Linear.Quaternion:$ccosh from linear-1.19.1.3

Percentage Accurate: 89.2% → 99.8%

Time: 13.1s

Alternatives: 19

Speedup: 1.4×

• 49.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 89.2% 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: 99.8% accurate, 1.0× speedup

Math

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

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(Float64(sinh(y) / x) * sin(x))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.0%

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

   1. lift-sin.f64 N/A

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

   2. lift-sinh.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-*l/ 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 99.8

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

4. Applied rewrites 99.8%

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

Alternative 2: 87.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y \cdot \sin x}{x}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\frac{\sinh y \cdot \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot -0.16666666666666666, x\right)}{x}\\ \mathbf{elif}\;t\_0 \leq 0.02:\\ \;\;\;\;\sin x \cdot \frac{\mathsf{fma}\left(y \cdot \left(y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(0.0001984126984126984, y \cdot y, 0.008333333333333333\right), 0.16666666666666666\right)\right), y, y\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sinh y \cdot \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.008333333333333333, -0.16666666666666666\right), x \cdot \left(x \cdot x\right), x\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sinh y) (sin x)) x)))
   (if (<= t_0 (- INFINITY))
     (/ (* (sinh y) (fma x (* (* x x) -0.16666666666666666) x)) x)
     (if (<= t_0 0.02)
       (*
        (sin x)
        (/
         (fma
          (*
           y
           (*
            y
            (fma
             y
             (* y (fma 0.0001984126984126984 (* y y) 0.008333333333333333))
             0.16666666666666666)))
          y
          y)
         x))
       (/
        (*
         (sinh y)
         (fma
          (fma (* x x) 0.008333333333333333 -0.16666666666666666)
          (* x (* x x))
          x))
        x)))))

C

double code(double x, double y) {
	double t_0 = (sinh(y) * sin(x)) / x;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (sinh(y) * fma(x, ((x * x) * -0.16666666666666666), x)) / x;
	} else if (t_0 <= 0.02) {
		tmp = sin(x) * (fma((y * (y * fma(y, (y * fma(0.0001984126984126984, (y * y), 0.008333333333333333)), 0.16666666666666666))), y, y) / x);
	} else {
		tmp = (sinh(y) * fma(fma((x * x), 0.008333333333333333, -0.16666666666666666), (x * (x * x)), x)) / 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(sinh(y) * fma(x, Float64(Float64(x * x) * -0.16666666666666666), x)) / x);
	elseif (t_0 <= 0.02)
		tmp = Float64(sin(x) * Float64(fma(Float64(y * Float64(y * fma(y, Float64(y * fma(0.0001984126984126984, Float64(y * y), 0.008333333333333333)), 0.16666666666666666))), y, y) / x));
	else
		tmp = Float64(Float64(sinh(y) * fma(fma(Float64(x * x), 0.008333333333333333, -0.16666666666666666), 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]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[Sinh[y], $MachinePrecision] * N[(x * N[(N[(x * x), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[t$95$0, 0.02], N[(N[Sin[x], $MachinePrecision] * N[(N[(N[(y * N[(y * N[(y * N[(y * N[(0.0001984126984126984 * N[(y * y), $MachinePrecision] + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y + y), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sinh[y], $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $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) < -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 69.6

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

   5. Applied rewrites 69.6%

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

   if -inf.0 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 0.0200000000000000004

   1. Initial program 80.8%

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

      1. lift-sin.f64 N/A

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

      2. lift-sinh.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l/ 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 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. unpow2 N/A

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

      7. unpow3 N/A

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

      8. lower-fma.f64 N/A

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

   7. Applied rewrites 98.2%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

   9. Applied rewrites 98.2%

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

       1. lift-*.f64 N/A

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

       2. lift-fma.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-fma.f64 N/A

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

       5. associate-*r* N/A

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

       6. lower-*.f64 N/A

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

   11. Applied rewrites 98.2%

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

   if 0.0200000000000000004 < (/.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 73.1

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

   5. Applied rewrites 73.1%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -\infty:\\ \;\;\;\;\frac{\sinh y \cdot \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot -0.16666666666666666, x\right)}{x}\\ \mathbf{elif}\;\frac{\sinh y \cdot \sin x}{x} \leq 0.02:\\ \;\;\;\;\sin x \cdot \frac{\mathsf{fma}\left(y \cdot \left(y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(0.0001984126984126984, y \cdot y, 0.008333333333333333\right), 0.16666666666666666\right)\right), y, y\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sinh y \cdot \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.008333333333333333, -0.16666666666666666\right), x \cdot \left(x \cdot x\right), x\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y \cdot \sin x}{x}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\frac{\sinh y \cdot \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot -0.16666666666666666, x\right)}{x}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(y, y \cdot \left(y \cdot \mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right)\right), y\right) \cdot \frac{\sin x}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sinh y \cdot \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.008333333333333333, -0.16666666666666666\right), x \cdot \left(x \cdot x\right), x\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sinh y) (sin x)) x)))
   (if (<= t_0 (- INFINITY))
     (/ (* (sinh y) (fma x (* (* x x) -0.16666666666666666) x)) x)
     (if (<= t_0 5e-6)
       (*
        (fma
         y
         (* y (* y (fma 0.008333333333333333 (* y y) 0.16666666666666666)))
         y)
        (/ (sin x) x))
       (/
        (*
         (sinh y)
         (fma
          (fma (* x x) 0.008333333333333333 -0.16666666666666666)
          (* x (* x x))
          x))
        x)))))

C

double code(double x, double y) {
	double t_0 = (sinh(y) * sin(x)) / x;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (sinh(y) * fma(x, ((x * x) * -0.16666666666666666), x)) / x;
	} else if (t_0 <= 5e-6) {
		tmp = fma(y, (y * (y * fma(0.008333333333333333, (y * y), 0.16666666666666666))), y) * (sin(x) / x);
	} else {
		tmp = (sinh(y) * fma(fma((x * x), 0.008333333333333333, -0.16666666666666666), (x * (x * x)), x)) / 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(sinh(y) * fma(x, Float64(Float64(x * x) * -0.16666666666666666), x)) / x);
	elseif (t_0 <= 5e-6)
		tmp = Float64(fma(y, Float64(y * Float64(y * fma(0.008333333333333333, Float64(y * y), 0.16666666666666666))), y) * Float64(sin(x) / x));
	else
		tmp = Float64(Float64(sinh(y) * fma(fma(Float64(x * x), 0.008333333333333333, -0.16666666666666666), 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]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[Sinh[y], $MachinePrecision] * N[(x * N[(N[(x * x), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[t$95$0, 5e-6], N[(N[(y * N[(y * N[(y * N[(0.008333333333333333 * N[(y * y), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sinh[y], $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $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) < -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 69.6

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

   5. Applied rewrites 69.6%

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

   if -inf.0 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 5.00000000000000041e-6

   1. Initial program 80.4%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 78.8

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

   5. Applied rewrites 78.8%

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

      1. lift-sin.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lift-/.f64 N/A

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

      10. lower-*.f64 98.2

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

   7. Applied rewrites 98.2%

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

   if 5.00000000000000041e-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 74.3

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

   5. Applied rewrites 74.3%

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

4. Final simplification 85.4%

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

Alternative 4: 86.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y \cdot \sin x}{x}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\frac{\sinh y \cdot \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot -0.16666666666666666, x\right)}{x}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{\sin x}{x} \cdot \mathsf{fma}\left(y \cdot y, y \cdot 0.16666666666666666, y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)}{x} \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right), x \cdot \left(x \cdot x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sinh y) (sin x)) x)))
   (if (<= t_0 (- INFINITY))
     (/ (* (sinh y) (fma x (* (* x x) -0.16666666666666666) x)) x)
     (if (<= t_0 5e-6)
       (* (/ (sin x) x) (fma (* y y) (* y 0.16666666666666666) y))
       (*
        (/
         (fma
          (fma
           y
           (* y (fma y (* y 0.0001984126984126984) 0.008333333333333333))
           0.16666666666666666)
          (* y (* y y))
          y)
         x)
        (fma
         (fma 0.008333333333333333 (* x x) -0.16666666666666666)
         (* x (* x x))
         x))))))

C

double code(double x, double y) {
	double t_0 = (sinh(y) * sin(x)) / x;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (sinh(y) * fma(x, ((x * x) * -0.16666666666666666), x)) / x;
	} else if (t_0 <= 5e-6) {
		tmp = (sin(x) / x) * fma((y * y), (y * 0.16666666666666666), y);
	} else {
		tmp = (fma(fma(y, (y * fma(y, (y * 0.0001984126984126984), 0.008333333333333333)), 0.16666666666666666), (y * (y * y)), y) / x) * fma(fma(0.008333333333333333, (x * x), -0.16666666666666666), (x * (x * x)), 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(sinh(y) * fma(x, Float64(Float64(x * x) * -0.16666666666666666), x)) / x);
	elseif (t_0 <= 5e-6)
		tmp = Float64(Float64(sin(x) / x) * fma(Float64(y * y), Float64(y * 0.16666666666666666), y));
	else
		tmp = Float64(Float64(fma(fma(y, Float64(y * fma(y, Float64(y * 0.0001984126984126984), 0.008333333333333333)), 0.16666666666666666), Float64(y * Float64(y * y)), y) / x) * fma(fma(0.008333333333333333, Float64(x * x), -0.16666666666666666), Float64(x * Float64(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]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[Sinh[y], $MachinePrecision] * N[(x * N[(N[(x * x), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[t$95$0, 5e-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[(N[(y * N[(y * N[(y * N[(y * 0.0001984126984126984), $MachinePrecision] + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] / x), $MachinePrecision] * N[(N[(0.008333333333333333 * N[(x * x), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(x * N[(x * x), $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 69.6

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

   5. Applied rewrites 69.6%

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

   if -inf.0 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 5.00000000000000041e-6

   1. Initial program 80.4%

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

   3. Taylor expanded in y around 0

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

      1. +-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 98.0%

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

   if 5.00000000000000041e-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-sin.f64 N/A

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

      2. lift-sinh.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l/ 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 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. unpow2 N/A

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

      7. unpow3 N/A

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

      8. lower-fma.f64 N/A

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

   7. Applied rewrites 89.1%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. unpow2 N/A

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

      7. unpow3 N/A

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

      8. lower-fma.f64 N/A

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. cube-mult N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 70.3

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

   10. Applied rewrites 70.3%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -\infty:\\ \;\;\;\;\frac{\sinh y \cdot \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot -0.16666666666666666, x\right)}{x}\\ \mathbf{elif}\;\frac{\sinh y \cdot \sin x}{x} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{\sin x}{x} \cdot \mathsf{fma}\left(y \cdot y, y \cdot 0.16666666666666666, y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)}{x} \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right), x \cdot \left(x \cdot x\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y \cdot \sin x}{x}\\ t_1 := \frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)}{x}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;t\_1 \cdot \mathsf{fma}\left(x, x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(-0.0001984126984126984, x \cdot x, 0.008333333333333333\right), -0.16666666666666666\right)\right), x\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{\sin x}{x} \cdot \mathsf{fma}\left(y \cdot y, y \cdot 0.16666666666666666, y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right), x \cdot \left(x \cdot x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sinh y) (sin x)) x))
        (t_1
         (/
          (fma
           (fma
            y
            (* y (fma y (* y 0.0001984126984126984) 0.008333333333333333))
            0.16666666666666666)
           (* y (* y y))
           y)
          x)))
   (if (<= t_0 (- INFINITY))
     (*
      t_1
      (fma
       x
       (*
        x
        (*
         x
         (fma
          (* x x)
          (fma -0.0001984126984126984 (* x x) 0.008333333333333333)
          -0.16666666666666666)))
       x))
     (if (<= t_0 5e-6)
       (* (/ (sin x) x) (fma (* y y) (* y 0.16666666666666666) y))
       (*
        t_1
        (fma
         (fma 0.008333333333333333 (* x x) -0.16666666666666666)
         (* x (* x x))
         x))))))

C

double code(double x, double y) {
	double t_0 = (sinh(y) * sin(x)) / x;
	double t_1 = fma(fma(y, (y * fma(y, (y * 0.0001984126984126984), 0.008333333333333333)), 0.16666666666666666), (y * (y * y)), y) / x;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = t_1 * fma(x, (x * (x * fma((x * x), fma(-0.0001984126984126984, (x * x), 0.008333333333333333), -0.16666666666666666))), x);
	} else if (t_0 <= 5e-6) {
		tmp = (sin(x) / x) * fma((y * y), (y * 0.16666666666666666), y);
	} else {
		tmp = t_1 * fma(fma(0.008333333333333333, (x * x), -0.16666666666666666), (x * (x * x)), x);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(sinh(y) * sin(x)) / x)
	t_1 = Float64(fma(fma(y, Float64(y * fma(y, Float64(y * 0.0001984126984126984), 0.008333333333333333)), 0.16666666666666666), Float64(y * Float64(y * y)), y) / x)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(t_1 * fma(x, Float64(x * Float64(x * fma(Float64(x * x), fma(-0.0001984126984126984, Float64(x * x), 0.008333333333333333), -0.16666666666666666))), x));
	elseif (t_0 <= 5e-6)
		tmp = Float64(Float64(sin(x) / x) * fma(Float64(y * y), Float64(y * 0.16666666666666666), y));
	else
		tmp = Float64(t_1 * fma(fma(0.008333333333333333, Float64(x * x), -0.16666666666666666), Float64(x * Float64(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[(N[(N[(y * N[(y * N[(y * N[(y * 0.0001984126984126984), $MachinePrecision] + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(t$95$1 * N[(x * N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * N[(-0.0001984126984126984 * N[(x * x), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e-6], N[(N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(y * 0.16666666666666666), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(N[(0.008333333333333333 * N[(x * x), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(x * N[(x * x), $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. Step-by-step derivation

      1. lift-sin.f64 N/A

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

      2. lift-sinh.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l/ 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 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. unpow2 N/A

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

      7. unpow3 N/A

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

      8. lower-fma.f64 N/A

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

   7. Applied rewrites 93.2%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

   10. Applied rewrites 66.3%

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

   if -inf.0 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 5.00000000000000041e-6

   1. Initial program 80.4%

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

   3. Taylor expanded in y around 0

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

      1. +-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 98.0%

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

   if 5.00000000000000041e-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-sin.f64 N/A

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

      2. lift-sinh.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l/ 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 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. unpow2 N/A

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

      7. unpow3 N/A

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

      8. lower-fma.f64 N/A

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

   7. Applied rewrites 89.1%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. unpow2 N/A

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

      7. unpow3 N/A

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

      8. lower-fma.f64 N/A

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. cube-mult N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 70.3

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

   10. Applied rewrites 70.3%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -\infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)}{x} \cdot \mathsf{fma}\left(x, x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(-0.0001984126984126984, x \cdot x, 0.008333333333333333\right), -0.16666666666666666\right)\right), x\right)\\ \mathbf{elif}\;\frac{\sinh y \cdot \sin x}{x} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{\sin x}{x} \cdot \mathsf{fma}\left(y \cdot y, y \cdot 0.16666666666666666, y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)}{x} \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right), x \cdot \left(x \cdot x\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y \cdot \sin x}{x}\\ t_1 := \frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)}{x}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;t\_1 \cdot \mathsf{fma}\left(x, x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(-0.0001984126984126984, x \cdot x, 0.008333333333333333\right), -0.16666666666666666\right)\right), x\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-31}:\\ \;\;\;\;y \cdot \frac{\sin x}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right), x \cdot \left(x \cdot x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sinh y) (sin x)) x))
        (t_1
         (/
          (fma
           (fma
            y
            (* y (fma y (* y 0.0001984126984126984) 0.008333333333333333))
            0.16666666666666666)
           (* y (* y y))
           y)
          x)))
   (if (<= t_0 (- INFINITY))
     (*
      t_1
      (fma
       x
       (*
        x
        (*
         x
         (fma
          (* x x)
          (fma -0.0001984126984126984 (* x x) 0.008333333333333333)
          -0.16666666666666666)))
       x))
     (if (<= t_0 5e-31)
       (* y (/ (sin x) x))
       (*
        t_1
        (fma
         (fma 0.008333333333333333 (* x x) -0.16666666666666666)
         (* x (* x x))
         x))))))

C

double code(double x, double y) {
	double t_0 = (sinh(y) * sin(x)) / x;
	double t_1 = fma(fma(y, (y * fma(y, (y * 0.0001984126984126984), 0.008333333333333333)), 0.16666666666666666), (y * (y * y)), y) / x;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = t_1 * fma(x, (x * (x * fma((x * x), fma(-0.0001984126984126984, (x * x), 0.008333333333333333), -0.16666666666666666))), x);
	} else if (t_0 <= 5e-31) {
		tmp = y * (sin(x) / x);
	} else {
		tmp = t_1 * fma(fma(0.008333333333333333, (x * x), -0.16666666666666666), (x * (x * x)), x);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(sinh(y) * sin(x)) / x)
	t_1 = Float64(fma(fma(y, Float64(y * fma(y, Float64(y * 0.0001984126984126984), 0.008333333333333333)), 0.16666666666666666), Float64(y * Float64(y * y)), y) / x)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(t_1 * fma(x, Float64(x * Float64(x * fma(Float64(x * x), fma(-0.0001984126984126984, Float64(x * x), 0.008333333333333333), -0.16666666666666666))), x));
	elseif (t_0 <= 5e-31)
		tmp = Float64(y * Float64(sin(x) / x));
	else
		tmp = Float64(t_1 * fma(fma(0.008333333333333333, Float64(x * x), -0.16666666666666666), Float64(x * Float64(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[(N[(N[(y * N[(y * N[(y * N[(y * 0.0001984126984126984), $MachinePrecision] + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(t$95$1 * N[(x * N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * N[(-0.0001984126984126984 * N[(x * x), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e-31], N[(y * N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(N[(0.008333333333333333 * N[(x * x), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(x * N[(x * x), $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. Step-by-step derivation

      1. lift-sin.f64 N/A

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

      2. lift-sinh.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l/ 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 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. unpow2 N/A

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

      7. unpow3 N/A

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

      8. lower-fma.f64 N/A

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

   7. Applied rewrites 93.2%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

   10. Applied rewrites 66.3%

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

   if -inf.0 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 5e-31

   1. Initial program 79.8%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 97.7

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

   5. Applied rewrites 97.7%

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

   if 5e-31 < (/.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-sin.f64 N/A

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

      2. lift-sinh.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l/ 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 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. unpow2 N/A

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

      7. unpow3 N/A

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

      8. lower-fma.f64 N/A

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

   7. Applied rewrites 89.7%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. unpow2 N/A

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

      7. unpow3 N/A

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

      8. lower-fma.f64 N/A

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. cube-mult N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 71.9

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

   10. Applied rewrites 71.9%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -\infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)}{x} \cdot \mathsf{fma}\left(x, x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(-0.0001984126984126984, x \cdot x, 0.008333333333333333\right), -0.16666666666666666\right)\right), x\right)\\ \mathbf{elif}\;\frac{\sinh y \cdot \sin x}{x} \leq 5 \cdot 10^{-31}:\\ \;\;\;\;y \cdot \frac{\sin x}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)}{x} \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right), x \cdot \left(x \cdot x\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 72.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot y\right)\\ t_1 := \frac{\sinh y \cdot \sin x}{x}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-124}:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(t\_0, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-31}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(0.16666666666666666, x \cdot x, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), t\_0, y\right)}{x} \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right), x \cdot \left(x \cdot x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y y))) (t_1 (/ (* (sinh y) (sin x)) x)))
   (if (<= t_1 -5e-124)
     (*
      (fma x (* x -0.16666666666666666) 1.0)
      (fma
       t_0
       (fma
        (* y y)
        (fma (* y y) 0.0001984126984126984 0.008333333333333333)
        0.16666666666666666)
       y))
     (if (<= t_1 5e-31)
       (/ y (fma 0.16666666666666666 (* x x) 1.0))
       (*
        (/
         (fma
          (fma
           y
           (* y (fma y (* y 0.0001984126984126984) 0.008333333333333333))
           0.16666666666666666)
          t_0
          y)
         x)
        (fma
         (fma 0.008333333333333333 (* x x) -0.16666666666666666)
         (* x (* x x))
         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 <= -5e-124) {
		tmp = fma(x, (x * -0.16666666666666666), 1.0) * fma(t_0, fma((y * y), fma((y * y), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666), y);
	} else if (t_1 <= 5e-31) {
		tmp = y / fma(0.16666666666666666, (x * x), 1.0);
	} else {
		tmp = (fma(fma(y, (y * fma(y, (y * 0.0001984126984126984), 0.008333333333333333)), 0.16666666666666666), t_0, y) / x) * fma(fma(0.008333333333333333, (x * x), -0.16666666666666666), (x * (x * x)), 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 <= -5e-124)
		tmp = Float64(fma(x, Float64(x * -0.16666666666666666), 1.0) * fma(t_0, fma(Float64(y * y), fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666), y));
	elseif (t_1 <= 5e-31)
		tmp = Float64(y / fma(0.16666666666666666, Float64(x * x), 1.0));
	else
		tmp = Float64(Float64(fma(fma(y, Float64(y * fma(y, Float64(y * 0.0001984126984126984), 0.008333333333333333)), 0.16666666666666666), t_0, y) / x) * fma(fma(0.008333333333333333, Float64(x * x), -0.16666666666666666), Float64(x * Float64(x * x)), 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, -5e-124], N[(N[(x * N[(x * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * N[(t$95$0 * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e-31], N[(y / N[(0.16666666666666666 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(y * N[(y * N[(y * N[(y * 0.0001984126984126984), $MachinePrecision] + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * t$95$0 + y), $MachinePrecision] / x), $MachinePrecision] * N[(N[(0.008333333333333333 * N[(x * x), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -5.0000000000000003e-124

   1. Initial program 99.0%

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

      1. lift-sin.f64 N/A

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

      2. lift-sinh.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l/ 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. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. unpow2 N/A

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

      7. unpow3 N/A

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

      8. lower-fma.f64 N/A

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

   7. Applied rewrites 92.8%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

   9. Applied rewrites 92.8%

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

   10. Taylor expanded in x around 0

       \[\leadsto \color{blue}{y + \left(\frac{-1}{6} \cdot \left({x}^{2} \cdot \left(y + {y}^{3} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right) + {y}^{3} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)} \]
   11. Step-by-step derivation

       1. +-commutative N/A

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

       2. associate-+r+ N/A

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

       3. associate-*r* N/A

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

       4. distribute-rgt1-in N/A

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

       5. +-commutative N/A

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

       6. lower-*.f64 N/A

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

   12. Applied rewrites 62.8%

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

   if -5.0000000000000003e-124 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 5e-31

   1. Initial program 77.7%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 99.1

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

   5. Applied rewrites 99.1%

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

      1. lift-sin.f64 N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 99.0

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

   7. Applied rewrites 99.0%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 71.6

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

   10. Applied rewrites 71.6%

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

   if 5e-31 < (/.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-sin.f64 N/A

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

      2. lift-sinh.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l/ 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 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. unpow2 N/A

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

      7. unpow3 N/A

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

      8. lower-fma.f64 N/A

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

   7. Applied rewrites 89.7%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. unpow2 N/A

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

      7. unpow3 N/A

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

      8. lower-fma.f64 N/A

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. cube-mult N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 71.9

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

   10. Applied rewrites 71.9%

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

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -5 \cdot 10^{-124}:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(y \cdot \left(y \cdot y\right), \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y\right)\\ \mathbf{elif}\;\frac{\sinh y \cdot \sin x}{x} \leq 5 \cdot 10^{-31}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(0.16666666666666666, x \cdot x, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)}{x} \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right), x \cdot \left(x \cdot x\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 72.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y \cdot \sin x}{x}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-124}:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(y \cdot \left(y \cdot y\right), \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-31}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(0.16666666666666666, x \cdot x, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right), x \cdot \left(x \cdot x\right), x\right) \cdot \frac{\mathsf{fma}\left(y \cdot y, y \cdot \mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sinh y) (sin x)) x)))
   (if (<= t_0 -5e-124)
     (*
      (fma x (* x -0.16666666666666666) 1.0)
      (fma
       (* y (* y y))
       (fma
        (* y y)
        (fma (* y y) 0.0001984126984126984 0.008333333333333333)
        0.16666666666666666)
       y))
     (if (<= t_0 5e-31)
       (/ y (fma 0.16666666666666666 (* x x) 1.0))
       (*
        (fma
         (fma 0.008333333333333333 (* x x) -0.16666666666666666)
         (* x (* x x))
         x)
        (/
         (fma
          (* y y)
          (* y (fma 0.008333333333333333 (* y y) 0.16666666666666666))
          y)
         x))))))

C

double code(double x, double y) {
	double t_0 = (sinh(y) * sin(x)) / x;
	double tmp;
	if (t_0 <= -5e-124) {
		tmp = fma(x, (x * -0.16666666666666666), 1.0) * fma((y * (y * y)), fma((y * y), fma((y * y), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666), y);
	} else if (t_0 <= 5e-31) {
		tmp = y / fma(0.16666666666666666, (x * x), 1.0);
	} else {
		tmp = fma(fma(0.008333333333333333, (x * x), -0.16666666666666666), (x * (x * x)), x) * (fma((y * y), (y * fma(0.008333333333333333, (y * y), 0.16666666666666666)), y) / x);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(sinh(y) * sin(x)) / x)
	tmp = 0.0
	if (t_0 <= -5e-124)
		tmp = Float64(fma(x, Float64(x * -0.16666666666666666), 1.0) * fma(Float64(y * Float64(y * y)), fma(Float64(y * y), fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666), y));
	elseif (t_0 <= 5e-31)
		tmp = Float64(y / fma(0.16666666666666666, Float64(x * x), 1.0));
	else
		tmp = Float64(fma(fma(0.008333333333333333, Float64(x * x), -0.16666666666666666), Float64(x * Float64(x * x)), x) * Float64(fma(Float64(y * y), Float64(y * fma(0.008333333333333333, Float64(y * y), 0.16666666666666666)), y) / 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, -5e-124], N[(N[(x * N[(x * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e-31], N[(y / N[(0.16666666666666666 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.008333333333333333 * N[(x * x), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] * N[(N[(N[(y * y), $MachinePrecision] * N[(y * N[(0.008333333333333333 * N[(y * y), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -5.0000000000000003e-124

   1. Initial program 99.0%

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

      1. lift-sin.f64 N/A

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

      2. lift-sinh.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l/ 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. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. unpow2 N/A

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

      7. unpow3 N/A

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

      8. lower-fma.f64 N/A

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

   7. Applied rewrites 92.8%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

   9. Applied rewrites 92.8%

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

   10. Taylor expanded in x around 0

       \[\leadsto \color{blue}{y + \left(\frac{-1}{6} \cdot \left({x}^{2} \cdot \left(y + {y}^{3} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right) + {y}^{3} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)} \]
   11. Step-by-step derivation

       1. +-commutative N/A

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

       2. associate-+r+ N/A

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

       3. associate-*r* N/A

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

       4. distribute-rgt1-in N/A

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

       5. +-commutative N/A

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

       6. lower-*.f64 N/A

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

   12. Applied rewrites 62.8%

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

   if -5.0000000000000003e-124 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 5e-31

   1. Initial program 77.7%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 99.1

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

   5. Applied rewrites 99.1%

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

      1. lift-sin.f64 N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 99.0

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

   7. Applied rewrites 99.0%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 71.6

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

   10. Applied rewrites 71.6%

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

   if 5e-31 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 80.5

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

   5. Applied rewrites 80.5%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-*l* N/A

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

      4. *-lft-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. sub-neg N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. metadata-eval N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-*.f64 65.3

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

   8. Applied rewrites 65.3%

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

   10. Applied rewrites 71.7%

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

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -5 \cdot 10^{-124}:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(y \cdot \left(y \cdot y\right), \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y\right)\\ \mathbf{elif}\;\frac{\sinh y \cdot \sin x}{x} \leq 5 \cdot 10^{-31}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(0.16666666666666666, x \cdot x, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right), x \cdot \left(x \cdot x\right), x\right) \cdot \frac{\mathsf{fma}\left(y \cdot y, y \cdot \mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 71.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot y\right)\\ t_1 := \frac{\sinh y \cdot \sin x}{x}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-124}:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(t\_0, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-48}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(0.16666666666666666, x \cdot x, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y y))) (t_1 (/ (* (sinh y) (sin x)) x)))
   (if (<= t_1 -5e-124)
     (*
      (fma x (* x -0.16666666666666666) 1.0)
      (fma
       t_0
       (fma
        (* y y)
        (fma (* y y) 0.0001984126984126984 0.008333333333333333)
        0.16666666666666666)
       y))
     (if (<= t_1 5e-48)
       (/ y (fma 0.16666666666666666 (* x x) 1.0))
       (fma
        t_0
        (fma
         y
         (* y (fma y (* y 0.0001984126984126984) 0.008333333333333333))
         0.16666666666666666)
        y)))))

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 <= -5e-124) {
		tmp = fma(x, (x * -0.16666666666666666), 1.0) * fma(t_0, fma((y * y), fma((y * y), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666), y);
	} else if (t_1 <= 5e-48) {
		tmp = y / fma(0.16666666666666666, (x * x), 1.0);
	} else {
		tmp = fma(t_0, fma(y, (y * fma(y, (y * 0.0001984126984126984), 0.008333333333333333)), 0.16666666666666666), y);
	}
	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 <= -5e-124)
		tmp = Float64(fma(x, Float64(x * -0.16666666666666666), 1.0) * fma(t_0, fma(Float64(y * y), fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666), y));
	elseif (t_1 <= 5e-48)
		tmp = Float64(y / fma(0.16666666666666666, Float64(x * x), 1.0));
	else
		tmp = fma(t_0, fma(y, Float64(y * fma(y, Float64(y * 0.0001984126984126984), 0.008333333333333333)), 0.16666666666666666), y);
	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, -5e-124], N[(N[(x * N[(x * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * N[(t$95$0 * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e-48], N[(y / N[(0.16666666666666666 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(y * N[(y * N[(y * N[(y * 0.0001984126984126984), $MachinePrecision] + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + y), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -5.0000000000000003e-124

   1. Initial program 99.0%

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

      1. lift-sin.f64 N/A

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

      2. lift-sinh.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l/ 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. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. unpow2 N/A

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

      7. unpow3 N/A

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

      8. lower-fma.f64 N/A

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

   7. Applied rewrites 92.8%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

   9. Applied rewrites 92.8%

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

   10. Taylor expanded in x around 0

       \[\leadsto \color{blue}{y + \left(\frac{-1}{6} \cdot \left({x}^{2} \cdot \left(y + {y}^{3} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right) + {y}^{3} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)} \]
   11. Step-by-step derivation

       1. +-commutative N/A

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

       2. associate-+r+ N/A

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

       3. associate-*r* N/A

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

       4. distribute-rgt1-in N/A

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

       5. +-commutative N/A

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

       6. lower-*.f64 N/A

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

   12. Applied rewrites 62.8%

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

   if -5.0000000000000003e-124 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 4.9999999999999999e-48

   1. Initial program 77.1%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 99.1

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

   5. Applied rewrites 99.1%

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

      1. lift-sin.f64 N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 99.0

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

   7. Applied rewrites 99.0%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 71.7

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

   10. Applied rewrites 71.7%

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

   if 4.9999999999999999e-48 < (/.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-sin.f64 N/A

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

      2. lift-sinh.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l/ 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. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. unpow2 N/A

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

      7. unpow3 N/A

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

      8. lower-fma.f64 N/A

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

   7. Applied rewrites 90.0%

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

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{y + {y}^{3} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)} \]
   9. Step-by-step derivation

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. cube-mult N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. unpow2 N/A

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

      16. associate-*l* N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 66.8

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

   10. Applied rewrites 66.8%

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

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -5 \cdot 10^{-124}:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(y \cdot \left(y \cdot y\right), \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y\right)\\ \mathbf{elif}\;\frac{\sinh y \cdot \sin x}{x} \leq 5 \cdot 10^{-48}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(0.16666666666666666, x \cdot x, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot \left(y \cdot y\right), \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 68.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y \cdot \sin x}{x}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-124}:\\ \;\;\;\;\mathsf{fma}\left(y, y \cdot 0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(y, \left(x \cdot x\right) \cdot -0.16666666666666666, y\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-48}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(0.16666666666666666, x \cdot x, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot \left(y \cdot y\right), \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sinh y) (sin x)) x)))
   (if (<= t_0 -5e-124)
     (*
      (fma y (* y 0.16666666666666666) 1.0)
      (fma y (* (* x x) -0.16666666666666666) y))
     (if (<= t_0 5e-48)
       (/ y (fma 0.16666666666666666 (* x x) 1.0))
       (fma
        (* y (* y y))
        (fma
         y
         (* y (fma y (* y 0.0001984126984126984) 0.008333333333333333))
         0.16666666666666666)
        y)))))

C

double code(double x, double y) {
	double t_0 = (sinh(y) * sin(x)) / x;
	double tmp;
	if (t_0 <= -5e-124) {
		tmp = fma(y, (y * 0.16666666666666666), 1.0) * fma(y, ((x * x) * -0.16666666666666666), y);
	} else if (t_0 <= 5e-48) {
		tmp = y / fma(0.16666666666666666, (x * x), 1.0);
	} else {
		tmp = fma((y * (y * y)), fma(y, (y * fma(y, (y * 0.0001984126984126984), 0.008333333333333333)), 0.16666666666666666), y);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(sinh(y) * sin(x)) / x)
	tmp = 0.0
	if (t_0 <= -5e-124)
		tmp = Float64(fma(y, Float64(y * 0.16666666666666666), 1.0) * fma(y, Float64(Float64(x * x) * -0.16666666666666666), y));
	elseif (t_0 <= 5e-48)
		tmp = Float64(y / fma(0.16666666666666666, Float64(x * x), 1.0));
	else
		tmp = fma(Float64(y * Float64(y * y)), fma(y, Float64(y * fma(y, Float64(y * 0.0001984126984126984), 0.008333333333333333)), 0.16666666666666666), y);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sinh[y], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-124], N[(N[(y * N[(y * 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * N[(y * N[(N[(x * x), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e-48], N[(y / N[(0.16666666666666666 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(y * N[(y * N[(y * N[(y * 0.0001984126984126984), $MachinePrecision] + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -5.0000000000000003e-124

   1. Initial program 99.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-lft-identity N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt-in N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 70.0

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

   5. Applied rewrites 70.0%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-out N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

   8. Applied rewrites 50.7%

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

   if -5.0000000000000003e-124 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 4.9999999999999999e-48

   1. Initial program 77.1%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 99.1

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

   5. Applied rewrites 99.1%

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

      1. lift-sin.f64 N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 99.0

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

   7. Applied rewrites 99.0%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 71.7

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

   10. Applied rewrites 71.7%

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

   if 4.9999999999999999e-48 < (/.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-sin.f64 N/A

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

      2. lift-sinh.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l/ 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. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. unpow2 N/A

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

      7. unpow3 N/A

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

      8. lower-fma.f64 N/A

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

   7. Applied rewrites 90.0%

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

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{y + {y}^{3} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)} \]
   9. Step-by-step derivation

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. cube-mult N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. unpow2 N/A

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

      16. associate-*l* N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 66.8

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

   10. Applied rewrites 66.8%

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

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -5 \cdot 10^{-124}:\\ \;\;\;\;\mathsf{fma}\left(y, y \cdot 0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(y, \left(x \cdot x\right) \cdot -0.16666666666666666, y\right)\\ \mathbf{elif}\;\frac{\sinh y \cdot \sin x}{x} \leq 5 \cdot 10^{-48}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(0.16666666666666666, x \cdot x, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot \left(y \cdot y\right), \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 67.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y \cdot \sin x}{x}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-124}:\\ \;\;\;\;\mathsf{fma}\left(y, y \cdot 0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(y, \left(x \cdot x\right) \cdot -0.16666666666666666, y\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-48}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(0.16666666666666666, x \cdot x, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sinh y) (sin x)) x)))
   (if (<= t_0 -5e-124)
     (*
      (fma y (* y 0.16666666666666666) 1.0)
      (fma y (* (* x x) -0.16666666666666666) y))
     (if (<= t_0 5e-48)
       (/ y (fma 0.16666666666666666 (* x x) 1.0))
       (fma
        (fma (* y y) 0.008333333333333333 0.16666666666666666)
        (* y (* y y))
        y)))))

C

double code(double x, double y) {
	double t_0 = (sinh(y) * sin(x)) / x;
	double tmp;
	if (t_0 <= -5e-124) {
		tmp = fma(y, (y * 0.16666666666666666), 1.0) * fma(y, ((x * x) * -0.16666666666666666), y);
	} else if (t_0 <= 5e-48) {
		tmp = y / fma(0.16666666666666666, (x * x), 1.0);
	} else {
		tmp = fma(fma((y * y), 0.008333333333333333, 0.16666666666666666), (y * (y * y)), y);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(sinh(y) * sin(x)) / x)
	tmp = 0.0
	if (t_0 <= -5e-124)
		tmp = Float64(fma(y, Float64(y * 0.16666666666666666), 1.0) * fma(y, Float64(Float64(x * x) * -0.16666666666666666), y));
	elseif (t_0 <= 5e-48)
		tmp = Float64(y / fma(0.16666666666666666, Float64(x * x), 1.0));
	else
		tmp = fma(fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666), Float64(y * Float64(y * y)), y);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sinh[y], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-124], N[(N[(y * N[(y * 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * N[(y * N[(N[(x * x), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e-48], N[(y / N[(0.16666666666666666 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision] * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -5.0000000000000003e-124

   1. Initial program 99.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-lft-identity N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt-in N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 70.0

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

   5. Applied rewrites 70.0%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-out N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

   8. Applied rewrites 50.7%

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

   if -5.0000000000000003e-124 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 4.9999999999999999e-48

   1. Initial program 77.1%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 99.1

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

   5. Applied rewrites 99.1%

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

      1. lift-sin.f64 N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 99.0

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

   7. Applied rewrites 99.0%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 71.7

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

   10. Applied rewrites 71.7%

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

   if 4.9999999999999999e-48 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 81.2

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

   5. Applied rewrites 81.2%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{y + {y}^{3} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. cube-mult N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 65.4

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

   8. Applied rewrites 65.4%

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

4. Final simplification 64.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -5 \cdot 10^{-124}:\\ \;\;\;\;\mathsf{fma}\left(y, y \cdot 0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(y, \left(x \cdot x\right) \cdot -0.16666666666666666, y\right)\\ \mathbf{elif}\;\frac{\sinh y \cdot \sin x}{x} \leq 5 \cdot 10^{-48}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(0.16666666666666666, x \cdot x, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 63.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y \cdot \sin x}{x}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-124}:\\ \;\;\;\;y \cdot \left(y \cdot \left(y \cdot \mathsf{fma}\left(x \cdot x, -0.027777777777777776, 0.16666666666666666\right)\right)\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-48}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(0.16666666666666666, x \cdot x, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sinh y) (sin x)) x)))
   (if (<= t_0 -5e-124)
     (* y (* y (* y (fma (* x x) -0.027777777777777776 0.16666666666666666))))
     (if (<= t_0 5e-48)
       (/ y (fma 0.16666666666666666 (* x x) 1.0))
       (fma
        (fma (* y y) 0.008333333333333333 0.16666666666666666)
        (* y (* y y))
        y)))))

C

double code(double x, double y) {
	double t_0 = (sinh(y) * sin(x)) / x;
	double tmp;
	if (t_0 <= -5e-124) {
		tmp = y * (y * (y * fma((x * x), -0.027777777777777776, 0.16666666666666666)));
	} else if (t_0 <= 5e-48) {
		tmp = y / fma(0.16666666666666666, (x * x), 1.0);
	} else {
		tmp = fma(fma((y * y), 0.008333333333333333, 0.16666666666666666), (y * (y * y)), y);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(sinh(y) * sin(x)) / x)
	tmp = 0.0
	if (t_0 <= -5e-124)
		tmp = Float64(y * Float64(y * Float64(y * fma(Float64(x * x), -0.027777777777777776, 0.16666666666666666))));
	elseif (t_0 <= 5e-48)
		tmp = Float64(y / fma(0.16666666666666666, Float64(x * x), 1.0));
	else
		tmp = fma(fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666), Float64(y * Float64(y * y)), y);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sinh[y], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-124], N[(y * N[(y * N[(y * N[(N[(x * x), $MachinePrecision] * -0.027777777777777776 + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e-48], N[(y / N[(0.16666666666666666 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision] * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -5.0000000000000003e-124

   1. Initial program 99.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-lft-identity N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt-in N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 70.0

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

   5. Applied rewrites 70.0%

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

   6. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{1}{6} \cdot \frac{{y}^{3} \cdot \sin x}{x}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{{y}^{3} \cdot \sin x}{x} \cdot \frac{1}{6}} \]

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. cube-mult N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. associate-*r/ N/A

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

      14. associate-*l/ N/A

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

      15. lower-/.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. lower-sin.f64 48.2

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

   8. Applied rewrites 48.2%

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

   9. Taylor expanded in x around 0

      \[\leadsto y \cdot \left(\color{blue}{\left(\frac{-1}{36} \cdot \left({x}^{2} \cdot y\right) + \frac{1}{6} \cdot y\right)} \cdot y\right) \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto y \cdot \left(\color{blue}{\left(\frac{1}{6} \cdot y + \frac{-1}{36} \cdot \left({x}^{2} \cdot y\right)\right)} \cdot y\right) \]

       2. associate-*r* N/A

          \[\leadsto y \cdot \left(\left(\frac{1}{6} \cdot y + \color{blue}{\left(\frac{-1}{36} \cdot {x}^{2}\right) \cdot y}\right) \cdot y\right) \]

       3. distribute-rgt-out N/A

          \[\leadsto y \cdot \left(\color{blue}{\left(y \cdot \left(\frac{1}{6} + \frac{-1}{36} \cdot {x}^{2}\right)\right)} \cdot y\right) \]

       4. lower-*.f64 N/A

          \[\leadsto y \cdot \left(\color{blue}{\left(y \cdot \left(\frac{1}{6} + \frac{-1}{36} \cdot {x}^{2}\right)\right)} \cdot y\right) \]

       5. +-commutative N/A

          \[\leadsto y \cdot \left(\left(y \cdot \color{blue}{\left(\frac{-1}{36} \cdot {x}^{2} + \frac{1}{6}\right)}\right) \cdot y\right) \]

       6. *-commutative N/A

          \[\leadsto y \cdot \left(\left(y \cdot \left(\color{blue}{{x}^{2} \cdot \frac{-1}{36}} + \frac{1}{6}\right)\right) \cdot y\right) \]

       7. lower-fma.f64 N/A

          \[\leadsto y \cdot \left(\left(y \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{-1}{36}, \frac{1}{6}\right)}\right) \cdot y\right) \]

       8. unpow2 N/A

          \[\leadsto y \cdot \left(\left(y \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{36}, \frac{1}{6}\right)\right) \cdot y\right) \]

       9. lower-*.f64 36.3

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

   11. Applied rewrites 36.3%

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

   if -5.0000000000000003e-124 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 4.9999999999999999e-48

   1. Initial program 77.1%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 99.1

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

   5. Applied rewrites 99.1%

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

      1. lift-sin.f64 N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 99.0

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

   7. Applied rewrites 99.0%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 71.7

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

   10. Applied rewrites 71.7%

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

   if 4.9999999999999999e-48 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 81.2

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

   5. Applied rewrites 81.2%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{y + {y}^{3} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. cube-mult N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 65.4

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

   8. Applied rewrites 65.4%

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

4. Final simplification 60.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -5 \cdot 10^{-124}:\\ \;\;\;\;y \cdot \left(y \cdot \left(y \cdot \mathsf{fma}\left(x \cdot x, -0.027777777777777776, 0.16666666666666666\right)\right)\right)\\ \mathbf{elif}\;\frac{\sinh y \cdot \sin x}{x} \leq 5 \cdot 10^{-48}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(0.16666666666666666, x \cdot x, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 61.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y \cdot \sin x}{x}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-124}:\\ \;\;\;\;y \cdot \left(y \cdot \left(y \cdot \mathsf{fma}\left(x \cdot x, -0.027777777777777776, 0.16666666666666666\right)\right)\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-48}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(0.16666666666666666, x \cdot x, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.16666666666666666, y \cdot \left(y \cdot y\right), y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sinh y) (sin x)) x)))
   (if (<= t_0 -5e-124)
     (* y (* y (* y (fma (* x x) -0.027777777777777776 0.16666666666666666))))
     (if (<= t_0 5e-48)
       (/ y (fma 0.16666666666666666 (* x x) 1.0))
       (fma 0.16666666666666666 (* y (* y y)) y)))))

C

double code(double x, double y) {
	double t_0 = (sinh(y) * sin(x)) / x;
	double tmp;
	if (t_0 <= -5e-124) {
		tmp = y * (y * (y * fma((x * x), -0.027777777777777776, 0.16666666666666666)));
	} else if (t_0 <= 5e-48) {
		tmp = y / fma(0.16666666666666666, (x * x), 1.0);
	} else {
		tmp = fma(0.16666666666666666, (y * (y * y)), y);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(sinh(y) * sin(x)) / x)
	tmp = 0.0
	if (t_0 <= -5e-124)
		tmp = Float64(y * Float64(y * Float64(y * fma(Float64(x * x), -0.027777777777777776, 0.16666666666666666))));
	elseif (t_0 <= 5e-48)
		tmp = Float64(y / fma(0.16666666666666666, Float64(x * x), 1.0));
	else
		tmp = fma(0.16666666666666666, Float64(y * Float64(y * y)), y);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sinh[y], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-124], N[(y * N[(y * N[(y * N[(N[(x * x), $MachinePrecision] * -0.027777777777777776 + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e-48], N[(y / N[(0.16666666666666666 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(0.16666666666666666 * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -5.0000000000000003e-124

   1. Initial program 99.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-lft-identity N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt-in N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 70.0

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

   5. Applied rewrites 70.0%

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

   6. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{1}{6} \cdot \frac{{y}^{3} \cdot \sin x}{x}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{{y}^{3} \cdot \sin x}{x} \cdot \frac{1}{6}} \]

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. cube-mult N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. associate-*r/ N/A

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

      14. associate-*l/ N/A

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

      15. lower-/.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. lower-sin.f64 48.2

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

   8. Applied rewrites 48.2%

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

   9. Taylor expanded in x around 0

      \[\leadsto y \cdot \left(\color{blue}{\left(\frac{-1}{36} \cdot \left({x}^{2} \cdot y\right) + \frac{1}{6} \cdot y\right)} \cdot y\right) \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto y \cdot \left(\color{blue}{\left(\frac{1}{6} \cdot y + \frac{-1}{36} \cdot \left({x}^{2} \cdot y\right)\right)} \cdot y\right) \]

       2. associate-*r* N/A

          \[\leadsto y \cdot \left(\left(\frac{1}{6} \cdot y + \color{blue}{\left(\frac{-1}{36} \cdot {x}^{2}\right) \cdot y}\right) \cdot y\right) \]

       3. distribute-rgt-out N/A

          \[\leadsto y \cdot \left(\color{blue}{\left(y \cdot \left(\frac{1}{6} + \frac{-1}{36} \cdot {x}^{2}\right)\right)} \cdot y\right) \]

       4. lower-*.f64 N/A

          \[\leadsto y \cdot \left(\color{blue}{\left(y \cdot \left(\frac{1}{6} + \frac{-1}{36} \cdot {x}^{2}\right)\right)} \cdot y\right) \]

       5. +-commutative N/A

          \[\leadsto y \cdot \left(\left(y \cdot \color{blue}{\left(\frac{-1}{36} \cdot {x}^{2} + \frac{1}{6}\right)}\right) \cdot y\right) \]

       6. *-commutative N/A

          \[\leadsto y \cdot \left(\left(y \cdot \left(\color{blue}{{x}^{2} \cdot \frac{-1}{36}} + \frac{1}{6}\right)\right) \cdot y\right) \]

       7. lower-fma.f64 N/A

          \[\leadsto y \cdot \left(\left(y \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{-1}{36}, \frac{1}{6}\right)}\right) \cdot y\right) \]

       8. unpow2 N/A

          \[\leadsto y \cdot \left(\left(y \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{36}, \frac{1}{6}\right)\right) \cdot y\right) \]

       9. lower-*.f64 36.3

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

   11. Applied rewrites 36.3%

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

   if -5.0000000000000003e-124 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 4.9999999999999999e-48

   1. Initial program 77.1%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 99.1

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

   5. Applied rewrites 99.1%

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

      1. lift-sin.f64 N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 99.0

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

   7. Applied rewrites 99.0%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 71.7

         \[\leadsto \frac{y}{\mathsf{fma}\left(0.16666666666666666, \color{blue}{x \cdot x}, 1\right)} \]

   10. Applied rewrites 71.7%

       \[\leadsto \frac{y}{\color{blue}{\mathsf{fma}\left(0.16666666666666666, x \cdot x, 1\right)}} \]

   if 4.9999999999999999e-48 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 100.0%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{y \cdot \left(\sin x + \frac{1}{6} \cdot \left({y}^{2} \cdot \sin x\right)\right)}}{x} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{y \cdot \left(\sin x + \frac{1}{6} \cdot \color{blue}{\left(\sin x \cdot {y}^{2}\right)}\right)}{x} \]

      2. associate-*r* N/A

         \[\leadsto \frac{y \cdot \left(\sin x + \color{blue}{\left(\frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}}\right)}{x} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(\sin x + \left(\frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}\right)}}{x} \]

      4. associate-*r* N/A

         \[\leadsto \frac{y \cdot \left(\sin x + \color{blue}{\frac{1}{6} \cdot \left(\sin x \cdot {y}^{2}\right)}\right)}{x} \]

      5. *-commutative N/A

         \[\leadsto \frac{y \cdot \left(\sin x + \frac{1}{6} \cdot \color{blue}{\left({y}^{2} \cdot \sin x\right)}\right)}{x} \]

      6. *-lft-identity N/A

         \[\leadsto \frac{y \cdot \left(\color{blue}{1 \cdot \sin x} + \frac{1}{6} \cdot \left({y}^{2} \cdot \sin x\right)\right)}{x} \]

      7. associate-*r* N/A

         \[\leadsto \frac{y \cdot \left(1 \cdot \sin x + \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \sin x}\right)}{x} \]

      8. distribute-rgt-in N/A

         \[\leadsto \frac{y \cdot \color{blue}{\left(\sin x \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)}}{x} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{y \cdot \color{blue}{\left(\sin x \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)}}{x} \]

      10. lower-sin.f64 N/A

          \[\leadsto \frac{y \cdot \left(\color{blue}{\sin x} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)}{x} \]

      11. +-commutative N/A

          \[\leadsto \frac{y \cdot \left(\sin x \cdot \color{blue}{\left(\frac{1}{6} \cdot {y}^{2} + 1\right)}\right)}{x} \]

      12. unpow2 N/A

          \[\leadsto \frac{y \cdot \left(\sin x \cdot \left(\frac{1}{6} \cdot \color{blue}{\left(y \cdot y\right)} + 1\right)\right)}{x} \]

      13. associate-*r* N/A

          \[\leadsto \frac{y \cdot \left(\sin x \cdot \left(\color{blue}{\left(\frac{1}{6} \cdot y\right) \cdot y} + 1\right)\right)}{x} \]

      14. *-commutative N/A

          \[\leadsto \frac{y \cdot \left(\sin x \cdot \left(\color{blue}{y \cdot \left(\frac{1}{6} \cdot y\right)} + 1\right)\right)}{x} \]

      15. lower-fma.f64 N/A

          \[\leadsto \frac{y \cdot \left(\sin x \cdot \color{blue}{\mathsf{fma}\left(y, \frac{1}{6} \cdot y, 1\right)}\right)}{x} \]

      16. *-commutative N/A

          \[\leadsto \frac{y \cdot \left(\sin x \cdot \mathsf{fma}\left(y, \color{blue}{y \cdot \frac{1}{6}}, 1\right)\right)}{x} \]

      17. lower-*.f64 72.0

          \[\leadsto \frac{y \cdot \left(\sin x \cdot \mathsf{fma}\left(y, \color{blue}{y \cdot 0.16666666666666666}, 1\right)\right)}{x} \]

   5. Applied rewrites 72.0%

      \[\leadsto \frac{\color{blue}{y \cdot \left(\sin x \cdot \mathsf{fma}\left(y, y \cdot 0.16666666666666666, 1\right)\right)}}{x} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{y \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto y \cdot \color{blue}{\left(\frac{1}{6} \cdot {y}^{2} + 1\right)} \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right) \cdot y + 1 \cdot y} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\frac{1}{6} \cdot \left({y}^{2} \cdot y\right)} + 1 \cdot y \]

      4. unpow2 N/A

         \[\leadsto \frac{1}{6} \cdot \left(\color{blue}{\left(y \cdot y\right)} \cdot y\right) + 1 \cdot y \]

      5. unpow3 N/A

         \[\leadsto \frac{1}{6} \cdot \color{blue}{{y}^{3}} + 1 \cdot y \]

      6. *-lft-identity N/A

         \[\leadsto \frac{1}{6} \cdot {y}^{3} + \color{blue}{y} \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{6}, {y}^{3}, y\right)} \]

      8. cube-mult N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{6}, \color{blue}{y \cdot \left(y \cdot y\right)}, y\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{6}, y \cdot \color{blue}{{y}^{2}}, y\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{6}, \color{blue}{y \cdot {y}^{2}}, y\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{6}, y \cdot \color{blue}{\left(y \cdot y\right)}, y\right) \]

      12. lower-*.f64 56.5

          \[\leadsto \mathsf{fma}\left(0.16666666666666666, y \cdot \color{blue}{\left(y \cdot y\right)}, y\right) \]

   8. Applied rewrites 56.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.16666666666666666, y \cdot \left(y \cdot y\right), y\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 57.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -5 \cdot 10^{-124}:\\ \;\;\;\;y \cdot \left(y \cdot \left(y \cdot \mathsf{fma}\left(x \cdot x, -0.027777777777777776, 0.16666666666666666\right)\right)\right)\\ \mathbf{elif}\;\frac{\sinh y \cdot \sin x}{x} \leq 5 \cdot 10^{-48}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(0.16666666666666666, x \cdot x, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.16666666666666666, y \cdot \left(y \cdot y\right), y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 56.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y \cdot \sin x}{x}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-124}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-48}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(0.16666666666666666, x \cdot x, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.16666666666666666, y \cdot \left(y \cdot y\right), y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sinh y) (sin x)) x)))
   (if (<= t_0 -5e-124)
     (* y (fma x (* x -0.16666666666666666) 1.0))
     (if (<= t_0 5e-48)
       (/ y (fma 0.16666666666666666 (* x x) 1.0))
       (fma 0.16666666666666666 (* y (* y y)) y)))))

C

double code(double x, double y) {
	double t_0 = (sinh(y) * sin(x)) / x;
	double tmp;
	if (t_0 <= -5e-124) {
		tmp = y * fma(x, (x * -0.16666666666666666), 1.0);
	} else if (t_0 <= 5e-48) {
		tmp = y / fma(0.16666666666666666, (x * x), 1.0);
	} else {
		tmp = fma(0.16666666666666666, (y * (y * y)), y);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(sinh(y) * sin(x)) / x)
	tmp = 0.0
	if (t_0 <= -5e-124)
		tmp = Float64(y * fma(x, Float64(x * -0.16666666666666666), 1.0));
	elseif (t_0 <= 5e-48)
		tmp = Float64(y / fma(0.16666666666666666, Float64(x * x), 1.0));
	else
		tmp = fma(0.16666666666666666, Float64(y * Float64(y * y)), y);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sinh[y], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-124], N[(y * N[(x * N[(x * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e-48], N[(y / N[(0.16666666666666666 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(0.16666666666666666 * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -5.0000000000000003e-124

   1. Initial program 99.0%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 21.7

         \[\leadsto y \cdot \frac{\color{blue}{\sin x}}{x} \]

   5. Applied rewrites 21.7%

      \[\leadsto \color{blue}{y \cdot \frac{\sin x}{x}} \]

   6. Taylor expanded in x around 0

      \[\leadsto y \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {x}^{2}\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto y \cdot \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2} + 1\right)} \]

      2. *-commutative N/A

         \[\leadsto y \cdot \left(\color{blue}{{x}^{2} \cdot \frac{-1}{6}} + 1\right) \]

      3. unpow2 N/A

         \[\leadsto y \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{-1}{6} + 1\right) \]

      4. associate-*l* N/A

         \[\leadsto y \cdot \left(\color{blue}{x \cdot \left(x \cdot \frac{-1}{6}\right)} + 1\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto y \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)} \]

      6. lower-*.f64 27.5

         \[\leadsto y \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot -0.16666666666666666}, 1\right) \]

   8. Applied rewrites 27.5%

      \[\leadsto y \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right)} \]

   if -5.0000000000000003e-124 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 4.9999999999999999e-48

   1. Initial program 77.1%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 99.1

         \[\leadsto y \cdot \frac{\color{blue}{\sin x}}{x} \]

   5. Applied rewrites 99.1%

      \[\leadsto \color{blue}{y \cdot \frac{\sin x}{x}} \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto y \cdot \frac{\color{blue}{\sin x}}{x} \]

      2. clear-num N/A

         \[\leadsto y \cdot \color{blue}{\frac{1}{\frac{x}{\sin x}}} \]

      3. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{y}{\frac{x}{\sin x}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{\frac{x}{\sin x}}} \]

      5. lower-/.f64 99.0

         \[\leadsto \frac{y}{\color{blue}{\frac{x}{\sin x}}} \]

   7. Applied rewrites 99.0%

      \[\leadsto \color{blue}{\frac{y}{\frac{x}{\sin x}}} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{y}{\color{blue}{1 + \frac{1}{6} \cdot {x}^{2}}} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{y}{\color{blue}{\frac{1}{6} \cdot {x}^{2} + 1}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{y}{\color{blue}{\mathsf{fma}\left(\frac{1}{6}, {x}^{2}, 1\right)}} \]

      3. unpow2 N/A

         \[\leadsto \frac{y}{\mathsf{fma}\left(\frac{1}{6}, \color{blue}{x \cdot x}, 1\right)} \]

      4. lower-*.f64 71.7

         \[\leadsto \frac{y}{\mathsf{fma}\left(0.16666666666666666, \color{blue}{x \cdot x}, 1\right)} \]

   10. Applied rewrites 71.7%

       \[\leadsto \frac{y}{\color{blue}{\mathsf{fma}\left(0.16666666666666666, x \cdot x, 1\right)}} \]

   if 4.9999999999999999e-48 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 100.0%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{y \cdot \left(\sin x + \frac{1}{6} \cdot \left({y}^{2} \cdot \sin x\right)\right)}}{x} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{y \cdot \left(\sin x + \frac{1}{6} \cdot \color{blue}{\left(\sin x \cdot {y}^{2}\right)}\right)}{x} \]

      2. associate-*r* N/A

         \[\leadsto \frac{y \cdot \left(\sin x + \color{blue}{\left(\frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}}\right)}{x} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(\sin x + \left(\frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}\right)}}{x} \]

      4. associate-*r* N/A

         \[\leadsto \frac{y \cdot \left(\sin x + \color{blue}{\frac{1}{6} \cdot \left(\sin x \cdot {y}^{2}\right)}\right)}{x} \]

      5. *-commutative N/A

         \[\leadsto \frac{y \cdot \left(\sin x + \frac{1}{6} \cdot \color{blue}{\left({y}^{2} \cdot \sin x\right)}\right)}{x} \]

      6. *-lft-identity N/A

         \[\leadsto \frac{y \cdot \left(\color{blue}{1 \cdot \sin x} + \frac{1}{6} \cdot \left({y}^{2} \cdot \sin x\right)\right)}{x} \]

      7. associate-*r* N/A

         \[\leadsto \frac{y \cdot \left(1 \cdot \sin x + \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \sin x}\right)}{x} \]

      8. distribute-rgt-in N/A

         \[\leadsto \frac{y \cdot \color{blue}{\left(\sin x \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)}}{x} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{y \cdot \color{blue}{\left(\sin x \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)}}{x} \]

      10. lower-sin.f64 N/A

          \[\leadsto \frac{y \cdot \left(\color{blue}{\sin x} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)}{x} \]

      11. +-commutative N/A

          \[\leadsto \frac{y \cdot \left(\sin x \cdot \color{blue}{\left(\frac{1}{6} \cdot {y}^{2} + 1\right)}\right)}{x} \]

      12. unpow2 N/A

          \[\leadsto \frac{y \cdot \left(\sin x \cdot \left(\frac{1}{6} \cdot \color{blue}{\left(y \cdot y\right)} + 1\right)\right)}{x} \]

      13. associate-*r* N/A

          \[\leadsto \frac{y \cdot \left(\sin x \cdot \left(\color{blue}{\left(\frac{1}{6} \cdot y\right) \cdot y} + 1\right)\right)}{x} \]

      14. *-commutative N/A

          \[\leadsto \frac{y \cdot \left(\sin x \cdot \left(\color{blue}{y \cdot \left(\frac{1}{6} \cdot y\right)} + 1\right)\right)}{x} \]

      15. lower-fma.f64 N/A

          \[\leadsto \frac{y \cdot \left(\sin x \cdot \color{blue}{\mathsf{fma}\left(y, \frac{1}{6} \cdot y, 1\right)}\right)}{x} \]

      16. *-commutative N/A

          \[\leadsto \frac{y \cdot \left(\sin x \cdot \mathsf{fma}\left(y, \color{blue}{y \cdot \frac{1}{6}}, 1\right)\right)}{x} \]

      17. lower-*.f64 72.0

          \[\leadsto \frac{y \cdot \left(\sin x \cdot \mathsf{fma}\left(y, \color{blue}{y \cdot 0.16666666666666666}, 1\right)\right)}{x} \]

   5. Applied rewrites 72.0%

      \[\leadsto \frac{\color{blue}{y \cdot \left(\sin x \cdot \mathsf{fma}\left(y, y \cdot 0.16666666666666666, 1\right)\right)}}{x} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{y \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto y \cdot \color{blue}{\left(\frac{1}{6} \cdot {y}^{2} + 1\right)} \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right) \cdot y + 1 \cdot y} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\frac{1}{6} \cdot \left({y}^{2} \cdot y\right)} + 1 \cdot y \]

      4. unpow2 N/A

         \[\leadsto \frac{1}{6} \cdot \left(\color{blue}{\left(y \cdot y\right)} \cdot y\right) + 1 \cdot y \]

      5. unpow3 N/A

         \[\leadsto \frac{1}{6} \cdot \color{blue}{{y}^{3}} + 1 \cdot y \]

      6. *-lft-identity N/A

         \[\leadsto \frac{1}{6} \cdot {y}^{3} + \color{blue}{y} \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{6}, {y}^{3}, y\right)} \]

      8. cube-mult N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{6}, \color{blue}{y \cdot \left(y \cdot y\right)}, y\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{6}, y \cdot \color{blue}{{y}^{2}}, y\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{6}, \color{blue}{y \cdot {y}^{2}}, y\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{6}, y \cdot \color{blue}{\left(y \cdot y\right)}, y\right) \]

      12. lower-*.f64 56.5

          \[\leadsto \mathsf{fma}\left(0.16666666666666666, y \cdot \color{blue}{\left(y \cdot y\right)}, y\right) \]

   8. Applied rewrites 56.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.16666666666666666, y \cdot \left(y \cdot y\right), y\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 54.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -5 \cdot 10^{-124}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right)\\ \mathbf{elif}\;\frac{\sinh y \cdot \sin x}{x} \leq 5 \cdot 10^{-48}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(0.16666666666666666, x \cdot x, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.16666666666666666, y \cdot \left(y \cdot y\right), y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 43.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq 0.02:\\ \;\;\;\;y \cdot \mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(y \cdot y\right) \cdot 0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (sinh y) (sin x)) x) 0.02)
   (* y (fma x (* x -0.16666666666666666) 1.0))
   (* y (* (* y y) 0.16666666666666666))))

C

double code(double x, double y) {
	double tmp;
	if (((sinh(y) * sin(x)) / x) <= 0.02) {
		tmp = 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 (Float64(Float64(sinh(y) * sin(x)) / x) <= 0.02)
		tmp = Float64(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[N[(N[(N[Sinh[y], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], 0.02], N[(y * 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 (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 0.0200000000000000004

   1. Initial program 86.5%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 69.3

         \[\leadsto y \cdot \frac{\color{blue}{\sin x}}{x} \]

   5. Applied rewrites 69.3%

      \[\leadsto \color{blue}{y \cdot \frac{\sin x}{x}} \]

   6. Taylor expanded in x around 0

      \[\leadsto y \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {x}^{2}\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto y \cdot \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2} + 1\right)} \]

      2. *-commutative N/A

         \[\leadsto y \cdot \left(\color{blue}{{x}^{2} \cdot \frac{-1}{6}} + 1\right) \]

      3. unpow2 N/A

         \[\leadsto y \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{-1}{6} + 1\right) \]

      4. associate-*l* N/A

         \[\leadsto y \cdot \left(\color{blue}{x \cdot \left(x \cdot \frac{-1}{6}\right)} + 1\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto y \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)} \]

      6. lower-*.f64 38.6

         \[\leadsto y \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot -0.16666666666666666}, 1\right) \]

   8. Applied rewrites 38.6%

      \[\leadsto y \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right)} \]

   if 0.0200000000000000004 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 100.0%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{y \cdot \left(\sin x + \frac{1}{6} \cdot \left({y}^{2} \cdot \sin x\right)\right)}}{x} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{y \cdot \left(\sin x + \frac{1}{6} \cdot \color{blue}{\left(\sin x \cdot {y}^{2}\right)}\right)}{x} \]

      2. associate-*r* N/A

         \[\leadsto \frac{y \cdot \left(\sin x + \color{blue}{\left(\frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}}\right)}{x} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(\sin x + \left(\frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}\right)}}{x} \]

      4. associate-*r* N/A

         \[\leadsto \frac{y \cdot \left(\sin x + \color{blue}{\frac{1}{6} \cdot \left(\sin x \cdot {y}^{2}\right)}\right)}{x} \]

      5. *-commutative N/A

         \[\leadsto \frac{y \cdot \left(\sin x + \frac{1}{6} \cdot \color{blue}{\left({y}^{2} \cdot \sin x\right)}\right)}{x} \]

      6. *-lft-identity N/A

         \[\leadsto \frac{y \cdot \left(\color{blue}{1 \cdot \sin x} + \frac{1}{6} \cdot \left({y}^{2} \cdot \sin x\right)\right)}{x} \]

      7. associate-*r* N/A

         \[\leadsto \frac{y \cdot \left(1 \cdot \sin x + \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \sin x}\right)}{x} \]

      8. distribute-rgt-in N/A

         \[\leadsto \frac{y \cdot \color{blue}{\left(\sin x \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)}}{x} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{y \cdot \color{blue}{\left(\sin x \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)}}{x} \]

      10. lower-sin.f64 N/A

          \[\leadsto \frac{y \cdot \left(\color{blue}{\sin x} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)}{x} \]

      11. +-commutative N/A

          \[\leadsto \frac{y \cdot \left(\sin x \cdot \color{blue}{\left(\frac{1}{6} \cdot {y}^{2} + 1\right)}\right)}{x} \]

      12. unpow2 N/A

          \[\leadsto \frac{y \cdot \left(\sin x \cdot \left(\frac{1}{6} \cdot \color{blue}{\left(y \cdot y\right)} + 1\right)\right)}{x} \]

      13. associate-*r* N/A

          \[\leadsto \frac{y \cdot \left(\sin x \cdot \left(\color{blue}{\left(\frac{1}{6} \cdot y\right) \cdot y} + 1\right)\right)}{x} \]

      14. *-commutative N/A

          \[\leadsto \frac{y \cdot \left(\sin x \cdot \left(\color{blue}{y \cdot \left(\frac{1}{6} \cdot y\right)} + 1\right)\right)}{x} \]

      15. lower-fma.f64 N/A

          \[\leadsto \frac{y \cdot \left(\sin x \cdot \color{blue}{\mathsf{fma}\left(y, \frac{1}{6} \cdot y, 1\right)}\right)}{x} \]

      16. *-commutative N/A

          \[\leadsto \frac{y \cdot \left(\sin x \cdot \mathsf{fma}\left(y, \color{blue}{y \cdot \frac{1}{6}}, 1\right)\right)}{x} \]

      17. lower-*.f64 68.4

          \[\leadsto \frac{y \cdot \left(\sin x \cdot \mathsf{fma}\left(y, \color{blue}{y \cdot 0.16666666666666666}, 1\right)\right)}{x} \]

   5. Applied rewrites 68.4%

      \[\leadsto \frac{\color{blue}{y \cdot \left(\sin x \cdot \mathsf{fma}\left(y, y \cdot 0.16666666666666666, 1\right)\right)}}{x} \]

   6. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{1}{6} \cdot \frac{{y}^{3} \cdot \sin x}{x}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{{y}^{3} \cdot \sin x}{x} \cdot \frac{1}{6}} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\left({y}^{3} \cdot \frac{\sin x}{x}\right)} \cdot \frac{1}{6} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{{y}^{3} \cdot \left(\frac{\sin x}{x} \cdot \frac{1}{6}\right)} \]

      4. *-commutative N/A

         \[\leadsto {y}^{3} \cdot \color{blue}{\left(\frac{1}{6} \cdot \frac{\sin x}{x}\right)} \]

      5. cube-mult N/A

         \[\leadsto \color{blue}{\left(y \cdot \left(y \cdot y\right)\right)} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right) \]

      6. unpow2 N/A

         \[\leadsto \left(y \cdot \color{blue}{{y}^{2}}\right) \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right) \]

      7. associate-*l* N/A

         \[\leadsto \color{blue}{y \cdot \left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right)\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{y \cdot \left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right)\right)} \]

      9. *-commutative N/A

         \[\leadsto y \cdot \color{blue}{\left(\left(\frac{1}{6} \cdot \frac{\sin x}{x}\right) \cdot {y}^{2}\right)} \]

      10. unpow2 N/A

          \[\leadsto y \cdot \left(\left(\frac{1}{6} \cdot \frac{\sin x}{x}\right) \cdot \color{blue}{\left(y \cdot y\right)}\right) \]

      11. associate-*r* N/A

          \[\leadsto y \cdot \color{blue}{\left(\left(\left(\frac{1}{6} \cdot \frac{\sin x}{x}\right) \cdot y\right) \cdot y\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto y \cdot \color{blue}{\left(\left(\left(\frac{1}{6} \cdot \frac{\sin x}{x}\right) \cdot y\right) \cdot y\right)} \]

      13. associate-*r/ N/A

          \[\leadsto y \cdot \left(\left(\color{blue}{\frac{\frac{1}{6} \cdot \sin x}{x}} \cdot y\right) \cdot y\right) \]

      14. associate-*l/ N/A

          \[\leadsto y \cdot \left(\color{blue}{\frac{\left(\frac{1}{6} \cdot \sin x\right) \cdot y}{x}} \cdot y\right) \]

      15. lower-/.f64 N/A

          \[\leadsto y \cdot \left(\color{blue}{\frac{\left(\frac{1}{6} \cdot \sin x\right) \cdot y}{x}} \cdot y\right) \]

      16. lower-*.f64 N/A

          \[\leadsto y \cdot \left(\frac{\color{blue}{\left(\frac{1}{6} \cdot \sin x\right) \cdot y}}{x} \cdot y\right) \]

      17. *-commutative N/A

          \[\leadsto y \cdot \left(\frac{\color{blue}{\left(\sin x \cdot \frac{1}{6}\right)} \cdot y}{x} \cdot y\right) \]

      18. lower-*.f64 N/A

          \[\leadsto y \cdot \left(\frac{\color{blue}{\left(\sin x \cdot \frac{1}{6}\right)} \cdot y}{x} \cdot y\right) \]

      19. lower-sin.f64 65.6

          \[\leadsto y \cdot \left(\frac{\left(\color{blue}{\sin x} \cdot 0.16666666666666666\right) \cdot y}{x} \cdot y\right) \]

   8. Applied rewrites 65.6%

      \[\leadsto \color{blue}{y \cdot \left(\frac{\left(\sin x \cdot 0.16666666666666666\right) \cdot y}{x} \cdot y\right)} \]

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{6} \cdot {y}^{3}} \]
   10. Step-by-step derivation

       1. unpow3 N/A

          \[\leadsto \frac{1}{6} \cdot \color{blue}{\left(\left(y \cdot y\right) \cdot y\right)} \]

       2. unpow2 N/A

          \[\leadsto \frac{1}{6} \cdot \left(\color{blue}{{y}^{2}} \cdot y\right) \]

       3. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right) \cdot y} \]

       4. *-commutative N/A

          \[\leadsto \color{blue}{y \cdot \left(\frac{1}{6} \cdot {y}^{2}\right)} \]

       5. lower-*.f64 N/A

          \[\leadsto \color{blue}{y \cdot \left(\frac{1}{6} \cdot {y}^{2}\right)} \]

       6. lower-*.f64 N/A

          \[\leadsto y \cdot \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right)} \]

       7. unpow2 N/A

          \[\leadsto y \cdot \left(\frac{1}{6} \cdot \color{blue}{\left(y \cdot y\right)}\right) \]

       8. lower-*.f64 51.9

          \[\leadsto y \cdot \left(0.16666666666666666 \cdot \color{blue}{\left(y \cdot y\right)}\right) \]

   11. Applied rewrites 51.9%

       \[\leadsto \color{blue}{y \cdot \left(0.16666666666666666 \cdot \left(y \cdot y\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq 0.02:\\ \;\;\;\;y \cdot \mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(y \cdot y\right) \cdot 0.16666666666666666\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 93.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y, y \cdot \left(y \cdot \mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right)\right), y\right) \cdot \frac{\sin x}{x}\\ \mathbf{if}\;y \leq 520:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{+62}:\\ \;\;\;\;\frac{\sinh y \cdot \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot -0.16666666666666666, x\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (*
          (fma
           y
           (* y (* y (fma 0.008333333333333333 (* y y) 0.16666666666666666)))
           y)
          (/ (sin x) x))))
   (if (<= y 520.0)
     t_0
     (if (<= y 1.15e+62)
       (/ (* (sinh y) (fma x (* (* x x) -0.16666666666666666) x)) x)
       t_0))))

C

double code(double x, double y) {
	double t_0 = fma(y, (y * (y * fma(0.008333333333333333, (y * y), 0.16666666666666666))), y) * (sin(x) / x);
	double tmp;
	if (y <= 520.0) {
		tmp = t_0;
	} else if (y <= 1.15e+62) {
		tmp = (sinh(y) * fma(x, ((x * x) * -0.16666666666666666), x)) / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(fma(y, Float64(y * Float64(y * fma(0.008333333333333333, Float64(y * y), 0.16666666666666666))), y) * Float64(sin(x) / x))
	tmp = 0.0
	if (y <= 520.0)
		tmp = t_0;
	elseif (y <= 1.15e+62)
		tmp = Float64(Float64(sinh(y) * fma(x, Float64(Float64(x * x) * -0.16666666666666666), x)) / x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * N[(y * N[(y * N[(0.008333333333333333 * N[(y * y), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 520.0], t$95$0, If[LessEqual[y, 1.15e+62], N[(N[(N[Sinh[y], $MachinePrecision] * N[(x * N[(N[(x * x), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < 520 or 1.14999999999999992e62 < y

   1. Initial program 89.6%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\sin x \cdot \color{blue}{\left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)}}{x} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\sin x \cdot \left(y \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) + 1\right)}\right)}{x} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\sin x \cdot \color{blue}{\left(y \cdot \left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) + y \cdot 1\right)}}{x} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\sin x \cdot \left(\color{blue}{\left(y \cdot {y}^{2}\right) \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)} + y \cdot 1\right)}{x} \]

      4. *-commutative N/A

         \[\leadsto \frac{\sin x \cdot \left(\color{blue}{\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) \cdot \left(y \cdot {y}^{2}\right)} + y \cdot 1\right)}{x} \]

      5. *-rgt-identity N/A

         \[\leadsto \frac{\sin x \cdot \left(\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) \cdot \left(y \cdot {y}^{2}\right) + \color{blue}{y}\right)}{x} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\sin x \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}, y \cdot {y}^{2}, y\right)}}{x} \]

      7. +-commutative N/A

         \[\leadsto \frac{\sin x \cdot \mathsf{fma}\left(\color{blue}{\frac{1}{120} \cdot {y}^{2} + \frac{1}{6}}, y \cdot {y}^{2}, y\right)}{x} \]

      8. *-commutative N/A

         \[\leadsto \frac{\sin x \cdot \mathsf{fma}\left(\color{blue}{{y}^{2} \cdot \frac{1}{120}} + \frac{1}{6}, y \cdot {y}^{2}, y\right)}{x} \]

      9. unpow2 N/A

         \[\leadsto \frac{\sin x \cdot \mathsf{fma}\left(\color{blue}{\left(y \cdot y\right)} \cdot \frac{1}{120} + \frac{1}{6}, y \cdot {y}^{2}, y\right)}{x} \]

      10. associate-*l* N/A

          \[\leadsto \frac{\sin x \cdot \mathsf{fma}\left(\color{blue}{y \cdot \left(y \cdot \frac{1}{120}\right)} + \frac{1}{6}, y \cdot {y}^{2}, y\right)}{x} \]

      11. lower-fma.f64 N/A

          \[\leadsto \frac{\sin x \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right)}, y \cdot {y}^{2}, y\right)}{x} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\sin x \cdot \mathsf{fma}\left(\mathsf{fma}\left(y, \color{blue}{y \cdot \frac{1}{120}}, \frac{1}{6}\right), y \cdot {y}^{2}, y\right)}{x} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\sin x \cdot \mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), \color{blue}{y \cdot {y}^{2}}, y\right)}{x} \]

      14. unpow2 N/A

          \[\leadsto \frac{\sin x \cdot \mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), y \cdot \color{blue}{\left(y \cdot y\right)}, y\right)}{x} \]

      15. lower-*.f64 82.2

          \[\leadsto \frac{\sin x \cdot \mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), y \cdot \color{blue}{\left(y \cdot y\right)}, y\right)}{x} \]

   5. Applied rewrites 82.2%

      \[\leadsto \frac{\sin x \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)}}{x} \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \frac{\color{blue}{\sin x} \cdot \left(\left(y \cdot \left(y \cdot \frac{1}{120}\right) + \frac{1}{6}\right) \cdot \left(y \cdot \left(y \cdot y\right)\right) + y\right)}{x} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\sin x \cdot \left(\left(y \cdot \color{blue}{\left(y \cdot \frac{1}{120}\right)} + \frac{1}{6}\right) \cdot \left(y \cdot \left(y \cdot y\right)\right) + y\right)}{x} \]

      3. lift-fma.f64 N/A

         \[\leadsto \frac{\sin x \cdot \left(\color{blue}{\mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right)} \cdot \left(y \cdot \left(y \cdot y\right)\right) + y\right)}{x} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\sin x \cdot \left(\mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right) \cdot \left(y \cdot \color{blue}{\left(y \cdot y\right)}\right) + y\right)}{x} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\sin x \cdot \left(\mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right) \cdot \color{blue}{\left(y \cdot \left(y \cdot y\right)\right)} + y\right)}{x} \]

      6. lift-fma.f64 N/A

         \[\leadsto \frac{\sin x \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), y \cdot \left(y \cdot y\right), y\right)}}{x} \]

      7. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), y \cdot \left(y \cdot y\right), y\right) \cdot \sin x}}{x} \]

      8. associate-/l* N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), y \cdot \left(y \cdot y\right), y\right) \cdot \frac{\sin x}{x}} \]

      9. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), y \cdot \left(y \cdot y\right), y\right) \cdot \color{blue}{\frac{\sin x}{x}} \]

      10. lower-*.f64 92.4

          \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right) \cdot \frac{\sin x}{x}} \]

   7. Applied rewrites 92.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, y \cdot \left(y \cdot \mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right)\right), y\right) \cdot \frac{\sin x}{x}} \]

   if 520 < y < 1.14999999999999992e62

   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 90.0

         \[\leadsto \frac{\mathsf{fma}\left(x, \color{blue}{\left(x \cdot x\right)} \cdot -0.16666666666666666, x\right) \cdot \sinh y}{x} \]

   5. Applied rewrites 90.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot -0.16666666666666666, x\right)} \cdot \sinh y}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 520:\\ \;\;\;\;\mathsf{fma}\left(y, y \cdot \left(y \cdot \mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right)\right), y\right) \cdot \frac{\sin x}{x}\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{+62}:\\ \;\;\;\;\frac{\sinh y \cdot \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot -0.16666666666666666, x\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, y \cdot \left(y \cdot \mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right)\right), y\right) \cdot \frac{\sin x}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 45.4% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(\left(y \cdot y\right) \cdot 0.16666666666666666\right)\\ \mathbf{if}\;y \leq 9.5 \cdot 10^{-251}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 62000000:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* (* y y) 0.16666666666666666))))
   (if (<= y 9.5e-251) t_0 (if (<= y 62000000.0) y t_0))))

C

double code(double x, double y) {
	double t_0 = y * ((y * y) * 0.16666666666666666);
	double tmp;
	if (y <= 9.5e-251) {
		tmp = t_0;
	} else if (y <= 62000000.0) {
		tmp = y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y * ((y * y) * 0.16666666666666666d0)
    if (y <= 9.5d-251) then
        tmp = t_0
    else if (y <= 62000000.0d0) then
        tmp = y
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = y * ((y * y) * 0.16666666666666666);
	double tmp;
	if (y <= 9.5e-251) {
		tmp = t_0;
	} else if (y <= 62000000.0) {
		tmp = y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * ((y * y) * 0.16666666666666666)
	tmp = 0
	if y <= 9.5e-251:
		tmp = t_0
	elif y <= 62000000.0:
		tmp = y
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * Float64(Float64(y * y) * 0.16666666666666666))
	tmp = 0.0
	if (y <= 9.5e-251)
		tmp = t_0;
	elseif (y <= 62000000.0)
		tmp = y;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y * ((y * y) * 0.16666666666666666);
	tmp = 0.0;
	if (y <= 9.5e-251)
		tmp = t_0;
	elseif (y <= 62000000.0)
		tmp = y;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 9.5e-251], t$95$0, If[LessEqual[y, 62000000.0], y, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < 9.49999999999999927e-251 or 6.2e7 < y

   1. Initial program 93.0%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{y \cdot \left(\sin x + \frac{1}{6} \cdot \left({y}^{2} \cdot \sin x\right)\right)}}{x} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{y \cdot \left(\sin x + \frac{1}{6} \cdot \color{blue}{\left(\sin x \cdot {y}^{2}\right)}\right)}{x} \]

      2. associate-*r* N/A

         \[\leadsto \frac{y \cdot \left(\sin x + \color{blue}{\left(\frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}}\right)}{x} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(\sin x + \left(\frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}\right)}}{x} \]

      4. associate-*r* N/A

         \[\leadsto \frac{y \cdot \left(\sin x + \color{blue}{\frac{1}{6} \cdot \left(\sin x \cdot {y}^{2}\right)}\right)}{x} \]

      5. *-commutative N/A

         \[\leadsto \frac{y \cdot \left(\sin x + \frac{1}{6} \cdot \color{blue}{\left({y}^{2} \cdot \sin x\right)}\right)}{x} \]

      6. *-lft-identity N/A

         \[\leadsto \frac{y \cdot \left(\color{blue}{1 \cdot \sin x} + \frac{1}{6} \cdot \left({y}^{2} \cdot \sin x\right)\right)}{x} \]

      7. associate-*r* N/A

         \[\leadsto \frac{y \cdot \left(1 \cdot \sin x + \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \sin x}\right)}{x} \]

      8. distribute-rgt-in N/A

         \[\leadsto \frac{y \cdot \color{blue}{\left(\sin x \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)}}{x} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{y \cdot \color{blue}{\left(\sin x \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)}}{x} \]

      10. lower-sin.f64 N/A

          \[\leadsto \frac{y \cdot \left(\color{blue}{\sin x} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)}{x} \]

      11. +-commutative N/A

          \[\leadsto \frac{y \cdot \left(\sin x \cdot \color{blue}{\left(\frac{1}{6} \cdot {y}^{2} + 1\right)}\right)}{x} \]

      12. unpow2 N/A

          \[\leadsto \frac{y \cdot \left(\sin x \cdot \left(\frac{1}{6} \cdot \color{blue}{\left(y \cdot y\right)} + 1\right)\right)}{x} \]

      13. associate-*r* N/A

          \[\leadsto \frac{y \cdot \left(\sin x \cdot \left(\color{blue}{\left(\frac{1}{6} \cdot y\right) \cdot y} + 1\right)\right)}{x} \]

      14. *-commutative N/A

          \[\leadsto \frac{y \cdot \left(\sin x \cdot \left(\color{blue}{y \cdot \left(\frac{1}{6} \cdot y\right)} + 1\right)\right)}{x} \]

      15. lower-fma.f64 N/A

          \[\leadsto \frac{y \cdot \left(\sin x \cdot \color{blue}{\mathsf{fma}\left(y, \frac{1}{6} \cdot y, 1\right)}\right)}{x} \]

      16. *-commutative N/A

          \[\leadsto \frac{y \cdot \left(\sin x \cdot \mathsf{fma}\left(y, \color{blue}{y \cdot \frac{1}{6}}, 1\right)\right)}{x} \]

      17. lower-*.f64 72.9

          \[\leadsto \frac{y \cdot \left(\sin x \cdot \mathsf{fma}\left(y, \color{blue}{y \cdot 0.16666666666666666}, 1\right)\right)}{x} \]

   5. Applied rewrites 72.9%

      \[\leadsto \frac{\color{blue}{y \cdot \left(\sin x \cdot \mathsf{fma}\left(y, y \cdot 0.16666666666666666, 1\right)\right)}}{x} \]

   6. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{1}{6} \cdot \frac{{y}^{3} \cdot \sin x}{x}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{{y}^{3} \cdot \sin x}{x} \cdot \frac{1}{6}} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\left({y}^{3} \cdot \frac{\sin x}{x}\right)} \cdot \frac{1}{6} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{{y}^{3} \cdot \left(\frac{\sin x}{x} \cdot \frac{1}{6}\right)} \]

      4. *-commutative N/A

         \[\leadsto {y}^{3} \cdot \color{blue}{\left(\frac{1}{6} \cdot \frac{\sin x}{x}\right)} \]

      5. cube-mult N/A

         \[\leadsto \color{blue}{\left(y \cdot \left(y \cdot y\right)\right)} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right) \]

      6. unpow2 N/A

         \[\leadsto \left(y \cdot \color{blue}{{y}^{2}}\right) \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right) \]

      7. associate-*l* N/A

         \[\leadsto \color{blue}{y \cdot \left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right)\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{y \cdot \left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right)\right)} \]

      9. *-commutative N/A

         \[\leadsto y \cdot \color{blue}{\left(\left(\frac{1}{6} \cdot \frac{\sin x}{x}\right) \cdot {y}^{2}\right)} \]

      10. unpow2 N/A

          \[\leadsto y \cdot \left(\left(\frac{1}{6} \cdot \frac{\sin x}{x}\right) \cdot \color{blue}{\left(y \cdot y\right)}\right) \]

      11. associate-*r* N/A

          \[\leadsto y \cdot \color{blue}{\left(\left(\left(\frac{1}{6} \cdot \frac{\sin x}{x}\right) \cdot y\right) \cdot y\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto y \cdot \color{blue}{\left(\left(\left(\frac{1}{6} \cdot \frac{\sin x}{x}\right) \cdot y\right) \cdot y\right)} \]

      13. associate-*r/ N/A

          \[\leadsto y \cdot \left(\left(\color{blue}{\frac{\frac{1}{6} \cdot \sin x}{x}} \cdot y\right) \cdot y\right) \]

      14. associate-*l/ N/A

          \[\leadsto y \cdot \left(\color{blue}{\frac{\left(\frac{1}{6} \cdot \sin x\right) \cdot y}{x}} \cdot y\right) \]

      15. lower-/.f64 N/A

          \[\leadsto y \cdot \left(\color{blue}{\frac{\left(\frac{1}{6} \cdot \sin x\right) \cdot y}{x}} \cdot y\right) \]

      16. lower-*.f64 N/A

          \[\leadsto y \cdot \left(\frac{\color{blue}{\left(\frac{1}{6} \cdot \sin x\right) \cdot y}}{x} \cdot y\right) \]

      17. *-commutative N/A

          \[\leadsto y \cdot \left(\frac{\color{blue}{\left(\sin x \cdot \frac{1}{6}\right)} \cdot y}{x} \cdot y\right) \]

      18. lower-*.f64 N/A

          \[\leadsto y \cdot \left(\frac{\color{blue}{\left(\sin x \cdot \frac{1}{6}\right)} \cdot y}{x} \cdot y\right) \]

      19. lower-sin.f64 50.9

          \[\leadsto y \cdot \left(\frac{\left(\color{blue}{\sin x} \cdot 0.16666666666666666\right) \cdot y}{x} \cdot y\right) \]

   8. Applied rewrites 50.9%

      \[\leadsto \color{blue}{y \cdot \left(\frac{\left(\sin x \cdot 0.16666666666666666\right) \cdot y}{x} \cdot y\right)} \]

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{6} \cdot {y}^{3}} \]
   10. Step-by-step derivation

       1. unpow3 N/A

          \[\leadsto \frac{1}{6} \cdot \color{blue}{\left(\left(y \cdot y\right) \cdot y\right)} \]

       2. unpow2 N/A

          \[\leadsto \frac{1}{6} \cdot \left(\color{blue}{{y}^{2}} \cdot y\right) \]

       3. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right) \cdot y} \]

       4. *-commutative N/A

          \[\leadsto \color{blue}{y \cdot \left(\frac{1}{6} \cdot {y}^{2}\right)} \]

       5. lower-*.f64 N/A

          \[\leadsto \color{blue}{y \cdot \left(\frac{1}{6} \cdot {y}^{2}\right)} \]

       6. lower-*.f64 N/A

          \[\leadsto y \cdot \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right)} \]

       7. unpow2 N/A

          \[\leadsto y \cdot \left(\frac{1}{6} \cdot \color{blue}{\left(y \cdot y\right)}\right) \]

       8. lower-*.f64 45.2

          \[\leadsto y \cdot \left(0.16666666666666666 \cdot \color{blue}{\left(y \cdot y\right)}\right) \]

   11. Applied rewrites 45.2%

       \[\leadsto \color{blue}{y \cdot \left(0.16666666666666666 \cdot \left(y \cdot y\right)\right)} \]

   if 9.49999999999999927e-251 < y < 6.2e7

   1. Initial program 79.6%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 92.7

         \[\leadsto y \cdot \frac{\color{blue}{\sin x}}{x} \]

   5. Applied rewrites 92.7%

      \[\leadsto \color{blue}{y \cdot \frac{\sin x}{x}} \]

   6. Taylor expanded in x around 0

      \[\leadsto y \cdot \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 57.4%

      \[\leadsto y \cdot \color{blue}{1} \]
   9. Step-by-step derivation

      1. *-rgt-identity 57.4

         \[\leadsto \color{blue}{y} \]

   10. Applied rewrites 57.4%

       \[\leadsto \color{blue}{y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 9.5 \cdot 10^{-251}:\\ \;\;\;\;y \cdot \left(\left(y \cdot y\right) \cdot 0.16666666666666666\right)\\ \mathbf{elif}\;y \leq 62000000:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(y \cdot y\right) \cdot 0.16666666666666666\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 56.6% accurate, 9.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 115000000:\\ \;\;\;\;\mathsf{fma}\left(0.16666666666666666, y \cdot \left(y \cdot y\right), y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(y \cdot y\right) \cdot 0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 115000000.0)
   (fma 0.16666666666666666 (* y (* y y)) y)
   (* y (* (* y y) 0.16666666666666666))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 115000000.0) {
		tmp = fma(0.16666666666666666, (y * (y * y)), y);
	} else {
		tmp = y * ((y * y) * 0.16666666666666666);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 115000000.0)
		tmp = fma(0.16666666666666666, Float64(y * Float64(y * y)), y);
	else
		tmp = Float64(y * Float64(Float64(y * y) * 0.16666666666666666));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 115000000.0], N[(0.16666666666666666 * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision], N[(y * N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.15e8

   1. Initial program 87.1%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{y \cdot \left(\sin x + \frac{1}{6} \cdot \left({y}^{2} \cdot \sin x\right)\right)}}{x} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{y \cdot \left(\sin x + \frac{1}{6} \cdot \color{blue}{\left(\sin x \cdot {y}^{2}\right)}\right)}{x} \]

      2. associate-*r* N/A

         \[\leadsto \frac{y \cdot \left(\sin x + \color{blue}{\left(\frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}}\right)}{x} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(\sin x + \left(\frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}\right)}}{x} \]

      4. associate-*r* N/A

         \[\leadsto \frac{y \cdot \left(\sin x + \color{blue}{\frac{1}{6} \cdot \left(\sin x \cdot {y}^{2}\right)}\right)}{x} \]

      5. *-commutative N/A

         \[\leadsto \frac{y \cdot \left(\sin x + \frac{1}{6} \cdot \color{blue}{\left({y}^{2} \cdot \sin x\right)}\right)}{x} \]

      6. *-lft-identity N/A

         \[\leadsto \frac{y \cdot \left(\color{blue}{1 \cdot \sin x} + \frac{1}{6} \cdot \left({y}^{2} \cdot \sin x\right)\right)}{x} \]

      7. associate-*r* N/A

         \[\leadsto \frac{y \cdot \left(1 \cdot \sin x + \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \sin x}\right)}{x} \]

      8. distribute-rgt-in N/A

         \[\leadsto \frac{y \cdot \color{blue}{\left(\sin x \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)}}{x} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{y \cdot \color{blue}{\left(\sin x \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)}}{x} \]

      10. lower-sin.f64 N/A

          \[\leadsto \frac{y \cdot \left(\color{blue}{\sin x} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)}{x} \]

      11. +-commutative N/A

          \[\leadsto \frac{y \cdot \left(\sin x \cdot \color{blue}{\left(\frac{1}{6} \cdot {y}^{2} + 1\right)}\right)}{x} \]

      12. unpow2 N/A

          \[\leadsto \frac{y \cdot \left(\sin x \cdot \left(\frac{1}{6} \cdot \color{blue}{\left(y \cdot y\right)} + 1\right)\right)}{x} \]

      13. associate-*r* N/A

          \[\leadsto \frac{y \cdot \left(\sin x \cdot \left(\color{blue}{\left(\frac{1}{6} \cdot y\right) \cdot y} + 1\right)\right)}{x} \]

      14. *-commutative N/A

          \[\leadsto \frac{y \cdot \left(\sin x \cdot \left(\color{blue}{y \cdot \left(\frac{1}{6} \cdot y\right)} + 1\right)\right)}{x} \]

      15. lower-fma.f64 N/A

          \[\leadsto \frac{y \cdot \left(\sin x \cdot \color{blue}{\mathsf{fma}\left(y, \frac{1}{6} \cdot y, 1\right)}\right)}{x} \]

      16. *-commutative N/A

          \[\leadsto \frac{y \cdot \left(\sin x \cdot \mathsf{fma}\left(y, \color{blue}{y \cdot \frac{1}{6}}, 1\right)\right)}{x} \]

      17. lower-*.f64 70.6

          \[\leadsto \frac{y \cdot \left(\sin x \cdot \mathsf{fma}\left(y, \color{blue}{y \cdot 0.16666666666666666}, 1\right)\right)}{x} \]

   5. Applied rewrites 70.6%

      \[\leadsto \frac{\color{blue}{y \cdot \left(\sin x \cdot \mathsf{fma}\left(y, y \cdot 0.16666666666666666, 1\right)\right)}}{x} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{y \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto y \cdot \color{blue}{\left(\frac{1}{6} \cdot {y}^{2} + 1\right)} \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right) \cdot y + 1 \cdot y} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\frac{1}{6} \cdot \left({y}^{2} \cdot y\right)} + 1 \cdot y \]

      4. unpow2 N/A

         \[\leadsto \frac{1}{6} \cdot \left(\color{blue}{\left(y \cdot y\right)} \cdot y\right) + 1 \cdot y \]

      5. unpow3 N/A

         \[\leadsto \frac{1}{6} \cdot \color{blue}{{y}^{3}} + 1 \cdot y \]

      6. *-lft-identity N/A

         \[\leadsto \frac{1}{6} \cdot {y}^{3} + \color{blue}{y} \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{6}, {y}^{3}, y\right)} \]

      8. cube-mult N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{6}, \color{blue}{y \cdot \left(y \cdot y\right)}, y\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{6}, y \cdot \color{blue}{{y}^{2}}, y\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{6}, \color{blue}{y \cdot {y}^{2}}, y\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{6}, y \cdot \color{blue}{\left(y \cdot y\right)}, y\right) \]

      12. lower-*.f64 60.6

          \[\leadsto \mathsf{fma}\left(0.16666666666666666, y \cdot \color{blue}{\left(y \cdot y\right)}, y\right) \]

   8. Applied rewrites 60.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.16666666666666666, y \cdot \left(y \cdot y\right), y\right)} \]

   if 1.15e8 < x

   1. Initial program 99.9%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{y \cdot \left(\sin x + \frac{1}{6} \cdot \left({y}^{2} \cdot \sin x\right)\right)}}{x} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{y \cdot \left(\sin x + \frac{1}{6} \cdot \color{blue}{\left(\sin x \cdot {y}^{2}\right)}\right)}{x} \]

      2. associate-*r* N/A

         \[\leadsto \frac{y \cdot \left(\sin x + \color{blue}{\left(\frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}}\right)}{x} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(\sin x + \left(\frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}\right)}}{x} \]

      4. associate-*r* N/A

         \[\leadsto \frac{y \cdot \left(\sin x + \color{blue}{\frac{1}{6} \cdot \left(\sin x \cdot {y}^{2}\right)}\right)}{x} \]

      5. *-commutative N/A

         \[\leadsto \frac{y \cdot \left(\sin x + \frac{1}{6} \cdot \color{blue}{\left({y}^{2} \cdot \sin x\right)}\right)}{x} \]

      6. *-lft-identity N/A

         \[\leadsto \frac{y \cdot \left(\color{blue}{1 \cdot \sin x} + \frac{1}{6} \cdot \left({y}^{2} \cdot \sin x\right)\right)}{x} \]

      7. associate-*r* N/A

         \[\leadsto \frac{y \cdot \left(1 \cdot \sin x + \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \sin x}\right)}{x} \]

      8. distribute-rgt-in N/A

         \[\leadsto \frac{y \cdot \color{blue}{\left(\sin x \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)}}{x} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{y \cdot \color{blue}{\left(\sin x \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)}}{x} \]

      10. lower-sin.f64 N/A

          \[\leadsto \frac{y \cdot \left(\color{blue}{\sin x} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)}{x} \]

      11. +-commutative N/A

          \[\leadsto \frac{y \cdot \left(\sin x \cdot \color{blue}{\left(\frac{1}{6} \cdot {y}^{2} + 1\right)}\right)}{x} \]

      12. unpow2 N/A

          \[\leadsto \frac{y \cdot \left(\sin x \cdot \left(\frac{1}{6} \cdot \color{blue}{\left(y \cdot y\right)} + 1\right)\right)}{x} \]

      13. associate-*r* N/A

          \[\leadsto \frac{y \cdot \left(\sin x \cdot \left(\color{blue}{\left(\frac{1}{6} \cdot y\right) \cdot y} + 1\right)\right)}{x} \]

      14. *-commutative N/A

          \[\leadsto \frac{y \cdot \left(\sin x \cdot \left(\color{blue}{y \cdot \left(\frac{1}{6} \cdot y\right)} + 1\right)\right)}{x} \]

      15. lower-fma.f64 N/A

          \[\leadsto \frac{y \cdot \left(\sin x \cdot \color{blue}{\mathsf{fma}\left(y, \frac{1}{6} \cdot y, 1\right)}\right)}{x} \]

      16. *-commutative N/A

          \[\leadsto \frac{y \cdot \left(\sin x \cdot \mathsf{fma}\left(y, \color{blue}{y \cdot \frac{1}{6}}, 1\right)\right)}{x} \]

      17. lower-*.f64 82.2

          \[\leadsto \frac{y \cdot \left(\sin x \cdot \mathsf{fma}\left(y, \color{blue}{y \cdot 0.16666666666666666}, 1\right)\right)}{x} \]

   5. Applied rewrites 82.2%

      \[\leadsto \frac{\color{blue}{y \cdot \left(\sin x \cdot \mathsf{fma}\left(y, y \cdot 0.16666666666666666, 1\right)\right)}}{x} \]

   6. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{1}{6} \cdot \frac{{y}^{3} \cdot \sin x}{x}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{{y}^{3} \cdot \sin x}{x} \cdot \frac{1}{6}} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\left({y}^{3} \cdot \frac{\sin x}{x}\right)} \cdot \frac{1}{6} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{{y}^{3} \cdot \left(\frac{\sin x}{x} \cdot \frac{1}{6}\right)} \]

      4. *-commutative N/A

         \[\leadsto {y}^{3} \cdot \color{blue}{\left(\frac{1}{6} \cdot \frac{\sin x}{x}\right)} \]

      5. cube-mult N/A

         \[\leadsto \color{blue}{\left(y \cdot \left(y \cdot y\right)\right)} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right) \]

      6. unpow2 N/A

         \[\leadsto \left(y \cdot \color{blue}{{y}^{2}}\right) \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right) \]

      7. associate-*l* N/A

         \[\leadsto \color{blue}{y \cdot \left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right)\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{y \cdot \left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right)\right)} \]

      9. *-commutative N/A

         \[\leadsto y \cdot \color{blue}{\left(\left(\frac{1}{6} \cdot \frac{\sin x}{x}\right) \cdot {y}^{2}\right)} \]

      10. unpow2 N/A

          \[\leadsto y \cdot \left(\left(\frac{1}{6} \cdot \frac{\sin x}{x}\right) \cdot \color{blue}{\left(y \cdot y\right)}\right) \]

      11. associate-*r* N/A

          \[\leadsto y \cdot \color{blue}{\left(\left(\left(\frac{1}{6} \cdot \frac{\sin x}{x}\right) \cdot y\right) \cdot y\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto y \cdot \color{blue}{\left(\left(\left(\frac{1}{6} \cdot \frac{\sin x}{x}\right) \cdot y\right) \cdot y\right)} \]

      13. associate-*r/ N/A

          \[\leadsto y \cdot \left(\left(\color{blue}{\frac{\frac{1}{6} \cdot \sin x}{x}} \cdot y\right) \cdot y\right) \]

      14. associate-*l/ N/A

          \[\leadsto y \cdot \left(\color{blue}{\frac{\left(\frac{1}{6} \cdot \sin x\right) \cdot y}{x}} \cdot y\right) \]

      15. lower-/.f64 N/A

          \[\leadsto y \cdot \left(\color{blue}{\frac{\left(\frac{1}{6} \cdot \sin x\right) \cdot y}{x}} \cdot y\right) \]

      16. lower-*.f64 N/A

          \[\leadsto y \cdot \left(\frac{\color{blue}{\left(\frac{1}{6} \cdot \sin x\right) \cdot y}}{x} \cdot y\right) \]

      17. *-commutative N/A

          \[\leadsto y \cdot \left(\frac{\color{blue}{\left(\sin x \cdot \frac{1}{6}\right)} \cdot y}{x} \cdot y\right) \]

      18. lower-*.f64 N/A

          \[\leadsto y \cdot \left(\frac{\color{blue}{\left(\sin x \cdot \frac{1}{6}\right)} \cdot y}{x} \cdot y\right) \]

      19. lower-sin.f64 57.8

          \[\leadsto y \cdot \left(\frac{\left(\color{blue}{\sin x} \cdot 0.16666666666666666\right) \cdot y}{x} \cdot y\right) \]

   8. Applied rewrites 57.8%

      \[\leadsto \color{blue}{y \cdot \left(\frac{\left(\sin x \cdot 0.16666666666666666\right) \cdot y}{x} \cdot y\right)} \]

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{6} \cdot {y}^{3}} \]
   10. Step-by-step derivation

       1. unpow3 N/A

          \[\leadsto \frac{1}{6} \cdot \color{blue}{\left(\left(y \cdot y\right) \cdot y\right)} \]

       2. unpow2 N/A

          \[\leadsto \frac{1}{6} \cdot \left(\color{blue}{{y}^{2}} \cdot y\right) \]

       3. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right) \cdot y} \]

       4. *-commutative N/A

          \[\leadsto \color{blue}{y \cdot \left(\frac{1}{6} \cdot {y}^{2}\right)} \]

       5. lower-*.f64 N/A

          \[\leadsto \color{blue}{y \cdot \left(\frac{1}{6} \cdot {y}^{2}\right)} \]

       6. lower-*.f64 N/A

          \[\leadsto y \cdot \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right)} \]

       7. unpow2 N/A

          \[\leadsto y \cdot \left(\frac{1}{6} \cdot \color{blue}{\left(y \cdot y\right)}\right) \]

       8. lower-*.f64 50.0

          \[\leadsto y \cdot \left(0.16666666666666666 \cdot \color{blue}{\left(y \cdot y\right)}\right) \]

   11. Applied rewrites 50.0%

       \[\leadsto \color{blue}{y \cdot \left(0.16666666666666666 \cdot \left(y \cdot y\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 115000000:\\ \;\;\;\;\mathsf{fma}\left(0.16666666666666666, y \cdot \left(y \cdot y\right), y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(y \cdot y\right) \cdot 0.16666666666666666\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 27.5% accurate, 217.0× speedup

Math

\[\begin{array}{l} \\ y \end{array} \]

FPCore

(FPCore (x y) :precision binary64 y)

C

double code(double x, double y) {
	return y;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = y
end function

Java

public static double code(double x, double y) {
	return y;
}

Python

def code(x, y):
	return y

Julia

function code(x, y)
	return y
end

MATLAB

function tmp = code(x, y)
	tmp = y;
end

Wolfram

code[x_, y_] := y

Derivation

1. Initial program 90.0%

   \[\frac{\sin x \cdot \sinh y}{x} \]
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 52.3

      \[\leadsto y \cdot \frac{\color{blue}{\sin x}}{x} \]

5. Applied rewrites 52.3%

   \[\leadsto \color{blue}{y \cdot \frac{\sin x}{x}} \]

6. Taylor expanded in x around 0

   \[\leadsto y \cdot \color{blue}{1} \]
7. Step-by-step derivation

8. Applied rewrites 28.1%

   \[\leadsto y \cdot \color{blue}{1} \]
9. Step-by-step derivation

   1. *-rgt-identity 28.1

      \[\leadsto \color{blue}{y} \]

10. Applied rewrites 28.1%

    \[\leadsto \color{blue}{y} \]
11. 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 2024216 
(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))