +-commutative N/A

         \[\leadsto y \cdot \color{blue}{\left(\frac{\sin x}{x} + \frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x}\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{y \cdot \frac{\sin x}{x} + y \cdot \left(\frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x}\right)} \]

      3. associate-*r/ N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + y \cdot \color{blue}{\frac{\frac{1}{6} \cdot \left({y}^{2} \cdot \sin x\right)}{x}} \]

      4. *-commutative N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + y \cdot \frac{\frac{1}{6} \cdot \color{blue}{\left(\sin x \cdot {y}^{2}\right)}}{x} \]

      5. associate-*r* N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + y \cdot \frac{\color{blue}{\left(\frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}}}{x} \]

      6. associate-*r/ N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + \color{blue}{\frac{y \cdot \left(\left(\frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}\right)}{x}} \]

      7. associate-*r* N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + \frac{y \cdot \color{blue}{\left(\frac{1}{6} \cdot \left(\sin x \cdot {y}^{2}\right)\right)}}{x} \]

      8. *-commutative N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + \frac{y \cdot \left(\frac{1}{6} \cdot \color{blue}{\left({y}^{2} \cdot \sin x\right)}\right)}{x} \]

      9. associate-*r* N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + \frac{y \cdot \color{blue}{\left(\left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \sin x\right)}}{x} \]

      10. associate-*r* N/A

          \[\leadsto y \cdot \frac{\sin x}{x} + \frac{\color{blue}{\left(y \cdot \left(\frac{1}{6} \cdot {y}^{2}\right)\right) \cdot \sin x}}{x} \]

      11. associate-/l* N/A

          \[\leadsto y \cdot \frac{\sin x}{x} + \color{blue}{\left(y \cdot \left(\frac{1}{6} \cdot {y}^{2}\right)\right) \cdot \frac{\sin x}{x}} \]

      12. distribute-rgt-out N/A

          \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot \left(y + y \cdot \left(\frac{1}{6} \cdot {y}^{2}\right)\right)} \]

      13. *-rgt-identity N/A

          \[\leadsto \frac{\sin x}{x} \cdot \left(\color{blue}{y \cdot 1} + y \cdot \left(\frac{1}{6} \cdot {y}^{2}\right)\right) \]

      14. distribute-lft-in N/A

          \[\leadsto \frac{\sin x}{x} \cdot \color{blue}{\left(y \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)} \]

   5. Applied rewrites 81.9%

      \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot \mathsf{fma}\left(y \cdot y, y \cdot 0.16666666666666666, y\right)} \]

   6. Taylor expanded in y around inf

      \[\leadsto \frac{1}{6} \cdot \color{blue}{\frac{{y}^{3} \cdot \sin x}{x}} \]
   7. Step-by-step derivation

   8. Applied rewrites 85.1%

      \[\leadsto \sin x \cdot \color{blue}{\frac{y \cdot \left(0.16666666666666666 \cdot \left(y \cdot y\right)\right)}{x}} \]

   9. Taylor expanded in x around 0

      \[\leadsto \frac{1}{6} \cdot {y}^{\color{blue}{3}} \]
   10. Step-by-step derivation

   11. Applied rewrites 69.7%

       \[\leadsto y \cdot \left(0.16666666666666666 \cdot \color{blue}{\left(y \cdot y\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq 4:\\ \;\;\;\;\mathsf{fma}\left(y, x \cdot \left(x \cdot -0.16666666666666666\right), y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(y \cdot y\right) \cdot 0.16666666666666666\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 39.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq 4:\\ \;\;\;\;y \cdot 1\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(y \cdot y\right) \cdot 0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (sinh y) (sin x)) x) 4.0)
   (* y 1.0)
   (* y (* (* y y) 0.16666666666666666))))

C

double code(double x, double y) {
	double tmp;
	if (((sinh(y) * sin(x)) / x) <= 4.0) {
		tmp = y * 1.0;
	} 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) <= 4.0d0) then
        tmp = y * 1.0d0
    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) <= 4.0) {
		tmp = y * 1.0;
	} else {
		tmp = y * ((y * y) * 0.16666666666666666);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((math.sinh(y) * math.sin(x)) / x) <= 4.0:
		tmp = y * 1.0
	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) <= 4.0)
		tmp = Float64(y * 1.0);
	else
		tmp = Float64(y * Float64(Float64(y * y) * 0.16666666666666666));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((sinh(y) * sin(x)) / x) <= 4.0)
		tmp = y * 1.0;
	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], 4.0], N[(y * 1.0), $MachinePrecision], N[(y * N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 4

   1. Initial program 83.0%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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. associate-/l* N/A

         \[\leadsto \color{blue}{y \cdot \frac{\sin x}{x}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{y \cdot \frac{\sin x}{x}} \]

      3. lower-/.f64 N/A

         \[\leadsto y \cdot \color{blue}{\frac{\sin x}{x}} \]

      4. lower-sin.f64 71.0

         \[\leadsto y \cdot \frac{\color{blue}{\sin x}}{x} \]

   5. Applied rewrites 71.0%

      \[\leadsto \color{blue}{y \cdot \frac{\sin x}{x}} \]

   6. Taylor expanded in x around 0

      \[\leadsto y \cdot 1 \]
   7. Step-by-step derivation

   8. Applied rewrites 38.7%

      \[\leadsto y \cdot 1 \]

   if 4 < (/.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. +-commutative N/A

         \[\leadsto y \cdot \color{blue}{\left(\frac{\sin x}{x} + \frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x}\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{y \cdot \frac{\sin x}{x} + y \cdot \left(\frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x}\right)} \]

      3. associate-*r/ N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + y \cdot \color{blue}{\frac{\frac{1}{6} \cdot \left({y}^{2} \cdot \sin x\right)}{x}} \]

      4. *-commutative N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + y \cdot \frac{\frac{1}{6} \cdot \color{blue}{\left(\sin x \cdot {y}^{2}\right)}}{x} \]

      5. associate-*r* N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + y \cdot \frac{\color{blue}{\left(\frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}}}{x} \]

      6. associate-*r/ N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + \color{blue}{\frac{y \cdot \left(\left(\frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}\right)}{x}} \]

      7. associate-*r* N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + \frac{y \cdot \color{blue}{\left(\frac{1}{6} \cdot \left(\sin x \cdot {y}^{2}\right)\right)}}{x} \]

      8. *-commutative N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + \frac{y \cdot \left(\frac{1}{6} \cdot \color{blue}{\left({y}^{2} \cdot \sin x\right)}\right)}{x} \]

      9. associate-*r* N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + \frac{y \cdot \color{blue}{\left(\left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \sin x\right)}}{x} \]

      10. associate-*r* N/A

          \[\leadsto y \cdot \frac{\sin x}{x} + \frac{\color{blue}{\left(y \cdot \left(\frac{1}{6} \cdot {y}^{2}\right)\right) \cdot \sin x}}{x} \]

      11. associate-/l* N/A

          \[\leadsto y \cdot \frac{\sin x}{x} + \color{blue}{\left(y \cdot \left(\frac{1}{6} \cdot {y}^{2}\right)\right) \cdot \frac{\sin x}{x}} \]

      12. distribute-rgt-out N/A

          \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot \left(y + y \cdot \left(\frac{1}{6} \cdot {y}^{2}\right)\right)} \]

      13. *-rgt-identity N/A

          \[\leadsto \frac{\sin x}{x} \cdot \left(\color{blue}{y \cdot 1} + y \cdot \left(\frac{1}{6} \cdot {y}^{2}\right)\right) \]

      14. distribute-lft-in N/A

          \[\leadsto \frac{\sin x}{x} \cdot \color{blue}{\left(y \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)} \]

   5. Applied rewrites 81.9%

      \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot \mathsf{fma}\left(y \cdot y, y \cdot 0.16666666666666666, y\right)} \]

   6. Taylor expanded in y around inf

      \[\leadsto \frac{1}{6} \cdot \color{blue}{\frac{{y}^{3} \cdot \sin x}{x}} \]
   7. Step-by-step derivation

   8. Applied rewrites 85.1%

      \[\leadsto \sin x \cdot \color{blue}{\frac{y \cdot \left(0.16666666666666666 \cdot \left(y \cdot y\right)\right)}{x}} \]

   9. Taylor expanded in x around 0

      \[\leadsto \frac{1}{6} \cdot {y}^{\color{blue}{3}} \]
   10. Step-by-step derivation

   11. Applied rewrites 69.7%

       \[\leadsto y \cdot \left(0.16666666666666666 \cdot \color{blue}{\left(y \cdot y\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 45.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq 4:\\ \;\;\;\;y \cdot 1\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(y \cdot y\right) \cdot 0.16666666666666666\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 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]

Derivation

1. Initial program 86.8%

   \[\frac{\sin x \cdot \sinh y}{x} \]
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

      \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]

   4. *-commutative N/A

      \[\leadsto \color{blue}{\frac{\sinh y}{x} \cdot \sin x} \]

   5. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{\sinh y}{x} \cdot \sin x} \]

   6. lower-/.f64 99.9

      \[\leadsto \color{blue}{\frac{\sinh y}{x}} \cdot \sin x \]

4. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\frac{\sinh y}{x} \cdot \sin x} \]

5. Final simplification 99.9%

   \[\leadsto \sin x \cdot \frac{\sinh y}{x} \]
6. Add Preprocessing

Alternative 19: 58.8% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.25 \cdot 10^{+67}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, y \cdot 0.16666666666666666, y\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.008333333333333333, -0.16666666666666666\right), 1\right)\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{+116}:\\ \;\;\;\;\left(y \cdot \left(y \cdot y\right)\right) \cdot \mathsf{fma}\left(x \cdot x, -0.027777777777777776, 0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(y \cdot y\right) \cdot 0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1.25e+67)
   (*
    (fma (* y y) (* y 0.16666666666666666) y)
    (fma (* x x) (fma x (* x 0.008333333333333333) -0.16666666666666666) 1.0))
   (if (<= x 1.7e+116)
     (* (* y (* y y)) (fma (* x x) -0.027777777777777776 0.16666666666666666))
     (* y (* (* y y) 0.16666666666666666)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.25e+67) {
		tmp = fma((y * y), (y * 0.16666666666666666), y) * fma((x * x), fma(x, (x * 0.008333333333333333), -0.16666666666666666), 1.0);
	} else if (x <= 1.7e+116) {
		tmp = (y * (y * y)) * fma((x * x), -0.027777777777777776, 0.16666666666666666);
	} else {
		tmp = y * ((y * y) * 0.16666666666666666);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.25e+67)
		tmp = Float64(fma(Float64(y * y), Float64(y * 0.16666666666666666), y) * fma(Float64(x * x), fma(x, Float64(x * 0.008333333333333333), -0.16666666666666666), 1.0));
	elseif (x <= 1.7e+116)
		tmp = Float64(Float64(y * Float64(y * y)) * fma(Float64(x * x), -0.027777777777777776, 0.16666666666666666));
	else
		tmp = Float64(y * Float64(Float64(y * y) * 0.16666666666666666));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.25e+67], N[(N[(N[(y * y), $MachinePrecision] * N[(y * 0.16666666666666666), $MachinePrecision] + y), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.7e+116], N[(N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.027777777777777776 + 0.16666666666666666), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.24999999999999994e67

   1. Initial program 84.5%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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. +-commutative N/A

         \[\leadsto y \cdot \color{blue}{\left(\frac{\sin x}{x} + \frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x}\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{y \cdot \frac{\sin x}{x} + y \cdot \left(\frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x}\right)} \]

      3. associate-*r/ N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + y \cdot \color{blue}{\frac{\frac{1}{6} \cdot \left({y}^{2} \cdot \sin x\right)}{x}} \]

      4. *-commutative N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + y \cdot \frac{\frac{1}{6} \cdot \color{blue}{\left(\sin x \cdot {y}^{2}\right)}}{x} \]

      5. associate-*r* N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + y \cdot \frac{\color{blue}{\left(\frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}}}{x} \]

      6. associate-*r/ N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + \color{blue}{\frac{y \cdot \left(\left(\frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}\right)}{x}} \]

      7. associate-*r* N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + \frac{y \cdot \color{blue}{\left(\frac{1}{6} \cdot \left(\sin x \cdot {y}^{2}\right)\right)}}{x} \]

      8. *-commutative N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + \frac{y \cdot \left(\frac{1}{6} \cdot \color{blue}{\left({y}^{2} \cdot \sin x\right)}\right)}{x} \]

      9. associate-*r* N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + \frac{y \cdot \color{blue}{\left(\left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \sin x\right)}}{x} \]

      10. associate-*r* N/A

          \[\leadsto y \cdot \frac{\sin x}{x} + \frac{\color{blue}{\left(y \cdot \left(\frac{1}{6} \cdot {y}^{2}\right)\right) \cdot \sin x}}{x} \]

      11. associate-/l* N/A

          \[\leadsto y \cdot \frac{\sin x}{x} + \color{blue}{\left(y \cdot \left(\frac{1}{6} \cdot {y}^{2}\right)\right) \cdot \frac{\sin x}{x}} \]

      12. distribute-rgt-out N/A

          \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot \left(y + y \cdot \left(\frac{1}{6} \cdot {y}^{2}\right)\right)} \]

      13. *-rgt-identity N/A

          \[\leadsto \frac{\sin x}{x} \cdot \left(\color{blue}{y \cdot 1} + y \cdot \left(\frac{1}{6} \cdot {y}^{2}\right)\right) \]

      14. distribute-lft-in N/A

          \[\leadsto \frac{\sin x}{x} \cdot \color{blue}{\left(y \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)} \]

   5. Applied rewrites 86.9%

      \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot \mathsf{fma}\left(y \cdot y, y \cdot 0.16666666666666666, y\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(1 + {x}^{2} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right) \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, y \cdot \frac{1}{6}, y\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 62.1%

      \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.008333333333333333, -0.16666666666666666\right), 1\right) \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, y \cdot 0.16666666666666666, y\right) \]

   if 1.24999999999999994e67 < x < 1.70000000000000011e116

   1. Initial program 99.8%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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. +-commutative N/A

         \[\leadsto y \cdot \color{blue}{\left(\frac{\sin x}{x} + \frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x}\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{y \cdot \frac{\sin x}{x} + y \cdot \left(\frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x}\right)} \]

      3. associate-*r/ N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + y \cdot \color{blue}{\frac{\frac{1}{6} \cdot \left({y}^{2} \cdot \sin x\right)}{x}} \]

      4. *-commutative N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + y \cdot \frac{\frac{1}{6} \cdot \color{blue}{\left(\sin x \cdot {y}^{2}\right)}}{x} \]

      5. associate-*r* N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + y \cdot \frac{\color{blue}{\left(\frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}}}{x} \]

      6. associate-*r/ N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + \color{blue}{\frac{y \cdot \left(\left(\frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}\right)}{x}} \]

      7. associate-*r* N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + \frac{y \cdot \color{blue}{\left(\frac{1}{6} \cdot \left(\sin x \cdot {y}^{2}\right)\right)}}{x} \]

      8. *-commutative N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + \frac{y \cdot \left(\frac{1}{6} \cdot \color{blue}{\left({y}^{2} \cdot \sin x\right)}\right)}{x} \]

      9. associate-*r* N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + \frac{y \cdot \color{blue}{\left(\left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \sin x\right)}}{x} \]

      10. associate-*r* N/A

          \[\leadsto y \cdot \frac{\sin x}{x} + \frac{\color{blue}{\left(y \cdot \left(\frac{1}{6} \cdot {y}^{2}\right)\right) \cdot \sin x}}{x} \]

      11. associate-/l* N/A

          \[\leadsto y \cdot \frac{\sin x}{x} + \color{blue}{\left(y \cdot \left(\frac{1}{6} \cdot {y}^{2}\right)\right) \cdot \frac{\sin x}{x}} \]

      12. distribute-rgt-out N/A

          \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot \left(y + y \cdot \left(\frac{1}{6} \cdot {y}^{2}\right)\right)} \]

      13. *-rgt-identity N/A

          \[\leadsto \frac{\sin x}{x} \cdot \left(\color{blue}{y \cdot 1} + y \cdot \left(\frac{1}{6} \cdot {y}^{2}\right)\right) \]

      14. distribute-lft-in N/A

          \[\leadsto \frac{\sin x}{x} \cdot \color{blue}{\left(y \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)} \]

   5. Applied rewrites 76.0%

      \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot \mathsf{fma}\left(y \cdot y, y \cdot 0.16666666666666666, y\right)} \]

   6. Taylor expanded in y around inf

      \[\leadsto \frac{1}{6} \cdot \color{blue}{\frac{{y}^{3} \cdot \sin x}{x}} \]
   7. Step-by-step derivation

   8. Applied rewrites 30.3%

      \[\leadsto \sin x \cdot \color{blue}{\frac{y \cdot \left(0.16666666666666666 \cdot \left(y \cdot y\right)\right)}{x}} \]

   9. Taylor expanded in x around 0

      \[\leadsto \frac{-1}{36} \cdot \left({x}^{2} \cdot {y}^{3}\right) + \frac{1}{6} \cdot \color{blue}{{y}^{3}} \]
   10. Step-by-step derivation

   11. Applied rewrites 53.1%

       \[\leadsto \left(y \cdot \left(y \cdot y\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{-0.027777777777777776}, 0.16666666666666666\right) \]

   if 1.70000000000000011e116 < 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. +-commutative N/A

         \[\leadsto y \cdot \color{blue}{\left(\frac{\sin x}{x} + \frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x}\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{y \cdot \frac{\sin x}{x} + y \cdot \left(\frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x}\right)} \]

      3. associate-*r/ N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + y \cdot \color{blue}{\frac{\frac{1}{6} \cdot \left({y}^{2} \cdot \sin x\right)}{x}} \]

      4. *-commutative N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + y \cdot \frac{\frac{1}{6} \cdot \color{blue}{\left(\sin x \cdot {y}^{2}\right)}}{x} \]

      5. associate-*r* N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + y \cdot \frac{\color{blue}{\left(\frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}}}{x} \]

      6. associate-*r/ N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + \color{blue}{\frac{y \cdot \left(\left(\frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}\right)}{x}} \]

      7. associate-*r* N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + \frac{y \cdot \color{blue}{\left(\frac{1}{6} \cdot \left(\sin x \cdot {y}^{2}\right)\right)}}{x} \]

      8. *-commutative N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + \frac{y \cdot \left(\frac{1}{6} \cdot \color{blue}{\left({y}^{2} \cdot \sin x\right)}\right)}{x} \]

      9. associate-*r* N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + \frac{y \cdot \color{blue}{\left(\left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \sin x\right)}}{x} \]

      10. associate-*r* N/A

          \[\leadsto y \cdot \frac{\sin x}{x} + \frac{\color{blue}{\left(y \cdot \left(\frac{1}{6} \cdot {y}^{2}\right)\right) \cdot \sin x}}{x} \]

      11. associate-/l* N/A

          \[\leadsto y \cdot \frac{\sin x}{x} + \color{blue}{\left(y \cdot \left(\frac{1}{6} \cdot {y}^{2}\right)\right) \cdot \frac{\sin x}{x}} \]

      12. distribute-rgt-out N/A

          \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot \left(y + y \cdot \left(\frac{1}{6} \cdot {y}^{2}\right)\right)} \]

      13. *-rgt-identity N/A

          \[\leadsto \frac{\sin x}{x} \cdot \left(\color{blue}{y \cdot 1} + y \cdot \left(\frac{1}{6} \cdot {y}^{2}\right)\right) \]

      14. distribute-lft-in N/A

          \[\leadsto \frac{\sin x}{x} \cdot \color{blue}{\left(y \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)} \]

   5. Applied rewrites 87.4%

      \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot \mathsf{fma}\left(y \cdot y, y \cdot 0.16666666666666666, y\right)} \]

   6. Taylor expanded in y around inf

      \[\leadsto \frac{1}{6} \cdot \color{blue}{\frac{{y}^{3} \cdot \sin x}{x}} \]
   7. Step-by-step derivation

   8. Applied rewrites 81.6%

      \[\leadsto \sin x \cdot \color{blue}{\frac{y \cdot \left(0.16666666666666666 \cdot \left(y \cdot y\right)\right)}{x}} \]

   9. Taylor expanded in x around 0

      \[\leadsto \frac{1}{6} \cdot {y}^{\color{blue}{3}} \]
   10. Step-by-step derivation

   11. Applied rewrites 49.0%

       \[\leadsto y \cdot \left(0.16666666666666666 \cdot \color{blue}{\left(y \cdot y\right)}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.25 \cdot 10^{+67}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, y \cdot 0.16666666666666666, y\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.008333333333333333, -0.16666666666666666\right), 1\right)\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{+116}:\\ \;\;\;\;\left(y \cdot \left(y \cdot y\right)\right) \cdot \mathsf{fma}\left(x \cdot x, -0.027777777777777776, 0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(y \cdot y\right) \cdot 0.16666666666666666\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 57.3% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y \cdot y\right) \cdot 0.16666666666666666\\ \mathbf{if}\;x \leq 7.2 \cdot 10^{+51}:\\ \;\;\;\;\mathsf{fma}\left(y, t\_0, y\right)\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{+116}:\\ \;\;\;\;\left(y \cdot \left(y \cdot y\right)\right) \cdot \mathsf{fma}\left(x \cdot x, -0.027777777777777776, 0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* y y) 0.16666666666666666)))
   (if (<= x 7.2e+51)
     (fma y t_0 y)
     (if (<= x 1.7e+116)
       (*
        (* y (* y y))
        (fma (* x x) -0.027777777777777776 0.16666666666666666))
       (* y t_0)))))

C

double code(double x, double y) {
	double t_0 = (y * y) * 0.16666666666666666;
	double tmp;
	if (x <= 7.2e+51) {
		tmp = fma(y, t_0, y);
	} else if (x <= 1.7e+116) {
		tmp = (y * (y * y)) * fma((x * x), -0.027777777777777776, 0.16666666666666666);
	} else {
		tmp = y * t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(y * y) * 0.16666666666666666)
	tmp = 0.0
	if (x <= 7.2e+51)
		tmp = fma(y, t_0, y);
	elseif (x <= 1.7e+116)
		tmp = Float64(Float64(y * Float64(y * y)) * fma(Float64(x * x), -0.027777777777777776, 0.16666666666666666));
	else
		tmp = Float64(y * t_0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]}, If[LessEqual[x, 7.2e+51], N[(y * t$95$0 + y), $MachinePrecision], If[LessEqual[x, 1.7e+116], N[(N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.027777777777777776 + 0.16666666666666666), $MachinePrecision]), $MachinePrecision], N[(y * t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < 7.20000000000000022e51

   1. Initial program 84.1%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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. +-commutative N/A

         \[\leadsto y \cdot \color{blue}{\left(\frac{\sin x}{x} + \frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x}\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{y \cdot \frac{\sin x}{x} + y \cdot \left(\frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x}\right)} \]

      3. associate-*r/ N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + y \cdot \color{blue}{\frac{\frac{1}{6} \cdot \left({y}^{2} \cdot \sin x\right)}{x}} \]

      4. *-commutative N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + y \cdot \frac{\frac{1}{6} \cdot \color{blue}{\left(\sin x \cdot {y}^{2}\right)}}{x} \]

      5. associate-*r* N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + y \cdot \frac{\color{blue}{\left(\frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}}}{x} \]

      6. associate-*r/ N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + \color{blue}{\frac{y \cdot \left(\left(\frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}\right)}{x}} \]

      7. associate-*r* N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + \frac{y \cdot \color{blue}{\left(\frac{1}{6} \cdot \left(\sin x \cdot {y}^{2}\right)\right)}}{x} \]

      8. *-commutative N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + \frac{y \cdot \left(\frac{1}{6} \cdot \color{blue}{\left({y}^{2} \cdot \sin x\right)}\right)}{x} \]

      9. associate-*r* N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + \frac{y \cdot \color{blue}{\left(\left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \sin x\right)}}{x} \]

      10. associate-*r* N/A

          \[\leadsto y \cdot \frac{\sin x}{x} + \frac{\color{blue}{\left(y \cdot \left(\frac{1}{6} \cdot {y}^{2}\right)\right) \cdot \sin x}}{x} \]

      11. associate-/l* N/A

          \[\leadsto y \cdot \frac{\sin x}{x} + \color{blue}{\left(y \cdot \left(\frac{1}{6} \cdot {y}^{2}\right)\right) \cdot \frac{\sin x}{x}} \]

      12. distribute-rgt-out N/A

          \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot \left(y + y \cdot \left(\frac{1}{6} \cdot {y}^{2}\right)\right)} \]

      13. *-rgt-identity N/A

          \[\leadsto \frac{\sin x}{x} \cdot \left(\color{blue}{y \cdot 1} + y \cdot \left(\frac{1}{6} \cdot {y}^{2}\right)\right) \]

      14. distribute-lft-in N/A

          \[\leadsto \frac{\sin x}{x} \cdot \color{blue}{\left(y \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)} \]

   5. Applied rewrites 87.9%

      \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot \mathsf{fma}\left(y \cdot y, y \cdot 0.16666666666666666, y\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto y + \color{blue}{\frac{1}{6} \cdot {y}^{3}} \]
   7. Step-by-step derivation

   8. Applied rewrites 62.1%

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{0.16666666666666666 \cdot \left(y \cdot y\right)}, y\right) \]

   if 7.20000000000000022e51 < x < 1.70000000000000011e116

   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. +-commutative N/A

         \[\leadsto y \cdot \color{blue}{\left(\frac{\sin x}{x} + \frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x}\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{y \cdot \frac{\sin x}{x} + y \cdot \left(\frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x}\right)} \]

      3. associate-*r/ N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + y \cdot \color{blue}{\frac{\frac{1}{6} \cdot \left({y}^{2} \cdot \sin x\right)}{x}} \]

      4. *-commutative N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + y \cdot \frac{\frac{1}{6} \cdot \color{blue}{\left(\sin x \cdot {y}^{2}\right)}}{x} \]

      5. associate-*r* N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + y \cdot \frac{\color{blue}{\left(\frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}}}{x} \]

      6. associate-*r/ N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + \color{blue}{\frac{y \cdot \left(\left(\frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}\right)}{x}} \]

      7. associate-*r* N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + \frac{y \cdot \color{blue}{\left(\frac{1}{6} \cdot \left(\sin x \cdot {y}^{2}\right)\right)}}{x} \]

      8. *-commutative N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + \frac{y \cdot \left(\frac{1}{6} \cdot \color{blue}{\left({y}^{2} \cdot \sin x\right)}\right)}{x} \]

      9. associate-*r* N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + \frac{y \cdot \color{blue}{\left(\left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \sin x\right)}}{x} \]

      10. associate-*r* N/A

          \[\leadsto y \cdot \frac{\sin x}{x} + \frac{\color{blue}{\left(y \cdot \left(\frac{1}{6} \cdot {y}^{2}\right)\right) \cdot \sin x}}{x} \]

      11. associate-/l* N/A

          \[\leadsto y \cdot \frac{\sin x}{x} + \color{blue}{\left(y \cdot \left(\frac{1}{6} \cdot {y}^{2}\right)\right) \cdot \frac{\sin x}{x}} \]

      12. distribute-rgt-out N/A

          \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot \left(y + y \cdot \left(\frac{1}{6} \cdot {y}^{2}\right)\right)} \]

      13. *-rgt-identity N/A

          \[\leadsto \frac{\sin x}{x} \cdot \left(\color{blue}{y \cdot 1} + y \cdot \left(\frac{1}{6} \cdot {y}^{2}\right)\right) \]

      14. distribute-lft-in N/A

          \[\leadsto \frac{\sin x}{x} \cdot \color{blue}{\left(y \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)} \]

   5. Applied rewrites 63.2%

      \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot \mathsf{fma}\left(y \cdot y, y \cdot 0.16666666666666666, y\right)} \]

   6. Taylor expanded in y around inf

      \[\leadsto \frac{1}{6} \cdot \color{blue}{\frac{{y}^{3} \cdot \sin x}{x}} \]
   7. Step-by-step derivation

   8. Applied rewrites 20.4%

      \[\leadsto \sin x \cdot \color{blue}{\frac{y \cdot \left(0.16666666666666666 \cdot \left(y \cdot y\right)\right)}{x}} \]

   9. Taylor expanded in x around 0

      \[\leadsto \frac{-1}{36} \cdot \left({x}^{2} \cdot {y}^{3}\right) + \frac{1}{6} \cdot \color{blue}{{y}^{3}} \]
   10. Step-by-step derivation

   11. Applied rewrites 33.8%

       \[\leadsto \left(y \cdot \left(y \cdot y\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{-0.027777777777777776}, 0.16666666666666666\right) \]

   if 1.70000000000000011e116 < 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. +-commutative N/A

         \[\leadsto y \cdot \color{blue}{\left(\frac{\sin x}{x} + \frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x}\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{y \cdot \frac{\sin x}{x} + y \cdot \left(\frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x}\right)} \]

      3. associate-*r/ N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + y \cdot \color{blue}{\frac{\frac{1}{6} \cdot \left({y}^{2} \cdot \sin x\right)}{x}} \]

      4. *-commutative N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + y \cdot \frac{\frac{1}{6} \cdot \color{blue}{\left(\sin x \cdot {y}^{2}\right)}}{x} \]

      5. associate-*r* N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + y \cdot \frac{\color{blue}{\left(\frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}}}{x} \]

      6. associate-*r/ N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + \color{blue}{\frac{y \cdot \left(\left(\frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}\right)}{x}} \]

      7. associate-*r* N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + \frac{y \cdot \color{blue}{\left(\frac{1}{6} \cdot \left(\sin x \cdot {y}^{2}\right)\right)}}{x} \]

      8. *-commutative N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + \frac{y \cdot \left(\frac{1}{6} \cdot \color{blue}{\left({y}^{2} \cdot \sin x\right)}\right)}{x} \]

      9. associate-*r* N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + \frac{y \cdot \color{blue}{\left(\left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \sin x\right)}}{x} \]

      10. associate-*r* N/A

          \[\leadsto y \cdot \frac{\sin x}{x} + \frac{\color{blue}{\left(y \cdot \left(\frac{1}{6} \cdot {y}^{2}\right)\right) \cdot \sin x}}{x} \]

      11. associate-/l* N/A

          \[\leadsto y \cdot \frac{\sin x}{x} + \color{blue}{\left(y \cdot \left(\frac{1}{6} \cdot {y}^{2}\right)\right) \cdot \frac{\sin x}{x}} \]

      12. distribute-rgt-out N/A

          \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot \left(y + y \cdot \left(\frac{1}{6} \cdot {y}^{2}\right)\right)} \]

      13. *-rgt-identity N/A

          \[\leadsto \frac{\sin x}{x} \cdot \left(\color{blue}{y \cdot 1} + y \cdot \left(\frac{1}{6} \cdot {y}^{2}\right)\right) \]

      14. distribute-lft-in N/A

          \[\leadsto \frac{\sin x}{x} \cdot \color{blue}{\left(y \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)} \]

   5. Applied rewrites 87.4%

      \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot \mathsf{fma}\left(y \cdot y, y \cdot 0.16666666666666666, y\right)} \]

   6. Taylor expanded in y around inf

      \[\leadsto \frac{1}{6} \cdot \color{blue}{\frac{{y}^{3} \cdot \sin x}{x}} \]
   7. Step-by-step derivation

   8. Applied rewrites 81.6%

      \[\leadsto \sin x \cdot \color{blue}{\frac{y \cdot \left(0.16666666666666666 \cdot \left(y \cdot y\right)\right)}{x}} \]

   9. Taylor expanded in x around 0

      \[\leadsto \frac{1}{6} \cdot {y}^{\color{blue}{3}} \]
   10. Step-by-step derivation

   11. Applied rewrites 49.0%

       \[\leadsto y \cdot \left(0.16666666666666666 \cdot \color{blue}{\left(y \cdot y\right)}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 59.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 7.2 \cdot 10^{+51}:\\ \;\;\;\;\mathsf{fma}\left(y, \left(y \cdot y\right) \cdot 0.16666666666666666, y\right)\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{+116}:\\ \;\;\;\;\left(y \cdot \left(y \cdot y\right)\right) \cdot \mathsf{fma}\left(x \cdot x, -0.027777777777777776, 0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(y \cdot y\right) \cdot 0.16666666666666666\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 58.4% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 18000000000000:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, y \cdot 0.16666666666666666, y\right) \cdot \mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(y \cdot y\right) \cdot 0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 18000000000000.0)
   (*
    (fma (* y y) (* y 0.16666666666666666) y)
    (fma x (* x -0.16666666666666666) 1.0))
   (* y (* (* y y) 0.16666666666666666))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 18000000000000.0) {
		tmp = fma((y * y), (y * 0.16666666666666666), y) * fma(x, (x * -0.16666666666666666), 1.0);
	} else {
		tmp = y * ((y * y) * 0.16666666666666666);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 18000000000000.0)
		tmp = Float64(fma(Float64(y * y), Float64(y * 0.16666666666666666), y) * fma(x, Float64(x * -0.16666666666666666), 1.0));
	else
		tmp = Float64(y * Float64(Float64(y * y) * 0.16666666666666666));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 18000000000000.0], N[(N[(N[(y * y), $MachinePrecision] * N[(y * 0.16666666666666666), $MachinePrecision] + y), $MachinePrecision] * N[(x * N[(x * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.8e13

   1. Initial program 83.3%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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. +-commutative N/A

         \[\leadsto y \cdot \color{blue}{\left(\frac{\sin x}{x} + \frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x}\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{y \cdot \frac{\sin x}{x} + y \cdot \left(\frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x}\right)} \]

      3. associate-*r/ N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + y \cdot \color{blue}{\frac{\frac{1}{6} \cdot \left({y}^{2} \cdot \sin x\right)}{x}} \]

      4. *-commutative N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + y \cdot \frac{\frac{1}{6} \cdot \color{blue}{\left(\sin x \cdot {y}^{2}\right)}}{x} \]

      5. associate-*r* N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + y \cdot \frac{\color{blue}{\left(\frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}}}{x} \]

      6. associate-*r/ N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + \color{blue}{\frac{y \cdot \left(\left(\frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}\right)}{x}} \]

      7. associate-*r* N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + \frac{y \cdot \color{blue}{\left(\frac{1}{6} \cdot \left(\sin x \cdot {y}^{2}\right)\right)}}{x} \]

      8. *-commutative N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + \frac{y \cdot \left(\frac{1}{6} \cdot \color{blue}{\left({y}^{2} \cdot \sin x\right)}\right)}{x} \]

      9. associate-*r* N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + \frac{y \cdot \color{blue}{\left(\left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \sin x\right)}}{x} \]

      10. associate-*r* N/A

          \[\leadsto y \cdot \frac{\sin x}{x} + \frac{\color{blue}{\left(y \cdot \left(\frac{1}{6} \cdot {y}^{2}\right)\right) \cdot \sin x}}{x} \]

      11. associate-/l* N/A

          \[\leadsto y \cdot \frac{\sin x}{x} + \color{blue}{\left(y \cdot \left(\frac{1}{6} \cdot {y}^{2}\right)\right) \cdot \frac{\sin x}{x}} \]

      12. distribute-rgt-out N/A

          \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot \left(y + y \cdot \left(\frac{1}{6} \cdot {y}^{2}\right)\right)} \]

      13. *-rgt-identity N/A

          \[\leadsto \frac{\sin x}{x} \cdot \left(\color{blue}{y \cdot 1} + y \cdot \left(\frac{1}{6} \cdot {y}^{2}\right)\right) \]

      14. distribute-lft-in N/A

          \[\leadsto \frac{\sin x}{x} \cdot \color{blue}{\left(y \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)} \]

   5. Applied rewrites 87.8%

      \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot \mathsf{fma}\left(y \cdot y, y \cdot 0.16666666666666666, y\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(1 + \frac{-1}{6} \cdot {x}^{2}\right) \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, y \cdot \frac{1}{6}, y\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 63.7%

      \[\leadsto \mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, y \cdot 0.16666666666666666, y\right) \]

   if 1.8e13 < 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. +-commutative N/A

         \[\leadsto y \cdot \color{blue}{\left(\frac{\sin x}{x} + \frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x}\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{y \cdot \frac{\sin x}{x} + y \cdot \left(\frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x}\right)} \]

      3. associate-*r/ N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + y \cdot \color{blue}{\frac{\frac{1}{6} \cdot \left({y}^{2} \cdot \sin x\right)}{x}} \]

      4. *-commutative N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + y \cdot \frac{\frac{1}{6} \cdot \color{blue}{\left(\sin x \cdot {y}^{2}\right)}}{x} \]

      5. associate-*r* N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + y \cdot \frac{\color{blue}{\left(\frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}}}{x} \]

      6. associate-*r/ N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + \color{blue}{\frac{y \cdot \left(\left(\frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}\right)}{x}} \]

      7. associate-*r* N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + \frac{y \cdot \color{blue}{\left(\frac{1}{6} \cdot \left(\sin x \cdot {y}^{2}\right)\right)}}{x} \]

      8. *-commutative N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + \frac{y \cdot \left(\frac{1}{6} \cdot \color{blue}{\left({y}^{2} \cdot \sin x\right)}\right)}{x} \]

      9. associate-*r* N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + \frac{y \cdot \color{blue}{\left(\left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \sin x\right)}}{x} \]

      10. associate-*r* N/A

          \[\leadsto y \cdot \frac{\sin x}{x} + \frac{\color{blue}{\left(y \cdot \left(\frac{1}{6} \cdot {y}^{2}\right)\right) \cdot \sin x}}{x} \]

      11. associate-/l* N/A

          \[\leadsto y \cdot \frac{\sin x}{x} + \color{blue}{\left(y \cdot \left(\frac{1}{6} \cdot {y}^{2}\right)\right) \cdot \frac{\sin x}{x}} \]

      12. distribute-rgt-out N/A

          \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot \left(y + y \cdot \left(\frac{1}{6} \cdot {y}^{2}\right)\right)} \]

      13. *-rgt-identity N/A

          \[\leadsto \frac{\sin x}{x} \cdot \left(\color{blue}{y \cdot 1} + y \cdot \left(\frac{1}{6} \cdot {y}^{2}\right)\right) \]

      14. distribute-lft-in N/A

          \[\leadsto \frac{\sin x}{x} \cdot \color{blue}{\left(y \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)} \]

   5. Applied rewrites 82.1%

      \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot \mathsf{fma}\left(y \cdot y, y \cdot 0.16666666666666666, y\right)} \]

   6. Taylor expanded in y around inf

      \[\leadsto \frac{1}{6} \cdot \color{blue}{\frac{{y}^{3} \cdot \sin x}{x}} \]
   7. Step-by-step derivation

   8. Applied rewrites 59.2%

      \[\leadsto \sin x \cdot \color{blue}{\frac{y \cdot \left(0.16666666666666666 \cdot \left(y \cdot y\right)\right)}{x}} \]

   9. Taylor expanded in x around 0

      \[\leadsto \frac{1}{6} \cdot {y}^{\color{blue}{3}} \]
   10. Step-by-step derivation

   11. Applied rewrites 38.5%

       \[\leadsto y \cdot \left(0.16666666666666666 \cdot \color{blue}{\left(y \cdot y\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 18000000000000:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, y \cdot 0.16666666666666666, y\right) \cdot \mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(y \cdot y\right) \cdot 0.16666666666666666\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 57.1% accurate, 9.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y \cdot y\right) \cdot 0.16666666666666666\\ \mathbf{if}\;x \leq 255000:\\ \;\;\;\;\mathsf{fma}\left(y, t\_0, y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* y y) 0.16666666666666666)))
   (if (<= x 255000.0) (fma y t_0 y) (* y t_0))))

C

double code(double x, double y) {
	double t_0 = (y * y) * 0.16666666666666666;
	double tmp;
	if (x <= 255000.0) {
		tmp = fma(y, t_0, y);
	} else {
		tmp = y * t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(y * y) * 0.16666666666666666)
	tmp = 0.0
	if (x <= 255000.0)
		tmp = fma(y, t_0, y);
	else
		tmp = Float64(y * t_0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]}, If[LessEqual[x, 255000.0], N[(y * t$95$0 + y), $MachinePrecision], N[(y * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 255000

   1. Initial program 83.2%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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. +-commutative N/A

         \[\leadsto y \cdot \color{blue}{\left(\frac{\sin x}{x} + \frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x}\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{y \cdot \frac{\sin x}{x} + y \cdot \left(\frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x}\right)} \]

      3. associate-*r/ N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + y \cdot \color{blue}{\frac{\frac{1}{6} \cdot \left({y}^{2} \cdot \sin x\right)}{x}} \]

      4. *-commutative N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + y \cdot \frac{\frac{1}{6} \cdot \color{blue}{\left(\sin x \cdot {y}^{2}\right)}}{x} \]

      5. associate-*r* N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + y \cdot \frac{\color{blue}{\left(\frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}}}{x} \]

      6. associate-*r/ N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + \color{blue}{\frac{y \cdot \left(\left(\frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}\right)}{x}} \]

      7. associate-*r* N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + \frac{y \cdot \color{blue}{\left(\frac{1}{6} \cdot \left(\sin x \cdot {y}^{2}\right)\right)}}{x} \]

      8. *-commutative N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + \frac{y \cdot \left(\frac{1}{6} \cdot \color{blue}{\left({y}^{2} \cdot \sin x\right)}\right)}{x} \]

      9. associate-*r* N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + \frac{y \cdot \color{blue}{\left(\left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \sin x\right)}}{x} \]

      10. associate-*r* N/A

          \[\leadsto y \cdot \frac{\sin x}{x} + \frac{\color{blue}{\left(y \cdot \left(\frac{1}{6} \cdot {y}^{2}\right)\right) \cdot \sin x}}{x} \]

      11. associate-/l* N/A

          \[\leadsto y \cdot \frac{\sin x}{x} + \color{blue}{\left(y \cdot \left(\frac{1}{6} \cdot {y}^{2}\right)\right) \cdot \frac{\sin x}{x}} \]

      12. distribute-rgt-out N/A

          \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot \left(y + y \cdot \left(\frac{1}{6} \cdot {y}^{2}\right)\right)} \]

      13. *-rgt-identity N/A

          \[\leadsto \frac{\sin x}{x} \cdot \left(\color{blue}{y \cdot 1} + y \cdot \left(\frac{1}{6} \cdot {y}^{2}\right)\right) \]

      14. distribute-lft-in N/A

          \[\leadsto \frac{\sin x}{x} \cdot \color{blue}{\left(y \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)} \]

   5. Applied rewrites 87.7%

      \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot \mathsf{fma}\left(y \cdot y, y \cdot 0.16666666666666666, y\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto y + \color{blue}{\frac{1}{6} \cdot {y}^{3}} \]
   7. Step-by-step derivation

   8. Applied rewrites 64.2%

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{0.16666666666666666 \cdot \left(y \cdot y\right)}, y\right) \]

   if 255000 < 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. +-commutative N/A

         \[\leadsto y \cdot \color{blue}{\left(\frac{\sin x}{x} + \frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x}\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{y \cdot \frac{\sin x}{x} + y \cdot \left(\frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x}\right)} \]

      3. associate-*r/ N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + y \cdot \color{blue}{\frac{\frac{1}{6} \cdot \left({y}^{2} \cdot \sin x\right)}{x}} \]

      4. *-commutative N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + y \cdot \frac{\frac{1}{6} \cdot \color{blue}{\left(\sin x \cdot {y}^{2}\right)}}{x} \]

      5. associate-*r* N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + y \cdot \frac{\color{blue}{\left(\frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}}}{x} \]

      6. associate-*r/ N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + \color{blue}{\frac{y \cdot \left(\left(\frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}\right)}{x}} \]

      7. associate-*r* N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + \frac{y \cdot \color{blue}{\left(\frac{1}{6} \cdot \left(\sin x \cdot {y}^{2}\right)\right)}}{x} \]

      8. *-commutative N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + \frac{y \cdot \left(\frac{1}{6} \cdot \color{blue}{\left({y}^{2} \cdot \sin x\right)}\right)}{x} \]

      9. associate-*r* N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + \frac{y \cdot \color{blue}{\left(\left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \sin x\right)}}{x} \]

      10. associate-*r* N/A

          \[\leadsto y \cdot \frac{\sin x}{x} + \frac{\color{blue}{\left(y \cdot \left(\frac{1}{6} \cdot {y}^{2}\right)\right) \cdot \sin x}}{x} \]

      11. associate-/l* N/A

          \[\leadsto y \cdot \frac{\sin x}{x} + \color{blue}{\left(y \cdot \left(\frac{1}{6} \cdot {y}^{2}\right)\right) \cdot \frac{\sin x}{x}} \]

      12. distribute-rgt-out N/A

          \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot \left(y + y \cdot \left(\frac{1}{6} \cdot {y}^{2}\right)\right)} \]

      13. *-rgt-identity N/A

          \[\leadsto \frac{\sin x}{x} \cdot \left(\color{blue}{y \cdot 1} + y \cdot \left(\frac{1}{6} \cdot {y}^{2}\right)\right) \]

      14. distribute-lft-in N/A

          \[\leadsto \frac{\sin x}{x} \cdot \color{blue}{\left(y \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)} \]

   5. Applied rewrites 82.4%

      \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot \mathsf{fma}\left(y \cdot y, y \cdot 0.16666666666666666, y\right)} \]

   6. Taylor expanded in y around inf

      \[\leadsto \frac{1}{6} \cdot \color{blue}{\frac{{y}^{3} \cdot \sin x}{x}} \]
   7. Step-by-step derivation

   8. Applied rewrites 58.3%

      \[\leadsto \sin x \cdot \color{blue}{\frac{y \cdot \left(0.16666666666666666 \cdot \left(y \cdot y\right)\right)}{x}} \]

   9. Taylor expanded in x around 0

      \[\leadsto \frac{1}{6} \cdot {y}^{\color{blue}{3}} \]
   10. Step-by-step derivation

   11. Applied rewrites 37.8%

       \[\leadsto y \cdot \left(0.16666666666666666 \cdot \color{blue}{\left(y \cdot y\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 255000:\\ \;\;\;\;\mathsf{fma}\left(y, \left(y \cdot y\right) \cdot 0.16666666666666666, y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(y \cdot y\right) \cdot 0.16666666666666666\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 27.9% accurate, 36.2× speedup

Math

\[\begin{array}{l} \\ y \cdot 1 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* y 1.0))

C

double code(double x, double y) {
	return y * 1.0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = y * 1.0d0
end function

Java

public static double code(double x, double y) {
	return y * 1.0;
}

Python

def code(x, y):
	return y * 1.0

Julia

function code(x, y)
	return Float64(y * 1.0)
end

MATLAB

function tmp = code(x, y)
	tmp = y * 1.0;
end

Wolfram

code[x_, y_] := N[(y * 1.0), $MachinePrecision]

Derivation

1. Initial program 86.8%

   \[\frac{\sin x \cdot \sinh y}{x} \]
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. associate-/l* N/A

      \[\leadsto \color{blue}{y \cdot \frac{\sin x}{x}} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{y \cdot \frac{\sin x}{x}} \]

   3. lower-/.f64 N/A

      \[\leadsto y \cdot \color{blue}{\frac{\sin x}{x}} \]

   4. lower-sin.f64 56.0

      \[\leadsto y \cdot \frac{\color{blue}{\sin x}}{x} \]

5. Applied rewrites 56.0%

   \[\leadsto \color{blue}{y \cdot \frac{\sin x}{x}} \]

6. Taylor expanded in x around 0

   \[\leadsto y \cdot 1 \]
7. Step-by-step derivation

8. Applied rewrites 31.0%

   \[\leadsto y \cdot 1 \]
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 2024233 
(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))