Linear.Quaternion:$ccosh from linear-1.19.1.3

Percentage Accurate: 89.2% → 99.8%

Time: 13.8s

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 86.0%

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

Alternative 2: 74.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y \cdot \sin x}{x}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\frac{\sinh y \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot -0.16666666666666666\right)\right)}{x}\\ \mathbf{elif}\;t\_0 \leq 0.005:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, y \cdot \mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), y\right) \cdot \frac{\sin x}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sinh y \cdot \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.008333333333333333, -0.16666666666666666\right), x \cdot \left(x \cdot x\right), x\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sinh y) (sin x)) x)))
   (if (<= t_0 (- INFINITY))
     (/ (* (sinh y) (* (* x x) (* x -0.16666666666666666))) x)
     (if (<= t_0 0.005)
       (*
        (fma
         (* y y)
         (* y (fma (* y y) 0.008333333333333333 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) * ((x * x) * (x * -0.16666666666666666))) / x;
	} else if (t_0 <= 0.005) {
		tmp = fma((y * y), (y * fma((y * y), 0.008333333333333333, 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) * Float64(Float64(x * x) * Float64(x * -0.16666666666666666))) / x);
	elseif (t_0 <= 0.005)
		tmp = Float64(fma(Float64(y * y), Float64(y * fma(Float64(y * y), 0.008333333333333333, 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[(N[(x * x), $MachinePrecision] * N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[t$95$0, 0.005], N[(N[(N[(y * y), $MachinePrecision] * N[(y * N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $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 69.1

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

   5. Applied rewrites 69.1%

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

   6. Taylor expanded in x around inf

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

      1. unpow3 N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 12.7

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

   8. Applied rewrites 12.7%

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

   if -inf.0 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 0.0050000000000000001

   1. Initial program 72.1%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 71.6

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

   5. Applied rewrites 71.6%

      \[\leadsto \frac{\sin x \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)}}{x} \]
   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 99.4

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

   7. Applied rewrites 99.4%

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

   if 0.0050000000000000001 < (/.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 75.3

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

   5. Applied rewrites 75.3%

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

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -\infty:\\ \;\;\;\;\frac{\sinh y \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot -0.16666666666666666\right)\right)}{x}\\ \mathbf{elif}\;\frac{\sinh y \cdot \sin x}{x} \leq 0.005:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, y \cdot \mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), y\right) \cdot \frac{\sin x}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sinh y \cdot \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.008333333333333333, -0.16666666666666666\right), x \cdot \left(x \cdot x\right), x\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 71.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y \cdot \sin x}{x}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\frac{\sinh y \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot -0.16666666666666666\right)\right)}{x}\\ \mathbf{elif}\;t\_0 \leq 10^{-66}:\\ \;\;\;\;\frac{\sin x}{x} \cdot \mathsf{fma}\left(y \cdot y, y \cdot 0.16666666666666666, y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right), x \cdot \left(x \cdot x\right), x\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sinh y) (sin x)) x)))
   (if (<= t_0 (- INFINITY))
     (/ (* (sinh y) (* (* x x) (* x -0.16666666666666666))) x)
     (if (<= t_0 1e-66)
       (* (/ (sin x) x) (fma (* y y) (* y 0.16666666666666666) y))
       (/
        (*
         (fma
          (fma 0.008333333333333333 (* x x) -0.16666666666666666)
          (* x (* x x))
          x)
         (fma
          (fma y (* y 0.008333333333333333) 0.16666666666666666)
          (* y (* y y))
          y))
        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) * ((x * x) * (x * -0.16666666666666666))) / x;
	} else if (t_0 <= 1e-66) {
		tmp = (sin(x) / x) * fma((y * y), (y * 0.16666666666666666), y);
	} else {
		tmp = (fma(fma(0.008333333333333333, (x * x), -0.16666666666666666), (x * (x * x)), x) * fma(fma(y, (y * 0.008333333333333333), 0.16666666666666666), (y * (y * y)), y)) / 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) * Float64(Float64(x * x) * Float64(x * -0.16666666666666666))) / x);
	elseif (t_0 <= 1e-66)
		tmp = Float64(Float64(sin(x) / x) * fma(Float64(y * y), Float64(y * 0.16666666666666666), y));
	else
		tmp = Float64(Float64(fma(fma(0.008333333333333333, Float64(x * x), -0.16666666666666666), Float64(x * Float64(x * x)), x) * fma(fma(y, Float64(y * 0.008333333333333333), 0.16666666666666666), Float64(y * Float64(y * y)), 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, (-Infinity)], N[(N[(N[Sinh[y], $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[t$95$0, 1e-66], 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[(0.008333333333333333 * N[(x * x), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] * N[(N[(y * N[(y * 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision] + y), $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 69.1

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

   5. Applied rewrites 69.1%

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

   6. Taylor expanded in x around inf

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

      1. unpow3 N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 12.7

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

   8. Applied rewrites 12.7%

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

   if -inf.0 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 9.9999999999999998e-67

   1. Initial program 70.7%

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

   3. Taylor expanded in y around 0

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

      1. +-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 98.9%

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

   if 9.9999999999999998e-67 < (/.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 83.4

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

   5. Applied rewrites 83.4%

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

   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. *-lft-identity N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. unpow2 N/A

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

      7. unpow3 N/A

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

      8. lower-fma.f64 N/A

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. cube-mult N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 68.0

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

   8. Applied rewrites 68.0%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -\infty:\\ \;\;\;\;\frac{\sinh y \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot -0.16666666666666666\right)\right)}{x}\\ \mathbf{elif}\;\frac{\sinh y \cdot \sin x}{x} \leq 10^{-66}:\\ \;\;\;\;\frac{\sin x}{x} \cdot \mathsf{fma}\left(y \cdot y, y \cdot 0.16666666666666666, y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right), x \cdot \left(x \cdot x\right), x\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 71.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y \cdot \sin x}{x}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\frac{\sinh y \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot -0.16666666666666666\right)\right)}{x}\\ \mathbf{elif}\;t\_0 \leq 10^{-66}:\\ \;\;\;\;y \cdot \frac{\sin x}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right), x \cdot \left(x \cdot x\right), x\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sinh y) (sin x)) x)))
   (if (<= t_0 (- INFINITY))
     (/ (* (sinh y) (* (* x x) (* x -0.16666666666666666))) x)
     (if (<= t_0 1e-66)
       (* y (/ (sin x) x))
       (/
        (*
         (fma
          (fma 0.008333333333333333 (* x x) -0.16666666666666666)
          (* x (* x x))
          x)
         (fma
          (fma y (* y 0.008333333333333333) 0.16666666666666666)
          (* y (* y y))
          y))
        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) * ((x * x) * (x * -0.16666666666666666))) / x;
	} else if (t_0 <= 1e-66) {
		tmp = y * (sin(x) / x);
	} else {
		tmp = (fma(fma(0.008333333333333333, (x * x), -0.16666666666666666), (x * (x * x)), x) * fma(fma(y, (y * 0.008333333333333333), 0.16666666666666666), (y * (y * y)), y)) / 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) * Float64(Float64(x * x) * Float64(x * -0.16666666666666666))) / x);
	elseif (t_0 <= 1e-66)
		tmp = Float64(y * Float64(sin(x) / x));
	else
		tmp = Float64(Float64(fma(fma(0.008333333333333333, Float64(x * x), -0.16666666666666666), Float64(x * Float64(x * x)), x) * fma(fma(y, Float64(y * 0.008333333333333333), 0.16666666666666666), Float64(y * Float64(y * y)), 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, (-Infinity)], N[(N[(N[Sinh[y], $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[t$95$0, 1e-66], N[(y * N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.008333333333333333 * N[(x * x), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] * N[(N[(y * N[(y * 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision] + y), $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 69.1

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

   5. Applied rewrites 69.1%

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

   6. Taylor expanded in x around inf

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

      1. unpow3 N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 12.7

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

   8. Applied rewrites 12.7%

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

   if -inf.0 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 9.9999999999999998e-67

   1. Initial program 70.7%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 98.3

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

   5. Applied rewrites 98.3%

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

   if 9.9999999999999998e-67 < (/.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 83.4

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

   5. Applied rewrites 83.4%

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

   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. *-lft-identity N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. unpow2 N/A

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

      7. unpow3 N/A

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

      8. lower-fma.f64 N/A

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. cube-mult N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 68.0

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

   8. Applied rewrites 68.0%

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

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -\infty:\\ \;\;\;\;\frac{\sinh y \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot -0.16666666666666666\right)\right)}{x}\\ \mathbf{elif}\;\frac{\sinh y \cdot \sin x}{x} \leq 10^{-66}:\\ \;\;\;\;y \cdot \frac{\sin x}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right), x \cdot \left(x \cdot x\right), x\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 82.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y \cdot \sin x}{x}\\ t_1 := x \cdot \left(x \cdot x\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), t\_1, x\right)}{\frac{x}{\mathsf{fma}\left(y \cdot y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y\right)}}\\ \mathbf{elif}\;t\_0 \leq 10^{-66}:\\ \;\;\;\;y \cdot \frac{\sin x}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right), t\_1, x\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sinh y) (sin x)) x)) (t_1 (* x (* x x))))
   (if (<= t_0 (- INFINITY))
     (/
      (fma
       (fma
        (* x x)
        (fma (* x x) -0.0001984126984126984 0.008333333333333333)
        -0.16666666666666666)
       t_1
       x)
      (/
       x
       (fma
        (* y y)
        (*
         y
         (fma
          y
          (* y (fma y (* y 0.0001984126984126984) 0.008333333333333333))
          0.16666666666666666))
        y)))
     (if (<= t_0 1e-66)
       (* y (/ (sin x) x))
       (/
        (*
         (fma (fma 0.008333333333333333 (* x x) -0.16666666666666666) t_1 x)
         (fma
          (fma y (* y 0.008333333333333333) 0.16666666666666666)
          (* y (* y y))
          y))
        x)))))

C

double code(double x, double y) {
	double t_0 = (sinh(y) * sin(x)) / x;
	double t_1 = x * (x * x);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma(fma((x * x), fma((x * x), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666), t_1, x) / (x / fma((y * y), (y * fma(y, (y * fma(y, (y * 0.0001984126984126984), 0.008333333333333333)), 0.16666666666666666)), y));
	} else if (t_0 <= 1e-66) {
		tmp = y * (sin(x) / x);
	} else {
		tmp = (fma(fma(0.008333333333333333, (x * x), -0.16666666666666666), t_1, x) * fma(fma(y, (y * 0.008333333333333333), 0.16666666666666666), (y * (y * y)), y)) / x;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(sinh(y) * sin(x)) / x)
	t_1 = Float64(x * Float64(x * x))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(fma(fma(Float64(x * x), fma(Float64(x * x), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666), t_1, x) / Float64(x / fma(Float64(y * y), Float64(y * fma(y, Float64(y * fma(y, Float64(y * 0.0001984126984126984), 0.008333333333333333)), 0.16666666666666666)), y)));
	elseif (t_0 <= 1e-66)
		tmp = Float64(y * Float64(sin(x) / x));
	else
		tmp = Float64(Float64(fma(fma(0.008333333333333333, Float64(x * x), -0.16666666666666666), t_1, x) * fma(fma(y, Float64(y * 0.008333333333333333), 0.16666666666666666), Float64(y * Float64(y * y)), 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]}, Block[{t$95$1 = N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * t$95$1 + x), $MachinePrecision] / N[(x / N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * N[(y * N[(y * N[(y * 0.0001984126984126984), $MachinePrecision] + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e-66], N[(y * N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.008333333333333333 * N[(x * x), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * t$95$1 + x), $MachinePrecision] * N[(N[(y * N[(y * 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision] + y), $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. 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

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. unpow2 N/A

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

      6. unpow3 N/A

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

      7. *-lft-identity N/A

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

      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 87.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. *-commutative N/A

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

   9. Applied rewrites 87.9%

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

   10. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. distribute-rgt-in N/A

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

       3. *-commutative N/A

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

       4. associate-*l* N/A

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

       5. unpow2 N/A

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

       6. unpow3 N/A

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

       7. *-lft-identity N/A

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

       8. lower-fma.f64 N/A

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

   12. Applied rewrites 64.0%

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

   if -inf.0 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 9.9999999999999998e-67

   1. Initial program 70.7%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 98.3

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

   5. Applied rewrites 98.3%

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

   if 9.9999999999999998e-67 < (/.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 83.4

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

   5. Applied rewrites 83.4%

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

   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. *-lft-identity N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. unpow2 N/A

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

      7. unpow3 N/A

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

      8. lower-fma.f64 N/A

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. cube-mult N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 68.0

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

   8. Applied rewrites 68.0%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -\infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), x \cdot \left(x \cdot x\right), x\right)}{\frac{x}{\mathsf{fma}\left(y \cdot y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y\right)}}\\ \mathbf{elif}\;\frac{\sinh y \cdot \sin x}{x} \leq 10^{-66}:\\ \;\;\;\;y \cdot \frac{\sin x}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right), x \cdot \left(x \cdot x\right), x\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 38.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y \cdot \sin x}{x}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-162}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(y \cdot \left(x \cdot x\right)\right)\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot \left(y \cdot \left(y \cdot y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sinh y) (sin x)) x)))
   (if (<= t_0 -2e-162)
     (* -0.16666666666666666 (* y (* x x)))
     (if (<= t_0 1.0) y (* 0.16666666666666666 (* y (* y y)))))))

C

double code(double x, double y) {
	double t_0 = (sinh(y) * sin(x)) / x;
	double tmp;
	if (t_0 <= -2e-162) {
		tmp = -0.16666666666666666 * (y * (x * x));
	} else if (t_0 <= 1.0) {
		tmp = y;
	} else {
		tmp = 0.16666666666666666 * (y * (y * y));
	}
	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 = (sinh(y) * sin(x)) / x
    if (t_0 <= (-2d-162)) then
        tmp = (-0.16666666666666666d0) * (y * (x * x))
    else if (t_0 <= 1.0d0) then
        tmp = y
    else
        tmp = 0.16666666666666666d0 * (y * (y * y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (Math.sinh(y) * Math.sin(x)) / x;
	double tmp;
	if (t_0 <= -2e-162) {
		tmp = -0.16666666666666666 * (y * (x * x));
	} else if (t_0 <= 1.0) {
		tmp = y;
	} else {
		tmp = 0.16666666666666666 * (y * (y * y));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (math.sinh(y) * math.sin(x)) / x
	tmp = 0
	if t_0 <= -2e-162:
		tmp = -0.16666666666666666 * (y * (x * x))
	elif t_0 <= 1.0:
		tmp = y
	else:
		tmp = 0.16666666666666666 * (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 <= -2e-162)
		tmp = Float64(-0.16666666666666666 * Float64(y * Float64(x * x)));
	elseif (t_0 <= 1.0)
		tmp = y;
	else
		tmp = Float64(0.16666666666666666 * Float64(y * Float64(y * y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (sinh(y) * sin(x)) / x;
	tmp = 0.0;
	if (t_0 <= -2e-162)
		tmp = -0.16666666666666666 * (y * (x * x));
	elseif (t_0 <= 1.0)
		tmp = y;
	else
		tmp = 0.16666666666666666 * (y * (y * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sinh[y], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-162], N[(-0.16666666666666666 * N[(y * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1.0], y, N[(0.16666666666666666 * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -1.99999999999999991e-162

   1. Initial program 98.9%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 31.0

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

   5. Applied rewrites 31.0%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. unpow3 N/A

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

      6. *-lft-identity N/A

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

      7. lower-fma.f64 N/A

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

      8. cube-mult N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 30.1

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

   8. Applied rewrites 30.1%

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

   9. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. cube-mult N/A

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

       3. unpow2 N/A

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

       4. associate-*r* N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 7.1

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

   11. Applied rewrites 7.1%

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

   12. Taylor expanded in y around 0

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

       1. lower-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 N/A

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

       4. unpow2 N/A

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

       5. lower-*.f64 7.1

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

   14. Applied rewrites 7.1%

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

   if -1.99999999999999991e-162 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 1

   1. Initial program 67.4%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 97.5

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

   5. Applied rewrites 97.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 54.1%

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

      1. *-rgt-identity 54.1

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

   10. Applied rewrites 54.1%

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

   if 1 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-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 66.8%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 53.9

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

   8. Applied rewrites 53.9%

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

   9. Taylor expanded in y around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. lower-*.f64 N/A

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

       5. cube-mult N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 N/A

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

       10. +-commutative N/A

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

       11. distribute-rgt-in N/A

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

       12. *-commutative N/A

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

       13. associate-*l* N/A

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

       14. metadata-eval N/A

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

   11. Applied rewrites 53.9%

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

   12. Taylor expanded in x around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. cube-mult N/A

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

       4. unpow2 N/A

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

       5. lower-*.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 47.5

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

   14. Applied rewrites 47.5%

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

4. Final simplification 38.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -2 \cdot 10^{-162}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(y \cdot \left(x \cdot x\right)\right)\\ \mathbf{elif}\;\frac{\sinh y \cdot \sin x}{x} \leq 1:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot \left(y \cdot \left(y \cdot y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 90.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq 10^{-66}:\\ \;\;\;\;\sin x \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sinh y \cdot \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.008333333333333333, -0.16666666666666666\right), x \cdot \left(x \cdot x\right), x\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (sinh y) (sin x)) x) 1e-66)
   (*
    (sin x)
    (/
     (fma
      (fma
       (* y y)
       (fma (* y y) 0.0001984126984126984 0.008333333333333333)
       0.16666666666666666)
      (* y (* y 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 tmp;
	if (((sinh(y) * sin(x)) / x) <= 1e-66) {
		tmp = sin(x) * (fma(fma((y * y), fma((y * y), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666), (y * (y * 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)
	tmp = 0.0
	if (Float64(Float64(sinh(y) * sin(x)) / x) <= 1e-66)
		tmp = Float64(sin(x) * Float64(fma(fma(Float64(y * y), fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666), Float64(y * Float64(y * 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_] := If[LessEqual[N[(N[(N[Sinh[y], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], 1e-66], N[(N[Sin[x], $MachinePrecision] * N[(N[(N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision] + 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 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 9.9999999999999998e-67

   1. Initial program 79.8%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 99.9

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

   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

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. unpow2 N/A

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

      6. unpow3 N/A

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

      7. *-lft-identity N/A

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

      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 96.0%

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

   if 9.9999999999999998e-67 < (/.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 77.2

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

   5. Applied rewrites 77.2%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq 10^{-66}:\\ \;\;\;\;\sin x \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sinh y \cdot \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.008333333333333333, -0.16666666666666666\right), x \cdot \left(x \cdot x\right), x\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 59.9% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 1

   1. Initial program 80.6%

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

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. unpow2 N/A

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

      6. unpow3 N/A

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

      7. *-lft-identity N/A

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

      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 95.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. *-commutative N/A

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

   9. Applied rewrites 95.1%

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

   10. Taylor expanded in x around 0

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

       1. distribute-rgt-in N/A

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

       2. *-lft-identity N/A

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

       3. associate-*r* N/A

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

       4. unpow2 N/A

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

       5. unpow3 N/A

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

       6. +-commutative N/A

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

       7. lower-fma.f64 N/A

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

       8. cube-mult N/A

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

       9. unpow2 N/A

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

       10. lower-*.f64 N/A

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

       11. unpow2 N/A

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

       12. lower-*.f64 60.1

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

   12. Applied rewrites 60.1%

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

   if 1 < (/.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 82.6

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

   5. Applied rewrites 82.6%

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

   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. *-lft-identity N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. unpow2 N/A

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

      7. unpow3 N/A

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

      8. lower-fma.f64 N/A

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. cube-mult N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 65.8

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

   8. Applied rewrites 65.8%

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

4. Final simplification 61.7%

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

Alternative 9: 58.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot y\right)\\ t_1 := \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right)\\ \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -1 \cdot 10^{-160}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_1, t\_0, y\right) \cdot \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot -0.16666666666666666, x\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, t\_1, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y y)))
        (t_1
         (fma
          (* y y)
          (fma (* y y) 0.0001984126984126984 0.008333333333333333)
          0.16666666666666666)))
   (if (<= (/ (* (sinh y) (sin x)) x) -1e-160)
     (/ (* (fma t_1 t_0 y) (fma x (* (* x x) -0.16666666666666666) x)) x)
     (fma t_0 t_1 y))))

C

double code(double x, double y) {
	double t_0 = y * (y * y);
	double t_1 = fma((y * y), fma((y * y), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666);
	double tmp;
	if (((sinh(y) * sin(x)) / x) <= -1e-160) {
		tmp = (fma(t_1, t_0, y) * fma(x, ((x * x) * -0.16666666666666666), x)) / x;
	} else {
		tmp = fma(t_0, t_1, y);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(y * Float64(y * y))
	t_1 = fma(Float64(y * y), fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666)
	tmp = 0.0
	if (Float64(Float64(sinh(y) * sin(x)) / x) <= -1e-160)
		tmp = Float64(Float64(fma(t_1, t_0, y) * fma(x, Float64(Float64(x * x) * -0.16666666666666666), x)) / x);
	else
		tmp = fma(t_0, t_1, 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[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sinh[y], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], -1e-160], N[(N[(N[(t$95$1 * t$95$0 + y), $MachinePrecision] * N[(x * N[(N[(x * x), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(t$95$0 * t$95$1 + y), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -9.9999999999999999e-161

   1. Initial program 98.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)} \cdot \sinh y}{x} \]
   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 71.7

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

   5. Applied rewrites 71.7%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. unpow2 N/A

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

      6. unpow3 N/A

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

      7. *-lft-identity N/A

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

      8. lower-fma.f64 N/A

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

   8. Applied rewrites 68.0%

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

   if -9.9999999999999999e-161 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 80.6%

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

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. unpow2 N/A

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

      6. unpow3 N/A

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

      7. *-lft-identity N/A

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

      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 95.7%

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

   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. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 58.4

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

   10. Applied rewrites 58.4%

       \[\leadsto \color{blue}{\mathsf{fma}\left(y \cdot \left(y \cdot y\right), \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -1 \cdot 10^{-160}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right) \cdot \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot -0.16666666666666666, x\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot \left(y \cdot y\right), \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 56.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -1 \cdot 10^{-160}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, y \cdot 0.16666666666666666, y\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot \left(y \cdot y\right), \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (sinh y) (sin x)) x) -1e-160)
   (*
    (fma (* y y) (* y 0.16666666666666666) y)
    (fma
     (* x x)
     (fma
      (* x x)
      (fma (* x x) -0.0001984126984126984 0.008333333333333333)
      -0.16666666666666666)
     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 tmp;
	if (((sinh(y) * sin(x)) / x) <= -1e-160) {
		tmp = fma((y * y), (y * 0.16666666666666666), y) * fma((x * x), fma((x * x), fma((x * x), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666), 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)
	tmp = 0.0
	if (Float64(Float64(sinh(y) * sin(x)) / x) <= -1e-160)
		tmp = Float64(fma(Float64(y * y), Float64(y * 0.16666666666666666), y) * fma(Float64(x * x), fma(Float64(x * x), fma(Float64(x * x), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666), 1.0));
	else
		tmp = fma(Float64(y * Float64(y * y)), fma(Float64(y * y), fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666), y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[Sinh[y], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], -1e-160], N[(N[(N[(y * y), $MachinePrecision] * N[(y * 0.16666666666666666), $MachinePrecision] + y), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $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]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -9.9999999999999999e-161

   1. Initial program 98.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*r/ N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. associate-*r* N/A

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

      11. associate-/l* N/A

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

      12. distribute-rgt-out N/A

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

      13. *-rgt-identity N/A

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

      14. distribute-lft-in N/A

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

   5. Applied rewrites 74.3%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 59.0

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

   8. Applied rewrites 59.0%

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

   if -9.9999999999999999e-161 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 80.6%

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

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. unpow2 N/A

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

      6. unpow3 N/A

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

      7. *-lft-identity N/A

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

      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 95.7%

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

   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. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 58.4

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

   10. Applied rewrites 58.4%

       \[\leadsto \color{blue}{\mathsf{fma}\left(y \cdot \left(y \cdot y\right), \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -1 \cdot 10^{-160}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, y \cdot 0.16666666666666666, y\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot \left(y \cdot y\right), \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 48.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -1 \cdot 10^{-160}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot \left(y \cdot y\right), \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (sinh y) (sin x)) x) -1e-160)
   (*
    y
    (fma
     (* x x)
     (fma
      (* x x)
      (fma (* x x) -0.0001984126984126984 0.008333333333333333)
      -0.16666666666666666)
     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 tmp;
	if (((sinh(y) * sin(x)) / x) <= -1e-160) {
		tmp = y * fma((x * x), fma((x * x), fma((x * x), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666), 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)
	tmp = 0.0
	if (Float64(Float64(sinh(y) * sin(x)) / x) <= -1e-160)
		tmp = Float64(y * fma(Float64(x * x), fma(Float64(x * x), fma(Float64(x * x), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666), 1.0));
	else
		tmp = fma(Float64(y * Float64(y * y)), fma(Float64(y * y), fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666), y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[Sinh[y], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], -1e-160], N[(y * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $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]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -9.9999999999999999e-161

   1. Initial program 98.9%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 30.1

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

   5. Applied rewrites 30.1%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 32.9

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

   8. Applied rewrites 32.9%

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

   if -9.9999999999999999e-161 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 80.6%

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

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. unpow2 N/A

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

      6. unpow3 N/A

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

      7. *-lft-identity N/A

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

      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 95.7%

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

   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. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 58.4

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

   10. Applied rewrites 58.4%

       \[\leadsto \color{blue}{\mathsf{fma}\left(y \cdot \left(y \cdot y\right), \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -1 \cdot 10^{-160}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot \left(y \cdot y\right), \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 47.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -1 \cdot 10^{-160}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, y \cdot \mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (sinh y) (sin x)) x) -1e-160)
   (*
    y
    (fma
     (* x x)
     (fma
      (* x x)
      (fma (* x x) -0.0001984126984126984 0.008333333333333333)
      -0.16666666666666666)
     1.0))
   (fma
    (* y y)
    (* y (fma (* y y) 0.008333333333333333 0.16666666666666666))
    y)))

C

double code(double x, double y) {
	double tmp;
	if (((sinh(y) * sin(x)) / x) <= -1e-160) {
		tmp = y * fma((x * x), fma((x * x), fma((x * x), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666), 1.0);
	} else {
		tmp = fma((y * y), (y * fma((y * y), 0.008333333333333333, 0.16666666666666666)), y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(sinh(y) * sin(x)) / x) <= -1e-160)
		tmp = Float64(y * fma(Float64(x * x), fma(Float64(x * x), fma(Float64(x * x), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666), 1.0));
	else
		tmp = fma(Float64(y * y), Float64(y * fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666)), y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[Sinh[y], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], -1e-160], N[(y * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(y * y), $MachinePrecision] * N[(y * N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -9.9999999999999999e-161

   1. Initial program 98.9%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 30.1

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

   5. Applied rewrites 30.1%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 32.9

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

   8. Applied rewrites 32.9%

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

   if -9.9999999999999999e-161 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 80.6%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 73.0

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

   5. Applied rewrites 73.0%

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

   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. unpow3 N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 57.8

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

   8. Applied rewrites 57.8%

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

4. Final simplification 50.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -1 \cdot 10^{-160}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, y \cdot \mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 49.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -1 \cdot 10^{-299}:\\ \;\;\;\;\left(y \cdot \left(y \cdot y\right)\right) \cdot \mathsf{fma}\left(x \cdot x, -0.027777777777777776, 0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, y \cdot \mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (sinh y) (sin x)) x) -1e-299)
   (* (* y (* y y)) (fma (* x x) -0.027777777777777776 0.16666666666666666))
   (fma
    (* y y)
    (* y (fma (* y y) 0.008333333333333333 0.16666666666666666))
    y)))

C

double code(double x, double y) {
	double tmp;
	if (((sinh(y) * sin(x)) / x) <= -1e-299) {
		tmp = (y * (y * y)) * fma((x * x), -0.027777777777777776, 0.16666666666666666);
	} else {
		tmp = fma((y * y), (y * fma((y * y), 0.008333333333333333, 0.16666666666666666)), y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(sinh(y) * sin(x)) / x) <= -1e-299)
		tmp = Float64(Float64(y * Float64(y * y)) * fma(Float64(x * x), -0.027777777777777776, 0.16666666666666666));
	else
		tmp = fma(Float64(y * y), Float64(y * fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666)), y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[Sinh[y], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], -1e-299], N[(N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.027777777777777776 + 0.16666666666666666), $MachinePrecision]), $MachinePrecision], N[(N[(y * y), $MachinePrecision] * N[(y * N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -9.99999999999999992e-300

   1. Initial program 98.3%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*r/ N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. associate-*r* N/A

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

      11. associate-/l* N/A

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

      12. distribute-rgt-out N/A

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

      13. *-rgt-identity N/A

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

      14. distribute-lft-in N/A

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

   5. Applied rewrites 78.5%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 55.5

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

   8. Applied rewrites 55.5%

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

   9. Taylor expanded in y around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. lower-*.f64 N/A

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

       5. cube-mult N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 N/A

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

       10. +-commutative N/A

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

       11. distribute-rgt-in N/A

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

       12. *-commutative N/A

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

       13. associate-*l* N/A

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

       14. metadata-eval N/A

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

   11. Applied rewrites 32.2%

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

   12. Taylor expanded in x around 0

       \[\leadsto \color{blue}{\frac{-1}{36} \cdot \left({x}^{2} \cdot {y}^{3}\right) + \frac{1}{6} \cdot {y}^{3}} \]
   13. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \color{blue}{\frac{1}{6} \cdot {y}^{3} + \frac{-1}{36} \cdot \left({x}^{2} \cdot {y}^{3}\right)} \]

       2. associate-*r* N/A

          \[\leadsto \frac{1}{6} \cdot {y}^{3} + \color{blue}{\left(\frac{-1}{36} \cdot {x}^{2}\right) \cdot {y}^{3}} \]

       3. distribute-rgt-out N/A

          \[\leadsto \color{blue}{{y}^{3} \cdot \left(\frac{1}{6} + \frac{-1}{36} \cdot {x}^{2}\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto \color{blue}{{y}^{3} \cdot \left(\frac{1}{6} + \frac{-1}{36} \cdot {x}^{2}\right)} \]

       5. cube-mult N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 N/A

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

       10. +-commutative N/A

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

       11. *-commutative N/A

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

       12. lower-fma.f64 N/A

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

       13. unpow2 N/A

           \[\leadsto \left(y \cdot \left(y \cdot y\right)\right) \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{36}, \frac{1}{6}\right) \]

       14. lower-*.f64 32.5

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

   14. Applied rewrites 32.5%

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

   if -9.99999999999999992e-300 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 79.3%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 71.0

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

   5. Applied rewrites 71.0%

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

   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. unpow3 N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 59.6

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

   8. Applied rewrites 59.6%

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

4. Final simplification 50.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -1 \cdot 10^{-299}:\\ \;\;\;\;\left(y \cdot \left(y \cdot y\right)\right) \cdot \mathsf{fma}\left(x \cdot x, -0.027777777777777776, 0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, y \cdot \mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 46.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -1 \cdot 10^{-299}:\\ \;\;\;\;\left(y \cdot \left(y \cdot y\right)\right) \cdot \mathsf{fma}\left(x \cdot x, -0.027777777777777776, 0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \left(y \cdot y\right) \cdot 0.16666666666666666, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (sinh y) (sin x)) x) -1e-299)
   (* (* y (* y y)) (fma (* x x) -0.027777777777777776 0.16666666666666666))
   (fma y (* (* y y) 0.16666666666666666) y)))

C

double code(double x, double y) {
	double tmp;
	if (((sinh(y) * sin(x)) / x) <= -1e-299) {
		tmp = (y * (y * y)) * fma((x * x), -0.027777777777777776, 0.16666666666666666);
	} else {
		tmp = fma(y, ((y * y) * 0.16666666666666666), y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(sinh(y) * sin(x)) / x) <= -1e-299)
		tmp = Float64(Float64(y * Float64(y * y)) * fma(Float64(x * x), -0.027777777777777776, 0.16666666666666666));
	else
		tmp = fma(y, Float64(Float64(y * y) * 0.16666666666666666), y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[Sinh[y], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], -1e-299], N[(N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.027777777777777776 + 0.16666666666666666), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -9.99999999999999992e-300

   1. Initial program 98.3%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*r/ N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. associate-*r* N/A

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

      11. associate-/l* N/A

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

      12. distribute-rgt-out N/A

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

      13. *-rgt-identity N/A

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

      14. distribute-lft-in N/A

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

   5. Applied rewrites 78.5%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 55.5

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

   8. Applied rewrites 55.5%

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

   9. Taylor expanded in y around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. lower-*.f64 N/A

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

       5. cube-mult N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 N/A

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

       10. +-commutative N/A

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

       11. distribute-rgt-in N/A

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

       12. *-commutative N/A

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

       13. associate-*l* N/A

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

       14. metadata-eval N/A

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

   11. Applied rewrites 32.2%

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

   12. Taylor expanded in x around 0

       \[\leadsto \color{blue}{\frac{-1}{36} \cdot \left({x}^{2} \cdot {y}^{3}\right) + \frac{1}{6} \cdot {y}^{3}} \]
   13. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \color{blue}{\frac{1}{6} \cdot {y}^{3} + \frac{-1}{36} \cdot \left({x}^{2} \cdot {y}^{3}\right)} \]

       2. associate-*r* N/A

          \[\leadsto \frac{1}{6} \cdot {y}^{3} + \color{blue}{\left(\frac{-1}{36} \cdot {x}^{2}\right) \cdot {y}^{3}} \]

       3. distribute-rgt-out N/A

          \[\leadsto \color{blue}{{y}^{3} \cdot \left(\frac{1}{6} + \frac{-1}{36} \cdot {x}^{2}\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto \color{blue}{{y}^{3} \cdot \left(\frac{1}{6} + \frac{-1}{36} \cdot {x}^{2}\right)} \]

       5. cube-mult N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 N/A

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

       10. +-commutative N/A

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

       11. *-commutative N/A

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

       12. lower-fma.f64 N/A

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

       13. unpow2 N/A

           \[\leadsto \left(y \cdot \left(y \cdot y\right)\right) \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{36}, \frac{1}{6}\right) \]

       14. lower-*.f64 32.5

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

   14. Applied rewrites 32.5%

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

   if -9.99999999999999992e-300 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 79.3%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*r/ N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. associate-*r* N/A

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

      11. associate-/l* N/A

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

      12. distribute-rgt-out N/A

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

      13. *-rgt-identity N/A

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

      14. distribute-lft-in N/A

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

   5. Applied rewrites 84.3%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow3 N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 52.6

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

   8. Applied rewrites 52.6%

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

4. Final simplification 45.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -1 \cdot 10^{-299}:\\ \;\;\;\;\left(y \cdot \left(y \cdot y\right)\right) \cdot \mathsf{fma}\left(x \cdot x, -0.027777777777777776, 0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \left(y \cdot y\right) \cdot 0.16666666666666666, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 44.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -2 \cdot 10^{-162}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \left(y \cdot y\right) \cdot 0.16666666666666666, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (sinh y) (sin x)) x) -2e-162)
   (* y (fma x (* x -0.16666666666666666) 1.0))
   (fma y (* (* y y) 0.16666666666666666) y)))

C

double code(double x, double y) {
	double tmp;
	if (((sinh(y) * sin(x)) / x) <= -2e-162) {
		tmp = y * fma(x, (x * -0.16666666666666666), 1.0);
	} else {
		tmp = fma(y, ((y * y) * 0.16666666666666666), y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(sinh(y) * sin(x)) / x) <= -2e-162)
		tmp = Float64(y * fma(x, Float64(x * -0.16666666666666666), 1.0));
	else
		tmp = fma(y, Float64(Float64(y * y) * 0.16666666666666666), y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[Sinh[y], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], -2e-162], N[(y * N[(x * N[(x * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -1.99999999999999991e-162

   1. Initial program 98.9%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 31.0

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

   5. Applied rewrites 31.0%

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

   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 30.2

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

   8. Applied rewrites 30.2%

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

   if -1.99999999999999991e-162 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 80.5%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*r/ N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. associate-*r* N/A

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

      11. associate-/l* N/A

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

      12. distribute-rgt-out N/A

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

      13. *-rgt-identity N/A

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

      14. distribute-lft-in N/A

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

   5. Applied rewrites 85.6%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow3 N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 51.6

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

   8. Applied rewrites 51.6%

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

4. Final simplification 45.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -2 \cdot 10^{-162}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \left(y \cdot y\right) \cdot 0.16666666666666666, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 43.3% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (sinh y) (sin x)) x) 1.0)
   (* y (fma x (* x -0.16666666666666666) 1.0))
   (* 0.16666666666666666 (* y (* y y)))))

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 1

   1. Initial program 80.6%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 69.7

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

   5. Applied rewrites 69.7%

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

   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 43.1

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

   8. Applied rewrites 43.1%

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

   if 1 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-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 66.8%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 53.9

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

   8. Applied rewrites 53.9%

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

   9. Taylor expanded in y around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. lower-*.f64 N/A

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

       5. cube-mult N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 N/A

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

       10. +-commutative N/A

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

       11. distribute-rgt-in N/A

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

       12. *-commutative N/A

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

       13. associate-*l* N/A

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

       14. metadata-eval N/A

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

   11. Applied rewrites 53.9%

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

   12. Taylor expanded in x around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. cube-mult N/A

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

       4. unpow2 N/A

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

       5. lower-*.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 47.5

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

   14. Applied rewrites 47.5%

       \[\leadsto \color{blue}{\left(y \cdot \left(y \cdot y\right)\right) \cdot 0.16666666666666666} \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.4%

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

Alternative 17: 26.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -2 \cdot 10^{-162}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(y \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (sinh y) (sin x)) x) -2e-162)
   (* -0.16666666666666666 (* y (* x x)))
   y))

C

double code(double x, double y) {
	double tmp;
	if (((sinh(y) * sin(x)) / x) <= -2e-162) {
		tmp = -0.16666666666666666 * (y * (x * x));
	} else {
		tmp = y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (((sinh(y) * sin(x)) / x) <= (-2d-162)) then
        tmp = (-0.16666666666666666d0) * (y * (x * x))
    else
        tmp = y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (((Math.sinh(y) * Math.sin(x)) / x) <= -2e-162) {
		tmp = -0.16666666666666666 * (y * (x * x));
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((math.sinh(y) * math.sin(x)) / x) <= -2e-162:
		tmp = -0.16666666666666666 * (y * (x * x))
	else:
		tmp = y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(sinh(y) * sin(x)) / x) <= -2e-162)
		tmp = Float64(-0.16666666666666666 * Float64(y * Float64(x * x)));
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((sinh(y) * sin(x)) / x) <= -2e-162)
		tmp = -0.16666666666666666 * (y * (x * x));
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[Sinh[y], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], -2e-162], N[(-0.16666666666666666 * N[(y * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], y]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -1.99999999999999991e-162

   1. Initial program 98.9%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 31.0

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

   5. Applied rewrites 31.0%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. unpow3 N/A

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

      6. *-lft-identity N/A

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

      7. lower-fma.f64 N/A

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

      8. cube-mult N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 30.1

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

   8. Applied rewrites 30.1%

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

   9. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. cube-mult N/A

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

       3. unpow2 N/A

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

       4. associate-*r* N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 7.1

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

   11. Applied rewrites 7.1%

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

   12. Taylor expanded in y around 0

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

       1. lower-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 N/A

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

       4. unpow2 N/A

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

       5. lower-*.f64 7.1

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

   14. Applied rewrites 7.1%

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

   if -1.99999999999999991e-162 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 80.5%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 59.9

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

   5. Applied rewrites 59.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 33.8%

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

      1. *-rgt-identity 33.8

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

   10. Applied rewrites 33.8%

       \[\leadsto \color{blue}{y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 25.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -2 \cdot 10^{-162}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(y \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 93.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y \cdot y, y \cdot \mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), y\right) \cdot \frac{\sin x}{x}\\ \mathbf{if}\;y \leq 0.49:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{+62}:\\ \;\;\;\;\frac{\sinh y \cdot \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot -0.16666666666666666, x\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (*
          (fma
           (* y y)
           (* y (fma (* y y) 0.008333333333333333 0.16666666666666666))
           y)
          (/ (sin x) x))))
   (if (<= y 0.49)
     t_0
     (if (<= y 1.2e+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((y * y), 0.008333333333333333, 0.16666666666666666)), y) * (sin(x) / x);
	double tmp;
	if (y <= 0.49) {
		tmp = t_0;
	} else if (y <= 1.2e+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(Float64(y * y), Float64(y * fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666)), y) * Float64(sin(x) / x))
	tmp = 0.0
	if (y <= 0.49)
		tmp = t_0;
	elseif (y <= 1.2e+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[(N[(y * y), $MachinePrecision] * N[(y * N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 0.49], t$95$0, If[LessEqual[y, 1.2e+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 < 0.48999999999999999 or 1.2e62 < y

   1. Initial program 85.4%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 79.3

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

   5. Applied rewrites 79.3%

      \[\leadsto \frac{\sin x \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)}}{x} \]
   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 93.8

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

   7. Applied rewrites 93.8%

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

   if 0.48999999999999999 < y < 1.2e62

   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 81.8

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

   5. Applied rewrites 81.8%

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

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.49:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, y \cdot \mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), y\right) \cdot \frac{\sin x}{x}\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{+62}:\\ \;\;\;\;\frac{\sinh y \cdot \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot -0.16666666666666666, x\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, y \cdot \mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), y\right) \cdot \frac{\sin x}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 27.7% 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 86.0%

   \[\frac{\sin x \cdot \sinh y}{x} \]
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 51.2

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

5. Applied rewrites 51.2%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 31.4%

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

   1. *-rgt-identity 31.4

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

10. Applied rewrites 31.4%

    \[\leadsto \color{blue}{y} \]
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 2024214 
(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))