Linear.Quaternion:$ccosh from linear-1.19.1.3

Percentage Accurate: 88.7% → 99.8%

Time: 14.6s

Alternatives: 24

Speedup: 1.0×

• 49.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 88.7% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.8%

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

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 99.9

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

4. Applied rewrites 99.9%

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

Alternative 2: 74.6% accurate, 0.4× speedup

Math

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

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-5)
       (* (/ (sin x) x) (fma (* y y) (* y 0.16666666666666666) y))
       (sinh y)))))

C

double code(double x, double y) {
	double t_0 = (sinh(y) * sin(x)) / x;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (sinh(y) * ((x * x) * (x * -0.16666666666666666))) / x;
	} else if (t_0 <= 1e-5) {
		tmp = (sin(x) / x) * fma((y * y), (y * 0.16666666666666666), y);
	} else {
		tmp = sinh(y);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(sinh(y) * sin(x)) / x)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(sinh(y) * Float64(Float64(x * x) * Float64(x * -0.16666666666666666))) / x);
	elseif (t_0 <= 1e-5)
		tmp = Float64(Float64(sin(x) / x) * fma(Float64(y * y), Float64(y * 0.16666666666666666), y));
	else
		tmp = sinh(y);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sinh[y], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[Sinh[y], $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[t$95$0, 1e-5], N[(N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(y * 0.16666666666666666), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision], N[Sinh[y], $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 77.6

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

   5. Applied rewrites 77.6%

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

   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 32.8

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

   8. Applied rewrites 32.8%

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

   if -inf.0 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 1.00000000000000008e-5

   1. Initial program 82.2%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*r/ N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. associate-*r* N/A

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

      11. associate-/l* N/A

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

      12. distribute-rgt-out N/A

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

      13. *-rgt-identity N/A

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

      14. distribute-lft-in N/A

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

   5. Applied rewrites 99.8%

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

   if 1.00000000000000008e-5 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 100.0%

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

      1. lift-sin.f64 N/A

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

      2. lift-sinh.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l/ N/A

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

      5. associate-/r/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 84.6%

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

4. Final simplification 80.8%

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

Alternative 3: 74.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y \cdot \sin x}{x}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\frac{\sinh y \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot -0.16666666666666666\right)\right)}{x}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-25}:\\ \;\;\;\;\frac{y}{\frac{x}{\sin x}}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \end{array} \]

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 2e-25) (/ y (/ x (sin x))) (sinh y)))))

C

double code(double x, double y) {
	double t_0 = (sinh(y) * sin(x)) / x;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (sinh(y) * ((x * x) * (x * -0.16666666666666666))) / x;
	} else if (t_0 <= 2e-25) {
		tmp = y / (x / sin(x));
	} else {
		tmp = sinh(y);
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double t_0 = (Math.sinh(y) * Math.sin(x)) / x;
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = (Math.sinh(y) * ((x * x) * (x * -0.16666666666666666))) / x;
	} else if (t_0 <= 2e-25) {
		tmp = y / (x / Math.sin(x));
	} else {
		tmp = Math.sinh(y);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (math.sinh(y) * math.sin(x)) / x
	tmp = 0
	if t_0 <= -math.inf:
		tmp = (math.sinh(y) * ((x * x) * (x * -0.16666666666666666))) / x
	elif t_0 <= 2e-25:
		tmp = y / (x / math.sin(x))
	else:
		tmp = math.sinh(y)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(sinh(y) * sin(x)) / x)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(sinh(y) * Float64(Float64(x * x) * Float64(x * -0.16666666666666666))) / x);
	elseif (t_0 <= 2e-25)
		tmp = Float64(y / Float64(x / sin(x)));
	else
		tmp = sinh(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 <= -Inf)
		tmp = (sinh(y) * ((x * x) * (x * -0.16666666666666666))) / x;
	elseif (t_0 <= 2e-25)
		tmp = y / (x / sin(x));
	else
		tmp = sinh(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, (-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, 2e-25], N[(y / N[(x / N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sinh[y], $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 77.6

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

   5. Applied rewrites 77.6%

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

   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 32.8

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

   8. Applied rewrites 32.8%

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

   if -inf.0 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 2.00000000000000008e-25

   1. Initial program 81.8%

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

   3. Taylor expanded in y around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sin.f64 99.4

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

   5. Applied rewrites 99.4%

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

      1. lift-sin.f64 N/A

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

      2. clear-num N/A

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

      3. lift-/.f64 N/A

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

      4. un-div-inv N/A

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

      5. lower-/.f64 99.4

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

   7. Applied rewrites 99.4%

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

   if 2.00000000000000008e-25 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 100.0%

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

      1. lift-sin.f64 N/A

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

      2. lift-sinh.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l/ N/A

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

      5. associate-/r/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 85.3%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -\infty:\\ \;\;\;\;\frac{\sinh y \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot -0.16666666666666666\right)\right)}{x}\\ \mathbf{elif}\;\frac{\sinh y \cdot \sin x}{x} \leq 2 \cdot 10^{-25}:\\ \;\;\;\;\frac{y}{\frac{x}{\sin x}}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y \cdot \sin x}{x}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(y \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right), \mathsf{fma}\left(y \cdot y, -3.306878306878307 \cdot 10^{-5}, -0.001388888888888889\right), \mathsf{fma}\left(y \cdot y, -0.027777777777777776, -0.16666666666666666\right)\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-25}:\\ \;\;\;\;\frac{y}{\frac{x}{\sin x}}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sinh y) (sin x)) x)))
   (if (<= t_0 (- INFINITY))
     (*
      (* y (* x x))
      (fma
       (* (* y y) (* y y))
       (fma (* y y) -3.306878306878307e-5 -0.001388888888888889)
       (fma (* y y) -0.027777777777777776 -0.16666666666666666)))
     (if (<= t_0 2e-25) (/ y (/ x (sin x))) (sinh y)))))

C

double code(double x, double y) {
	double t_0 = (sinh(y) * sin(x)) / x;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (y * (x * x)) * fma(((y * y) * (y * y)), fma((y * y), -3.306878306878307e-5, -0.001388888888888889), fma((y * y), -0.027777777777777776, -0.16666666666666666));
	} else if (t_0 <= 2e-25) {
		tmp = y / (x / sin(x));
	} else {
		tmp = sinh(y);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(sinh(y) * sin(x)) / x)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(y * Float64(x * x)) * fma(Float64(Float64(y * y) * Float64(y * y)), fma(Float64(y * y), -3.306878306878307e-5, -0.001388888888888889), fma(Float64(y * y), -0.027777777777777776, -0.16666666666666666)));
	elseif (t_0 <= 2e-25)
		tmp = Float64(y / Float64(x / sin(x)));
	else
		tmp = sinh(y);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sinh[y], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(y * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(y * y), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * -3.306878306878307e-5 + -0.001388888888888889), $MachinePrecision] + N[(N[(y * y), $MachinePrecision] * -0.027777777777777776 + -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e-25], N[(y / N[(x / N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sinh[y], $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 77.6

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

   5. Applied rewrites 77.6%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 62.9%

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

   9. Taylor expanded in x around inf

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

       1. associate-*r* N/A

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

       2. lower-*.f64 N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 N/A

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

       7. *-commutative N/A

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

       8. associate-*l* N/A

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

       9. *-commutative N/A

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

       10. lower-fma.f64 N/A

           \[\leadsto \left(y \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\mathsf{fma}\left({y}^{4}, \frac{-1}{6} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right), \frac{-1}{6} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)} \]

   11. Applied rewrites 29.6%

       \[\leadsto \color{blue}{\left(y \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right), \mathsf{fma}\left(y \cdot y, -3.306878306878307 \cdot 10^{-5}, -0.001388888888888889\right), \mathsf{fma}\left(y \cdot y, -0.027777777777777776, -0.16666666666666666\right)\right)} \]

   if -inf.0 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 2.00000000000000008e-25

   1. Initial program 81.8%

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

   3. Taylor expanded in y around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sin.f64 99.4

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

   5. Applied rewrites 99.4%

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

      1. lift-sin.f64 N/A

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

      2. clear-num N/A

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

      3. lift-/.f64 N/A

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

      4. un-div-inv N/A

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

      5. lower-/.f64 99.4

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

   7. Applied rewrites 99.4%

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

   if 2.00000000000000008e-25 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 100.0%

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

      1. lift-sin.f64 N/A

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

      2. lift-sinh.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l/ N/A

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

      5. associate-/r/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 85.3%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -\infty:\\ \;\;\;\;\left(y \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right), \mathsf{fma}\left(y \cdot y, -3.306878306878307 \cdot 10^{-5}, -0.001388888888888889\right), \mathsf{fma}\left(y \cdot y, -0.027777777777777776, -0.16666666666666666\right)\right)\\ \mathbf{elif}\;\frac{\sinh y \cdot \sin x}{x} \leq 2 \cdot 10^{-25}:\\ \;\;\;\;\frac{y}{\frac{x}{\sin x}}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y \cdot \sin x}{x}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(y \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right), \mathsf{fma}\left(y \cdot y, -3.306878306878307 \cdot 10^{-5}, -0.001388888888888889\right), \mathsf{fma}\left(y \cdot y, -0.027777777777777776, -0.16666666666666666\right)\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-25}:\\ \;\;\;\;y \cdot \frac{\sin x}{x}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sinh y) (sin x)) x)))
   (if (<= t_0 (- INFINITY))
     (*
      (* y (* x x))
      (fma
       (* (* y y) (* y y))
       (fma (* y y) -3.306878306878307e-5 -0.001388888888888889)
       (fma (* y y) -0.027777777777777776 -0.16666666666666666)))
     (if (<= t_0 2e-25) (* y (/ (sin x) x)) (sinh y)))))

C

double code(double x, double y) {
	double t_0 = (sinh(y) * sin(x)) / x;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (y * (x * x)) * fma(((y * y) * (y * y)), fma((y * y), -3.306878306878307e-5, -0.001388888888888889), fma((y * y), -0.027777777777777776, -0.16666666666666666));
	} else if (t_0 <= 2e-25) {
		tmp = y * (sin(x) / x);
	} else {
		tmp = sinh(y);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(sinh(y) * sin(x)) / x)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(y * Float64(x * x)) * fma(Float64(Float64(y * y) * Float64(y * y)), fma(Float64(y * y), -3.306878306878307e-5, -0.001388888888888889), fma(Float64(y * y), -0.027777777777777776, -0.16666666666666666)));
	elseif (t_0 <= 2e-25)
		tmp = Float64(y * Float64(sin(x) / x));
	else
		tmp = sinh(y);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sinh[y], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(y * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(y * y), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * -3.306878306878307e-5 + -0.001388888888888889), $MachinePrecision] + N[(N[(y * y), $MachinePrecision] * -0.027777777777777776 + -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e-25], N[(y * N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[Sinh[y], $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 77.6

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

   5. Applied rewrites 77.6%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 62.9%

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

   9. Taylor expanded in x around inf

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

       1. associate-*r* N/A

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

       2. lower-*.f64 N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 N/A

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

       7. *-commutative N/A

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

       8. associate-*l* N/A

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

       9. *-commutative N/A

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

       10. lower-fma.f64 N/A

           \[\leadsto \left(y \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\mathsf{fma}\left({y}^{4}, \frac{-1}{6} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right), \frac{-1}{6} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)} \]

   11. Applied rewrites 29.6%

       \[\leadsto \color{blue}{\left(y \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right), \mathsf{fma}\left(y \cdot y, -3.306878306878307 \cdot 10^{-5}, -0.001388888888888889\right), \mathsf{fma}\left(y \cdot y, -0.027777777777777776, -0.16666666666666666\right)\right)} \]

   if -inf.0 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 2.00000000000000008e-25

   1. Initial program 81.8%

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

   3. Taylor expanded in y around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sin.f64 99.4

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

   5. Applied rewrites 99.4%

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

   if 2.00000000000000008e-25 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 100.0%

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

      1. lift-sin.f64 N/A

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

      2. lift-sinh.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l/ N/A

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

      5. associate-/r/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 85.3%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -\infty:\\ \;\;\;\;\left(y \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right), \mathsf{fma}\left(y \cdot y, -3.306878306878307 \cdot 10^{-5}, -0.001388888888888889\right), \mathsf{fma}\left(y \cdot y, -0.027777777777777776, -0.16666666666666666\right)\right)\\ \mathbf{elif}\;\frac{\sinh y \cdot \sin x}{x} \leq 2 \cdot 10^{-25}:\\ \;\;\;\;y \cdot \frac{\sin x}{x}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 52.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y \cdot \sin x}{x}\\ t_1 := x \cdot \mathsf{fma}\left(x \cdot x, 0.008333333333333333, -0.16666666666666666\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-206}:\\ \;\;\;\;\frac{y \cdot \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, x \cdot -0.16666666666666666, x\right) \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), \left(y \cdot y\right) \cdot \left(y \cdot y\right), \left(y \cdot y\right) \cdot \left(0.16666666666666666 \cdot \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot -0.16666666666666666, x\right)\right)\right)}{x}\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;y \cdot \left(\left(y \cdot y\right) \cdot 0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, t\_1, x\right), \frac{y \cdot y}{x} \cdot \mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), x \cdot t\_1\right), y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sinh y) (sin x)) x))
        (t_1 (* x (fma (* x x) 0.008333333333333333 -0.16666666666666666))))
   (if (<= t_0 -2e-206)
     (/
      (*
       y
       (fma
        (*
         (fma (* x x) (* x -0.16666666666666666) x)
         (fma (* y y) 0.0001984126984126984 0.008333333333333333))
        (* (* y y) (* y y))
        (*
         (* y y)
         (* 0.16666666666666666 (fma x (* (* x x) -0.16666666666666666) x)))))
      x)
     (if (<= t_0 0.0)
       (* y (* (* y y) 0.16666666666666666))
       (fma
        y
        (fma
         (fma (* x x) t_1 x)
         (*
          (/ (* y y) x)
          (fma (* y y) 0.008333333333333333 0.16666666666666666))
         (* x t_1))
        y)))))

C

double code(double x, double y) {
	double t_0 = (sinh(y) * sin(x)) / x;
	double t_1 = x * fma((x * x), 0.008333333333333333, -0.16666666666666666);
	double tmp;
	if (t_0 <= -2e-206) {
		tmp = (y * fma((fma((x * x), (x * -0.16666666666666666), x) * fma((y * y), 0.0001984126984126984, 0.008333333333333333)), ((y * y) * (y * y)), ((y * y) * (0.16666666666666666 * fma(x, ((x * x) * -0.16666666666666666), x))))) / x;
	} else if (t_0 <= 0.0) {
		tmp = y * ((y * y) * 0.16666666666666666);
	} else {
		tmp = fma(y, fma(fma((x * x), t_1, x), (((y * y) / x) * fma((y * y), 0.008333333333333333, 0.16666666666666666)), (x * t_1)), y);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(sinh(y) * sin(x)) / x)
	t_1 = Float64(x * fma(Float64(x * x), 0.008333333333333333, -0.16666666666666666))
	tmp = 0.0
	if (t_0 <= -2e-206)
		tmp = Float64(Float64(y * fma(Float64(fma(Float64(x * x), Float64(x * -0.16666666666666666), x) * fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333)), Float64(Float64(y * y) * Float64(y * y)), Float64(Float64(y * y) * Float64(0.16666666666666666 * fma(x, Float64(Float64(x * x) * -0.16666666666666666), x))))) / x);
	elseif (t_0 <= 0.0)
		tmp = Float64(y * Float64(Float64(y * y) * 0.16666666666666666));
	else
		tmp = fma(y, fma(fma(Float64(x * x), t_1, x), Float64(Float64(Float64(y * y) / x) * fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666)), Float64(x * t_1)), y);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sinh[y], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, Block[{t$95$1 = N[(x * N[(N[(x * x), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-206], N[(N[(y * N[(N[(N[(N[(x * x), $MachinePrecision] * N[(x * -0.16666666666666666), $MachinePrecision] + x), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision] + N[(N[(y * y), $MachinePrecision] * N[(0.16666666666666666 * N[(x * N[(N[(x * x), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(y * N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[(N[(x * x), $MachinePrecision] * t$95$1 + x), $MachinePrecision] * N[(N[(N[(y * y), $MachinePrecision] / x), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + N[(x * t$95$1), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -2.00000000000000006e-206

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)} \cdot \sinh y}{x} \]
   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 70.2

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

   5. Applied rewrites 70.2%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 60.9%

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

   9. Taylor expanded in y around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. lower-*.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. +-commutative N/A

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

       9. *-commutative N/A

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

       10. cube-mult N/A

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

       11. unpow2 N/A

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

       12. associate-*r* N/A

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

       13. *-commutative N/A

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

       14. lower-fma.f64 N/A

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

       15. lower-*.f64 N/A

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

       16. unpow2 N/A

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

       17. lower-*.f64 41.5

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

   11. Applied rewrites 41.5%

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

   if -2.00000000000000006e-206 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 0.0

   1. Initial program 65.5%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*r/ N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. associate-*r* N/A

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

      11. associate-/l* N/A

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

      12. distribute-rgt-out N/A

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

      13. *-rgt-identity N/A

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

      14. distribute-lft-in N/A

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)} \cdot \mathsf{fma}\left(y \cdot y, y \cdot \frac{1}{6}, y\right) \]
   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 44.9

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

   8. Applied rewrites 44.9%

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

   9. Taylor expanded in y around inf

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

       1. lower-*.f64 N/A

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

       2. cube-mult N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 3.5

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

   11. Applied rewrites 3.5%

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

   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. cube-mult N/A

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

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 46.5

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

   14. Applied rewrites 46.5%

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

   if 0.0 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. sub-neg N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 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%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 70.3%

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

   10. Applied rewrites 73.3%

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

4. Final simplification 54.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -2 \cdot 10^{-206}:\\ \;\;\;\;\frac{y \cdot \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, x \cdot -0.16666666666666666, x\right) \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), \left(y \cdot y\right) \cdot \left(y \cdot y\right), \left(y \cdot y\right) \cdot \left(0.16666666666666666 \cdot \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot -0.16666666666666666, x\right)\right)\right)}{x}\\ \mathbf{elif}\;\frac{\sinh y \cdot \sin x}{x} \leq 0:\\ \;\;\;\;y \cdot \left(\left(y \cdot y\right) \cdot 0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, x \cdot \mathsf{fma}\left(x \cdot x, 0.008333333333333333, -0.16666666666666666\right), x\right), \frac{y \cdot y}{x} \cdot \mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, 0.008333333333333333, -0.16666666666666666\right)\right)\right), y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 40.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y \cdot \sin x}{x}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-180}:\\ \;\;\;\;\left(y \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right), \mathsf{fma}\left(y \cdot y, -3.306878306878307 \cdot 10^{-5}, -0.001388888888888889\right), \mathsf{fma}\left(y \cdot y, -0.027777777777777776, -0.16666666666666666\right)\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;y \cdot \left(\left(y \cdot y\right) \cdot 0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.008333333333333333, -0.16666666666666666\right), x \cdot \left(x \cdot x\right), x\right) \cdot \mathsf{fma}\left(y \cdot y, y \cdot \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), y\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sinh y) (sin x)) x)))
   (if (<= t_0 -2e-180)
     (*
      (* y (* x x))
      (fma
       (* (* y y) (* y y))
       (fma (* y y) -3.306878306878307e-5 -0.001388888888888889)
       (fma (* y y) -0.027777777777777776 -0.16666666666666666)))
     (if (<= t_0 0.0)
       (* y (* (* y y) 0.16666666666666666))
       (/
        (*
         (fma
          (fma (* x x) 0.008333333333333333 -0.16666666666666666)
          (* x (* x x))
          x)
         (fma
          (* y y)
          (* y (fma y (* y 0.008333333333333333) 0.16666666666666666))
          y))
        x)))))

C

double code(double x, double y) {
	double t_0 = (sinh(y) * sin(x)) / x;
	double tmp;
	if (t_0 <= -2e-180) {
		tmp = (y * (x * x)) * fma(((y * y) * (y * y)), fma((y * y), -3.306878306878307e-5, -0.001388888888888889), fma((y * y), -0.027777777777777776, -0.16666666666666666));
	} else if (t_0 <= 0.0) {
		tmp = y * ((y * y) * 0.16666666666666666);
	} else {
		tmp = (fma(fma((x * x), 0.008333333333333333, -0.16666666666666666), (x * (x * x)), x) * fma((y * y), (y * fma(y, (y * 0.008333333333333333), 0.16666666666666666)), y)) / x;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(sinh(y) * sin(x)) / x)
	tmp = 0.0
	if (t_0 <= -2e-180)
		tmp = Float64(Float64(y * Float64(x * x)) * fma(Float64(Float64(y * y) * Float64(y * y)), fma(Float64(y * y), -3.306878306878307e-5, -0.001388888888888889), fma(Float64(y * y), -0.027777777777777776, -0.16666666666666666)));
	elseif (t_0 <= 0.0)
		tmp = Float64(y * Float64(Float64(y * y) * 0.16666666666666666));
	else
		tmp = Float64(Float64(fma(fma(Float64(x * x), 0.008333333333333333, -0.16666666666666666), Float64(x * Float64(x * x)), x) * fma(Float64(y * y), Float64(y * fma(y, Float64(y * 0.008333333333333333), 0.16666666666666666)), y)) / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sinh[y], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-180], N[(N[(y * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(y * y), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * -3.306878306878307e-5 + -0.001388888888888889), $MachinePrecision] + N[(N[(y * y), $MachinePrecision] * -0.027777777777777776 + -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(y * N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(x * x), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision] * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * N[(y * 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $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) < -2e-180

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)} \cdot \sinh y}{x} \]
   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 72.6

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

   5. Applied rewrites 72.6%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 62.6%

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

   9. Taylor expanded in x around inf

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

       1. associate-*r* N/A

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

       2. lower-*.f64 N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 N/A

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

       7. *-commutative N/A

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

       8. associate-*l* N/A

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

       9. *-commutative N/A

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

       10. lower-fma.f64 N/A

           \[\leadsto \left(y \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\mathsf{fma}\left({y}^{4}, \frac{-1}{6} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right), \frac{-1}{6} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)} \]

   11. Applied rewrites 21.2%

       \[\leadsto \color{blue}{\left(y \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right), \mathsf{fma}\left(y \cdot y, -3.306878306878307 \cdot 10^{-5}, -0.001388888888888889\right), \mathsf{fma}\left(y \cdot y, -0.027777777777777776, -0.16666666666666666\right)\right)} \]

   if -2e-180 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 0.0

   1. Initial program 68.2%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*r/ N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. associate-*r* N/A

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

      11. associate-/l* N/A

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

      12. distribute-rgt-out N/A

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

      13. *-rgt-identity N/A

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

      14. distribute-lft-in N/A

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

   5. Applied rewrites 99.8%

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

   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 44.0

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

   8. Applied rewrites 44.0%

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

   9. Taylor expanded in y around inf

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

       1. lower-*.f64 N/A

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

       2. cube-mult N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 3.4

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

   11. Applied rewrites 3.4%

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

   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. cube-mult N/A

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

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 43.2

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

   14. Applied rewrites 43.2%

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

   if 0.0 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. sub-neg N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 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%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-*l* N/A

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

      4. *-lft-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 69.4

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

   8. Applied rewrites 69.4%

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

4. Final simplification 45.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -2 \cdot 10^{-180}:\\ \;\;\;\;\left(y \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right), \mathsf{fma}\left(y \cdot y, -3.306878306878307 \cdot 10^{-5}, -0.001388888888888889\right), \mathsf{fma}\left(y \cdot y, -0.027777777777777776, -0.16666666666666666\right)\right)\\ \mathbf{elif}\;\frac{\sinh y \cdot \sin x}{x} \leq 0:\\ \;\;\;\;y \cdot \left(\left(y \cdot y\right) \cdot 0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.008333333333333333, -0.16666666666666666\right), x \cdot \left(x \cdot x\right), x\right) \cdot \mathsf{fma}\left(y \cdot y, y \cdot \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), y\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 40.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y \cdot \sin x}{x}\\ t_1 := \left(y \cdot y\right) \cdot \left(y \cdot y\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-180}:\\ \;\;\;\;\left(y \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(t\_1, \mathsf{fma}\left(y \cdot y, -3.306878306878307 \cdot 10^{-5}, -0.001388888888888889\right), \mathsf{fma}\left(y \cdot y, -0.027777777777777776, -0.16666666666666666\right)\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;y \cdot \left(\left(y \cdot y\right) \cdot 0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(y \cdot x\right) \cdot \mathsf{fma}\left(t\_1, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), \mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right)\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sinh y) (sin x)) x)) (t_1 (* (* y y) (* y y))))
   (if (<= t_0 -2e-180)
     (*
      (* y (* x x))
      (fma
       t_1
       (fma (* y y) -3.306878306878307e-5 -0.001388888888888889)
       (fma (* y y) -0.027777777777777776 -0.16666666666666666)))
     (if (<= t_0 0.0)
       (* y (* (* y y) 0.16666666666666666))
       (/
        (*
         (* y x)
         (fma
          t_1
          (fma (* y y) 0.0001984126984126984 0.008333333333333333)
          (fma 0.16666666666666666 (* y y) 1.0)))
        x)))))

C

double code(double x, double y) {
	double t_0 = (sinh(y) * sin(x)) / x;
	double t_1 = (y * y) * (y * y);
	double tmp;
	if (t_0 <= -2e-180) {
		tmp = (y * (x * x)) * fma(t_1, fma((y * y), -3.306878306878307e-5, -0.001388888888888889), fma((y * y), -0.027777777777777776, -0.16666666666666666));
	} else if (t_0 <= 0.0) {
		tmp = y * ((y * y) * 0.16666666666666666);
	} else {
		tmp = ((y * x) * fma(t_1, fma((y * y), 0.0001984126984126984, 0.008333333333333333), fma(0.16666666666666666, (y * y), 1.0))) / x;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(sinh(y) * sin(x)) / x)
	t_1 = Float64(Float64(y * y) * Float64(y * y))
	tmp = 0.0
	if (t_0 <= -2e-180)
		tmp = Float64(Float64(y * Float64(x * x)) * fma(t_1, fma(Float64(y * y), -3.306878306878307e-5, -0.001388888888888889), fma(Float64(y * y), -0.027777777777777776, -0.16666666666666666)));
	elseif (t_0 <= 0.0)
		tmp = Float64(y * Float64(Float64(y * y) * 0.16666666666666666));
	else
		tmp = Float64(Float64(Float64(y * x) * fma(t_1, fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333), fma(0.16666666666666666, Float64(y * y), 1.0))) / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sinh[y], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, Block[{t$95$1 = N[(N[(y * y), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-180], N[(N[(y * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * N[(N[(y * y), $MachinePrecision] * -3.306878306878307e-5 + -0.001388888888888889), $MachinePrecision] + N[(N[(y * y), $MachinePrecision] * -0.027777777777777776 + -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(y * N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y * x), $MachinePrecision] * N[(t$95$1 * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + N[(0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -2e-180

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)} \cdot \sinh y}{x} \]
   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 72.6

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

   5. Applied rewrites 72.6%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 62.6%

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

   9. Taylor expanded in x around inf

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

       1. associate-*r* N/A

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

       2. lower-*.f64 N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 N/A

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

       7. *-commutative N/A

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

       8. associate-*l* N/A

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

       9. *-commutative N/A

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

       10. lower-fma.f64 N/A

           \[\leadsto \left(y \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\mathsf{fma}\left({y}^{4}, \frac{-1}{6} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right), \frac{-1}{6} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)} \]

   11. Applied rewrites 21.2%

       \[\leadsto \color{blue}{\left(y \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right), \mathsf{fma}\left(y \cdot y, -3.306878306878307 \cdot 10^{-5}, -0.001388888888888889\right), \mathsf{fma}\left(y \cdot y, -0.027777777777777776, -0.16666666666666666\right)\right)} \]

   if -2e-180 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 0.0

   1. Initial program 68.2%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*r/ N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. associate-*r* N/A

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

      11. associate-/l* N/A

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

      12. distribute-rgt-out N/A

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

      13. *-rgt-identity N/A

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

      14. distribute-lft-in N/A

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

   5. Applied rewrites 99.8%

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

   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 44.0

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

   8. Applied rewrites 44.0%

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

   9. Taylor expanded in y around inf

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

       1. lower-*.f64 N/A

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

       2. cube-mult N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 3.4

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

   11. Applied rewrites 3.4%

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

   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. cube-mult N/A

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

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 43.2

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

   14. Applied rewrites 43.2%

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

   if 0.0 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)} \cdot \sinh y}{x} \]
   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 66.1

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

   5. Applied rewrites 66.1%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 63.1%

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

   9. Taylor expanded in x around 0

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

       1. associate-*r* N/A

          \[\leadsto \frac{\color{blue}{\left(x \cdot y\right) \cdot \left(1 + \left(\frac{1}{6} \cdot {y}^{2} + {y}^{4} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)}}{x} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\left(x \cdot y\right) \cdot \left(1 + \left(\frac{1}{6} \cdot {y}^{2} + {y}^{4} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)}}{x} \]

       3. *-commutative N/A

          \[\leadsto \frac{\color{blue}{\left(y \cdot x\right)} \cdot \left(1 + \left(\frac{1}{6} \cdot {y}^{2} + {y}^{4} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)}{x} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\left(y \cdot x\right)} \cdot \left(1 + \left(\frac{1}{6} \cdot {y}^{2} + {y}^{4} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)}{x} \]

       5. associate-+r+ N/A

          \[\leadsto \frac{\left(y \cdot x\right) \cdot \color{blue}{\left(\left(1 + \frac{1}{6} \cdot {y}^{2}\right) + {y}^{4} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)}}{x} \]

       6. +-commutative N/A

          \[\leadsto \frac{\left(y \cdot x\right) \cdot \color{blue}{\left({y}^{4} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right) + \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)}}{x} \]

       7. lower-fma.f64 N/A

          \[\leadsto \frac{\left(y \cdot x\right) \cdot \color{blue}{\mathsf{fma}\left({y}^{4}, \frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}, 1 + \frac{1}{6} \cdot {y}^{2}\right)}}{x} \]

       8. metadata-eval N/A

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

       9. pow-sqr N/A

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

       10. lower-*.f64 N/A

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

       11. unpow2 N/A

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

       12. lower-*.f64 N/A

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

       13. unpow2 N/A

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

       14. lower-*.f64 N/A

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

       15. +-commutative N/A

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

       16. *-commutative N/A

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

       17. lower-fma.f64 N/A

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

       18. unpow2 N/A

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

       19. lower-*.f64 N/A

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

       20. +-commutative N/A

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

       21. lower-fma.f64 N/A

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

       22. unpow2 N/A

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

       23. lower-*.f64 69.6

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

   11. Applied rewrites 69.6%

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

4. Final simplification 45.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -2 \cdot 10^{-180}:\\ \;\;\;\;\left(y \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right), \mathsf{fma}\left(y \cdot y, -3.306878306878307 \cdot 10^{-5}, -0.001388888888888889\right), \mathsf{fma}\left(y \cdot y, -0.027777777777777776, -0.16666666666666666\right)\right)\\ \mathbf{elif}\;\frac{\sinh y \cdot \sin x}{x} \leq 0:\\ \;\;\;\;y \cdot \left(\left(y \cdot y\right) \cdot 0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(y \cdot x\right) \cdot \mathsf{fma}\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right), \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), \mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right)\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 40.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y \cdot \sin x}{x}\\ t_1 := \left(y \cdot y\right) \cdot 0.16666666666666666\\ t_2 := \left(y \cdot y\right) \cdot \left(y \cdot y\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-180}:\\ \;\;\;\;\left(y \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(t\_2, \mathsf{fma}\left(y \cdot y, -3.306878306878307 \cdot 10^{-5}, -0.001388888888888889\right), \mathsf{fma}\left(y \cdot y, -0.027777777777777776, -0.16666666666666666\right)\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;y \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(t\_2, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), t\_1\right), x\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sinh y) (sin x)) x))
        (t_1 (* (* y y) 0.16666666666666666))
        (t_2 (* (* y y) (* y y))))
   (if (<= t_0 -2e-180)
     (*
      (* y (* x x))
      (fma
       t_2
       (fma (* y y) -3.306878306878307e-5 -0.001388888888888889)
       (fma (* y y) -0.027777777777777776 -0.16666666666666666)))
     (if (<= t_0 0.0)
       (* y t_1)
       (/
        (*
         y
         (fma
          x
          (fma
           t_2
           (fma (* y y) 0.0001984126984126984 0.008333333333333333)
           t_1)
          x))
        x)))))

C

double code(double x, double y) {
	double t_0 = (sinh(y) * sin(x)) / x;
	double t_1 = (y * y) * 0.16666666666666666;
	double t_2 = (y * y) * (y * y);
	double tmp;
	if (t_0 <= -2e-180) {
		tmp = (y * (x * x)) * fma(t_2, fma((y * y), -3.306878306878307e-5, -0.001388888888888889), fma((y * y), -0.027777777777777776, -0.16666666666666666));
	} else if (t_0 <= 0.0) {
		tmp = y * t_1;
	} else {
		tmp = (y * fma(x, fma(t_2, fma((y * y), 0.0001984126984126984, 0.008333333333333333), t_1), x)) / x;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(sinh(y) * sin(x)) / x)
	t_1 = Float64(Float64(y * y) * 0.16666666666666666)
	t_2 = Float64(Float64(y * y) * Float64(y * y))
	tmp = 0.0
	if (t_0 <= -2e-180)
		tmp = Float64(Float64(y * Float64(x * x)) * fma(t_2, fma(Float64(y * y), -3.306878306878307e-5, -0.001388888888888889), fma(Float64(y * y), -0.027777777777777776, -0.16666666666666666)));
	elseif (t_0 <= 0.0)
		tmp = Float64(y * t_1);
	else
		tmp = Float64(Float64(y * fma(x, fma(t_2, fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333), t_1), x)) / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sinh[y], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, Block[{t$95$1 = N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y * y), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-180], N[(N[(y * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(t$95$2 * N[(N[(y * y), $MachinePrecision] * -3.306878306878307e-5 + -0.001388888888888889), $MachinePrecision] + N[(N[(y * y), $MachinePrecision] * -0.027777777777777776 + -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(y * t$95$1), $MachinePrecision], N[(N[(y * N[(x * N[(t$95$2 * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + t$95$1), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -2e-180

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)} \cdot \sinh y}{x} \]
   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 72.6

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

   5. Applied rewrites 72.6%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 62.6%

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

   9. Taylor expanded in x around inf

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

       1. associate-*r* N/A

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

       2. lower-*.f64 N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 N/A

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

       7. *-commutative N/A

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

       8. associate-*l* N/A

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

       9. *-commutative N/A

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

       10. lower-fma.f64 N/A

           \[\leadsto \left(y \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\mathsf{fma}\left({y}^{4}, \frac{-1}{6} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right), \frac{-1}{6} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)} \]

   11. Applied rewrites 21.2%

       \[\leadsto \color{blue}{\left(y \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right), \mathsf{fma}\left(y \cdot y, -3.306878306878307 \cdot 10^{-5}, -0.001388888888888889\right), \mathsf{fma}\left(y \cdot y, -0.027777777777777776, -0.16666666666666666\right)\right)} \]

   if -2e-180 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 0.0

   1. Initial program 68.2%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*r/ N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. associate-*r* N/A

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

      11. associate-/l* N/A

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

      12. distribute-rgt-out N/A

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

      13. *-rgt-identity N/A

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

      14. distribute-lft-in N/A

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

   5. Applied rewrites 99.8%

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

   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 44.0

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

   8. Applied rewrites 44.0%

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

   9. Taylor expanded in y around inf

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

       1. lower-*.f64 N/A

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

       2. cube-mult N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 3.4

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

   11. Applied rewrites 3.4%

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

   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. cube-mult N/A

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

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 43.2

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

   14. Applied rewrites 43.2%

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

   if 0.0 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)} \cdot \sinh y}{x} \]
   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 66.1

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

   5. Applied rewrites 66.1%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 63.1%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

          \[\leadsto \frac{y \cdot \left(x \cdot \color{blue}{\left(\left(\frac{1}{6} \cdot {y}^{2} + {y}^{4} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) + 1\right)}\right)}{x} \]

       2. distribute-lft-in N/A

          \[\leadsto \frac{y \cdot \color{blue}{\left(x \cdot \left(\frac{1}{6} \cdot {y}^{2} + {y}^{4} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) + x \cdot 1\right)}}{x} \]

       3. *-rgt-identity N/A

          \[\leadsto \frac{y \cdot \left(x \cdot \left(\frac{1}{6} \cdot {y}^{2} + {y}^{4} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) + \color{blue}{x}\right)}{x} \]

       4. lower-fma.f64 N/A

          \[\leadsto \frac{y \cdot \color{blue}{\mathsf{fma}\left(x, \frac{1}{6} \cdot {y}^{2} + {y}^{4} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right), x\right)}}{x} \]

   11. Applied rewrites 69.6%

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

4. Final simplification 45.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -2 \cdot 10^{-180}:\\ \;\;\;\;\left(y \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right), \mathsf{fma}\left(y \cdot y, -3.306878306878307 \cdot 10^{-5}, -0.001388888888888889\right), \mathsf{fma}\left(y \cdot y, -0.027777777777777776, -0.16666666666666666\right)\right)\\ \mathbf{elif}\;\frac{\sinh y \cdot \sin x}{x} \leq 0:\\ \;\;\;\;y \cdot \left(\left(y \cdot y\right) \cdot 0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right), \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), \left(y \cdot y\right) \cdot 0.16666666666666666\right), x\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 37.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y \cdot \sin x}{x}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-235}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(y \cdot \left(x \cdot x\right)\right)\\ \mathbf{elif}\;t\_0 \leq 10^{-5}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(y \cdot y\right) \cdot 0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sinh y) (sin x)) x)))
   (if (<= t_0 -5e-235)
     (* -0.16666666666666666 (* y (* x x)))
     (if (<= t_0 1e-5) y (* y (* (* y y) 0.16666666666666666))))))

C

double code(double x, double y) {
	double t_0 = (sinh(y) * sin(x)) / x;
	double tmp;
	if (t_0 <= -5e-235) {
		tmp = -0.16666666666666666 * (y * (x * x));
	} else if (t_0 <= 1e-5) {
		tmp = y;
	} else {
		tmp = y * ((y * y) * 0.16666666666666666);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (sinh(y) * sin(x)) / x
    if (t_0 <= (-5d-235)) then
        tmp = (-0.16666666666666666d0) * (y * (x * x))
    else if (t_0 <= 1d-5) then
        tmp = y
    else
        tmp = y * ((y * y) * 0.16666666666666666d0)
    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 <= -5e-235) {
		tmp = -0.16666666666666666 * (y * (x * x));
	} else if (t_0 <= 1e-5) {
		tmp = y;
	} else {
		tmp = y * ((y * y) * 0.16666666666666666);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (math.sinh(y) * math.sin(x)) / x
	tmp = 0
	if t_0 <= -5e-235:
		tmp = -0.16666666666666666 * (y * (x * x))
	elif t_0 <= 1e-5:
		tmp = y
	else:
		tmp = y * ((y * y) * 0.16666666666666666)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(sinh(y) * sin(x)) / x)
	tmp = 0.0
	if (t_0 <= -5e-235)
		tmp = Float64(-0.16666666666666666 * Float64(y * Float64(x * x)));
	elseif (t_0 <= 1e-5)
		tmp = y;
	else
		tmp = Float64(y * Float64(Float64(y * y) * 0.16666666666666666));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (sinh(y) * sin(x)) / x;
	tmp = 0.0;
	if (t_0 <= -5e-235)
		tmp = -0.16666666666666666 * (y * (x * x));
	elseif (t_0 <= 1e-5)
		tmp = y;
	else
		tmp = y * ((y * y) * 0.16666666666666666);
	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, -5e-235], N[(-0.16666666666666666 * N[(y * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e-5], y, N[(y * N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -4.9999999999999998e-235

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)} \cdot \sinh y}{x} \]
   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 68.7

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

   5. Applied rewrites 68.7%

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

   6. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. unpow3 N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 40.3

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

   8. Applied rewrites 40.3%

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

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{-1}{6} \cdot \left({x}^{2} \cdot y\right)} \]
   10. 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 16.6

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

   11. Applied rewrites 16.6%

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

   if -4.9999999999999998e-235 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 1.00000000000000008e-5

   1. Initial program 74.9%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 99.2

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

   5. Applied rewrites 99.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 49.9%

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

      1. *-rgt-identity 49.9

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

   10. Applied rewrites 49.9%

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

   if 1.00000000000000008e-5 < (/.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 73.8%

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

   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 53.5

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

   8. Applied rewrites 53.5%

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

   9. Taylor expanded in y around inf

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

       1. lower-*.f64 N/A

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

       2. cube-mult N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 53.5

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

   11. Applied rewrites 53.5%

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

   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. cube-mult N/A

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

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 61.4

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

   14. Applied rewrites 61.4%

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

4. Final simplification 40.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -5 \cdot 10^{-235}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(y \cdot \left(x \cdot x\right)\right)\\ \mathbf{elif}\;\frac{\sinh y \cdot \sin x}{x} \leq 10^{-5}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(y \cdot y\right) \cdot 0.16666666666666666\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 55.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -1 \cdot 10^{-291}:\\ \;\;\;\;\frac{y \cdot \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, x \cdot -0.16666666666666666, x\right) \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), \left(y \cdot y\right) \cdot \left(y \cdot y\right), \left(y \cdot y\right) \cdot \left(0.16666666666666666 \cdot \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot -0.16666666666666666, x\right)\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (sinh y) (sin x)) x) -1e-291)
   (/
    (*
     y
     (fma
      (*
       (fma (* x x) (* x -0.16666666666666666) x)
       (fma (* y y) 0.0001984126984126984 0.008333333333333333))
      (* (* y y) (* y y))
      (*
       (* y y)
       (* 0.16666666666666666 (fma x (* (* x x) -0.16666666666666666) x)))))
    x)
   (sinh y)))

C

double code(double x, double y) {
	double tmp;
	if (((sinh(y) * sin(x)) / x) <= -1e-291) {
		tmp = (y * fma((fma((x * x), (x * -0.16666666666666666), x) * fma((y * y), 0.0001984126984126984, 0.008333333333333333)), ((y * y) * (y * y)), ((y * y) * (0.16666666666666666 * fma(x, ((x * x) * -0.16666666666666666), x))))) / x;
	} else {
		tmp = sinh(y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(sinh(y) * sin(x)) / x) <= -1e-291)
		tmp = Float64(Float64(y * fma(Float64(fma(Float64(x * x), Float64(x * -0.16666666666666666), x) * fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333)), Float64(Float64(y * y) * Float64(y * y)), Float64(Float64(y * y) * Float64(0.16666666666666666 * fma(x, Float64(Float64(x * x) * -0.16666666666666666), x))))) / x);
	else
		tmp = sinh(y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[Sinh[y], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], -1e-291], N[(N[(y * N[(N[(N[(N[(x * x), $MachinePrecision] * N[(x * -0.16666666666666666), $MachinePrecision] + x), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision] + N[(N[(y * y), $MachinePrecision] * N[(0.16666666666666666 * N[(x * N[(N[(x * x), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[Sinh[y], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -9.99999999999999962e-292

   1. Initial program 99.8%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 65.9

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

   5. Applied rewrites 65.9%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 57.8%

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

   9. Taylor expanded in y around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. lower-*.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. +-commutative N/A

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

       9. *-commutative N/A

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

       10. cube-mult N/A

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

       11. unpow2 N/A

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

       12. associate-*r* N/A

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

       13. *-commutative N/A

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

       14. lower-fma.f64 N/A

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

       15. lower-*.f64 N/A

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

       16. unpow2 N/A

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

       17. lower-*.f64 37.1

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

   11. Applied rewrites 37.1%

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

   if -9.99999999999999962e-292 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 84.5%

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

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 65.9%

      \[\leadsto \frac{\sinh y}{\color{blue}{1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -1 \cdot 10^{-291}:\\ \;\;\;\;\frac{y \cdot \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, x \cdot -0.16666666666666666, x\right) \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), \left(y \cdot y\right) \cdot \left(y \cdot y\right), \left(y \cdot y\right) \cdot \left(0.16666666666666666 \cdot \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot -0.16666666666666666, x\right)\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 60.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \mathsf{fma}\left(x \cdot x, 0.008333333333333333, -0.16666666666666666\right)\\ \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq 10^{-238}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.16666666666666666, x \cdot \left(x \cdot x\right), x\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right), y \cdot \left(y \cdot \left(y \cdot y\right)\right), \mathsf{fma}\left(y, y \cdot 0.16666666666666666, 1\right)\right)\right) \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, t\_0, x\right), \frac{y \cdot y}{x} \cdot \mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), x \cdot t\_0\right), y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* x (fma (* x x) 0.008333333333333333 -0.16666666666666666))))
   (if (<= (/ (* (sinh y) (sin x)) x) 1e-238)
     (*
      (*
       (fma -0.16666666666666666 (* x (* x x)) x)
       (fma
        (fma y (* y 0.0001984126984126984) 0.008333333333333333)
        (* y (* y (* y y)))
        (fma y (* y 0.16666666666666666) 1.0)))
      (/ y x))
     (fma
      y
      (fma
       (fma (* x x) t_0 x)
       (* (/ (* y y) x) (fma (* y y) 0.008333333333333333 0.16666666666666666))
       (* x t_0))
      y))))

C

double code(double x, double y) {
	double t_0 = x * fma((x * x), 0.008333333333333333, -0.16666666666666666);
	double tmp;
	if (((sinh(y) * sin(x)) / x) <= 1e-238) {
		tmp = (fma(-0.16666666666666666, (x * (x * x)), x) * fma(fma(y, (y * 0.0001984126984126984), 0.008333333333333333), (y * (y * (y * y))), fma(y, (y * 0.16666666666666666), 1.0))) * (y / x);
	} else {
		tmp = fma(y, fma(fma((x * x), t_0, x), (((y * y) / x) * fma((y * y), 0.008333333333333333, 0.16666666666666666)), (x * t_0)), y);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(x * fma(Float64(x * x), 0.008333333333333333, -0.16666666666666666))
	tmp = 0.0
	if (Float64(Float64(sinh(y) * sin(x)) / x) <= 1e-238)
		tmp = Float64(Float64(fma(-0.16666666666666666, Float64(x * Float64(x * x)), x) * fma(fma(y, Float64(y * 0.0001984126984126984), 0.008333333333333333), Float64(y * Float64(y * Float64(y * y))), fma(y, Float64(y * 0.16666666666666666), 1.0))) * Float64(y / x));
	else
		tmp = fma(y, fma(fma(Float64(x * x), t_0, x), Float64(Float64(Float64(y * y) / x) * fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666)), Float64(x * t_0)), y);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x * N[(N[(x * x), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sinh[y], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], 1e-238], N[(N[(N[(-0.16666666666666666 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] * N[(N[(y * N[(y * 0.0001984126984126984), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] * N[(y * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * N[(y * 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[(N[(x * x), $MachinePrecision] * t$95$0 + x), $MachinePrecision] * N[(N[(N[(y * y), $MachinePrecision] / x), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + N[(x * t$95$0), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 9.9999999999999999e-239

   1. Initial program 85.9%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 43.4

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

   5. Applied rewrites 43.4%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 38.1%

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

   10. Applied rewrites 53.3%

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

   if 9.9999999999999999e-239 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. sub-neg N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 79.9

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

   5. Applied rewrites 79.9%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 74.6%

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

   10. Applied rewrites 77.8%

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

4. Final simplification 61.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq 10^{-238}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.16666666666666666, x \cdot \left(x \cdot x\right), x\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right), y \cdot \left(y \cdot \left(y \cdot y\right)\right), \mathsf{fma}\left(y, y \cdot 0.16666666666666666, 1\right)\right)\right) \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, x \cdot \mathsf{fma}\left(x \cdot x, 0.008333333333333333, -0.16666666666666666\right), x\right), \frac{y \cdot y}{x} \cdot \mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, 0.008333333333333333, -0.16666666666666666\right)\right)\right), y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 60.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot y\right)\\ t_1 := x \cdot \left(x \cdot x\right)\\ \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq 10^{-259}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.16666666666666666, t\_1, x\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right), y \cdot t\_0, \mathsf{fma}\left(y, y \cdot 0.16666666666666666, 1\right)\right)\right) \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.008333333333333333, -0.16666666666666666\right), t\_1, x\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), t\_0, y\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y y))) (t_1 (* x (* x x))))
   (if (<= (/ (* (sinh y) (sin x)) x) 1e-259)
     (*
      (*
       (fma -0.16666666666666666 t_1 x)
       (fma
        (fma y (* y 0.0001984126984126984) 0.008333333333333333)
        (* y t_0)
        (fma y (* y 0.16666666666666666) 1.0)))
      (/ y x))
     (/
      (*
       (fma (fma (* x x) 0.008333333333333333 -0.16666666666666666) t_1 x)
       (fma
        (fma
         y
         (* y (fma (* y y) 0.0001984126984126984 0.008333333333333333))
         0.16666666666666666)
        t_0
        y))
      x))))

C

double code(double x, double y) {
	double t_0 = y * (y * y);
	double t_1 = x * (x * x);
	double tmp;
	if (((sinh(y) * sin(x)) / x) <= 1e-259) {
		tmp = (fma(-0.16666666666666666, t_1, x) * fma(fma(y, (y * 0.0001984126984126984), 0.008333333333333333), (y * t_0), fma(y, (y * 0.16666666666666666), 1.0))) * (y / x);
	} else {
		tmp = (fma(fma((x * x), 0.008333333333333333, -0.16666666666666666), t_1, x) * fma(fma(y, (y * fma((y * y), 0.0001984126984126984, 0.008333333333333333)), 0.16666666666666666), t_0, y)) / x;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(y * Float64(y * y))
	t_1 = Float64(x * Float64(x * x))
	tmp = 0.0
	if (Float64(Float64(sinh(y) * sin(x)) / x) <= 1e-259)
		tmp = Float64(Float64(fma(-0.16666666666666666, t_1, x) * fma(fma(y, Float64(y * 0.0001984126984126984), 0.008333333333333333), Float64(y * t_0), fma(y, Float64(y * 0.16666666666666666), 1.0))) * Float64(y / x));
	else
		tmp = Float64(Float64(fma(fma(Float64(x * x), 0.008333333333333333, -0.16666666666666666), t_1, x) * fma(fma(y, Float64(y * fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333)), 0.16666666666666666), t_0, y)) / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sinh[y], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], 1e-259], N[(N[(N[(-0.16666666666666666 * t$95$1 + x), $MachinePrecision] * N[(N[(y * N[(y * 0.0001984126984126984), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] * N[(y * t$95$0), $MachinePrecision] + N[(y * N[(y * 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(x * x), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision] * t$95$1 + x), $MachinePrecision] * N[(N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * t$95$0 + y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 1.0000000000000001e-259

   1. Initial program 85.8%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 43.7

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

   5. Applied rewrites 43.7%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 38.3%

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

   10. Applied rewrites 53.6%

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

   if 1.0000000000000001e-259 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. sub-neg N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 79.1

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

   5. Applied rewrites 79.1%

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

   6. Taylor expanded in y around 0

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. unpow2 N/A

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

      7. unpow3 N/A

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

      8. lower-fma.f64 N/A

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

   8. Applied rewrites 76.0%

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

4. Final simplification 61.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq 10^{-259}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.16666666666666666, x \cdot \left(x \cdot x\right), x\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right), y \cdot \left(y \cdot \left(y \cdot y\right)\right), \mathsf{fma}\left(y, y \cdot 0.16666666666666666, 1\right)\right)\right) \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.008333333333333333, -0.16666666666666666\right), x \cdot \left(x \cdot x\right), x\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 42.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y \cdot y\right) \cdot \left(y \cdot y\right)\\ \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -5 \cdot 10^{-235}:\\ \;\;\;\;\left(y \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(t\_0, \mathsf{fma}\left(y \cdot y, -3.306878306878307 \cdot 10^{-5}, -0.001388888888888889\right), \mathsf{fma}\left(y \cdot y, -0.027777777777777776, -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(t\_0, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), \left(y \cdot y\right) \cdot 0.16666666666666666\right), y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* y y) (* y y))))
   (if (<= (/ (* (sinh y) (sin x)) x) -5e-235)
     (*
      (* y (* x x))
      (fma
       t_0
       (fma (* y y) -3.306878306878307e-5 -0.001388888888888889)
       (fma (* y y) -0.027777777777777776 -0.16666666666666666)))
     (fma
      y
      (fma
       t_0
       (fma (* y y) 0.0001984126984126984 0.008333333333333333)
       (* (* y y) 0.16666666666666666))
      y))))

C

double code(double x, double y) {
	double t_0 = (y * y) * (y * y);
	double tmp;
	if (((sinh(y) * sin(x)) / x) <= -5e-235) {
		tmp = (y * (x * x)) * fma(t_0, fma((y * y), -3.306878306878307e-5, -0.001388888888888889), fma((y * y), -0.027777777777777776, -0.16666666666666666));
	} else {
		tmp = fma(y, fma(t_0, fma((y * y), 0.0001984126984126984, 0.008333333333333333), ((y * y) * 0.16666666666666666)), y);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(y * y) * Float64(y * y))
	tmp = 0.0
	if (Float64(Float64(sinh(y) * sin(x)) / x) <= -5e-235)
		tmp = Float64(Float64(y * Float64(x * x)) * fma(t_0, fma(Float64(y * y), -3.306878306878307e-5, -0.001388888888888889), fma(Float64(y * y), -0.027777777777777776, -0.16666666666666666)));
	else
		tmp = fma(y, fma(t_0, fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333), Float64(Float64(y * y) * 0.16666666666666666)), y);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * y), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sinh[y], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], -5e-235], N[(N[(y * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[(N[(y * y), $MachinePrecision] * -3.306878306878307e-5 + -0.001388888888888889), $MachinePrecision] + N[(N[(y * y), $MachinePrecision] * -0.027777777777777776 + -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(t$95$0 * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -4.9999999999999998e-235

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)} \cdot \sinh y}{x} \]
   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 68.7

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

   5. Applied rewrites 68.7%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 60.0%

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

   9. Taylor expanded in x around inf

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

       1. associate-*r* N/A

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

       2. lower-*.f64 N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 N/A

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

       7. *-commutative N/A

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

       8. associate-*l* N/A

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

       9. *-commutative N/A

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

       10. lower-fma.f64 N/A

           \[\leadsto \left(y \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\mathsf{fma}\left({y}^{4}, \frac{-1}{6} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right), \frac{-1}{6} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)} \]

   11. Applied rewrites 19.3%

       \[\leadsto \color{blue}{\left(y \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right), \mathsf{fma}\left(y \cdot y, -3.306878306878307 \cdot 10^{-5}, -0.001388888888888889\right), \mathsf{fma}\left(y \cdot y, -0.027777777777777776, -0.16666666666666666\right)\right)} \]

   if -4.9999999999999998e-235 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 85.2%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 42.9

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

   5. Applied rewrites 42.9%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 40.8%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

          \[\leadsto y \cdot \color{blue}{\left(\left(\frac{1}{6} \cdot {y}^{2} + {y}^{4} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) + 1\right)} \]

       2. distribute-lft-in N/A

          \[\leadsto \color{blue}{y \cdot \left(\frac{1}{6} \cdot {y}^{2} + {y}^{4} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) + y \cdot 1} \]

       3. *-rgt-identity N/A

          \[\leadsto y \cdot \left(\frac{1}{6} \cdot {y}^{2} + {y}^{4} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) + \color{blue}{y} \]

       4. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{1}{6} \cdot {y}^{2} + {y}^{4} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right), y\right)} \]

   11. Applied rewrites 60.1%

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

4. Final simplification 44.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -5 \cdot 10^{-235}:\\ \;\;\;\;\left(y \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right), \mathsf{fma}\left(y \cdot y, -3.306878306878307 \cdot 10^{-5}, -0.001388888888888889\right), \mathsf{fma}\left(y \cdot y, -0.027777777777777776, -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right), \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), \left(y \cdot y\right) \cdot 0.16666666666666666\right), y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 43.8% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (sinh y) (sin x)) x) 1e-5)
   (* y (fma (* x x) -0.16666666666666666 1.0))
   (* y (* (* y y) 0.16666666666666666))))

C

double code(double x, double y) {
	double tmp;
	if (((sinh(y) * sin(x)) / x) <= 1e-5) {
		tmp = y * fma((x * x), -0.16666666666666666, 1.0);
	} else {
		tmp = y * ((y * y) * 0.16666666666666666);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(sinh(y) * sin(x)) / x) <= 1e-5)
		tmp = Float64(y * fma(Float64(x * x), -0.16666666666666666, 1.0));
	else
		tmp = Float64(y * Float64(Float64(y * y) * 0.16666666666666666));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[Sinh[y], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], 1e-5], N[(y * N[(N[(x * x), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 1.00000000000000008e-5

   1. Initial program 87.6%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 70.3

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

   5. Applied rewrites 70.3%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 43.4

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

   8. Applied rewrites 43.4%

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

   if 1.00000000000000008e-5 < (/.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 73.8%

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

   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 53.5

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

   8. Applied rewrites 53.5%

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

   9. Taylor expanded in y around inf

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

       1. lower-*.f64 N/A

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

       2. cube-mult N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 53.5

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

   11. Applied rewrites 53.5%

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

   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. cube-mult N/A

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

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 61.4

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

   14. Applied rewrites 61.4%

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

4. Final simplification 47.9%

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

Alternative 16: 26.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -5 \cdot 10^{-235}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(y \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (sinh y) (sin x)) x) -5e-235)
   (* -0.16666666666666666 (* y (* x x)))
   y))

C

double code(double x, double y) {
	double tmp;
	if (((sinh(y) * sin(x)) / x) <= -5e-235) {
		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) <= (-5d-235)) 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) <= -5e-235) {
		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) <= -5e-235:
		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) <= -5e-235)
		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) <= -5e-235)
		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], -5e-235], 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) < -4.9999999999999998e-235

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)} \cdot \sinh y}{x} \]
   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 68.7

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

   5. Applied rewrites 68.7%

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

   6. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. unpow3 N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 40.3

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

   8. Applied rewrites 40.3%

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

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{-1}{6} \cdot \left({x}^{2} \cdot y\right)} \]
   10. 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 16.6

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

   11. Applied rewrites 16.6%

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

   if -4.9999999999999998e-235 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 85.2%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 60.5

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

   5. Applied rewrites 60.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 31.5%

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

      1. *-rgt-identity 31.5

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

   10. Applied rewrites 31.5%

       \[\leadsto \color{blue}{y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 25.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -5 \cdot 10^{-235}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(y \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.8%

   \[\frac{\sin x \cdot \sinh y}{x} \]
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 99.8

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

4. Applied rewrites 99.8%

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

5. Final simplification 99.8%

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

Alternative 18: 63.1% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.3 \cdot 10^{+92}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.16666666666666666, x \cdot \left(x \cdot x\right), x\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right), y \cdot \left(y \cdot \left(y \cdot y\right)\right), \mathsf{fma}\left(y, y \cdot 0.16666666666666666, 1\right)\right)\right) \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(y \cdot y\right) \cdot 0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1.3e+92)
   (*
    (*
     (fma -0.16666666666666666 (* x (* x x)) x)
     (fma
      (fma y (* y 0.0001984126984126984) 0.008333333333333333)
      (* y (* y (* y y)))
      (fma y (* y 0.16666666666666666) 1.0)))
    (/ y x))
   (* y (* (* y y) 0.16666666666666666))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.3e+92) {
		tmp = (fma(-0.16666666666666666, (x * (x * x)), x) * fma(fma(y, (y * 0.0001984126984126984), 0.008333333333333333), (y * (y * (y * y))), fma(y, (y * 0.16666666666666666), 1.0))) * (y / x);
	} else {
		tmp = y * ((y * y) * 0.16666666666666666);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.3e+92)
		tmp = Float64(Float64(fma(-0.16666666666666666, Float64(x * Float64(x * x)), x) * fma(fma(y, Float64(y * 0.0001984126984126984), 0.008333333333333333), Float64(y * Float64(y * Float64(y * y))), fma(y, Float64(y * 0.16666666666666666), 1.0))) * Float64(y / x));
	else
		tmp = Float64(y * Float64(Float64(y * y) * 0.16666666666666666));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.3e+92], N[(N[(N[(-0.16666666666666666 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] * N[(N[(y * N[(y * 0.0001984126984126984), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] * N[(y * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * N[(y * 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.2999999999999999e92

   1. Initial program 89.0%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 59.3

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

   5. Applied rewrites 59.3%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 53.9%

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

   10. Applied rewrites 65.7%

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

   if 1.2999999999999999e92 < 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 83.4%

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

   6. Taylor expanded in 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 17.8

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

   8. Applied rewrites 17.8%

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

   9. Taylor expanded in y around inf

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

       1. lower-*.f64 N/A

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

       2. cube-mult N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 17.2

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

   11. Applied rewrites 17.2%

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

   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. cube-mult N/A

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

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 57.5

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

   14. Applied rewrites 57.5%

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

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.3 \cdot 10^{+92}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.16666666666666666, x \cdot \left(x \cdot x\right), x\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right), y \cdot \left(y \cdot \left(y \cdot y\right)\right), \mathsf{fma}\left(y, y \cdot 0.16666666666666666, 1\right)\right)\right) \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(y \cdot y\right) \cdot 0.16666666666666666\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 60.5% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.7 \cdot 10^{+86}:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, x \cdot x, 1\right), y \cdot \left(y \cdot \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right)\right), \left(x \cdot x\right) \cdot -0.16666666666666666\right), y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(y \cdot y\right) \cdot 0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 2.7e+86)
   (fma
    y
    (fma
     (fma -0.16666666666666666 (* x x) 1.0)
     (* y (* y (fma y (* y 0.008333333333333333) 0.16666666666666666)))
     (* (* x x) -0.16666666666666666))
    y)
   (* y (* (* y y) 0.16666666666666666))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 2.7e+86) {
		tmp = fma(y, fma(fma(-0.16666666666666666, (x * x), 1.0), (y * (y * fma(y, (y * 0.008333333333333333), 0.16666666666666666))), ((x * x) * -0.16666666666666666)), y);
	} else {
		tmp = y * ((y * y) * 0.16666666666666666);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 2.7e+86)
		tmp = fma(y, fma(fma(-0.16666666666666666, Float64(x * x), 1.0), Float64(y * Float64(y * fma(y, Float64(y * 0.008333333333333333), 0.16666666666666666))), Float64(Float64(x * x) * -0.16666666666666666)), y);
	else
		tmp = Float64(y * Float64(Float64(y * y) * 0.16666666666666666));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 2.7e+86], N[(y * N[(N[(-0.16666666666666666 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(y * N[(y * N[(y * N[(y * 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(x * x), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision], N[(y * N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.70000000000000018e86

   1. Initial program 88.8%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 57.9

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

   5. Applied rewrites 57.9%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 63.5%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

       4. associate-*l* N/A

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

       5. cube-mult N/A

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

       6. unpow2 N/A

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

       7. associate-*l* N/A

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

       8. distribute-lft-out N/A

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

       9. lower-fma.f64 N/A

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

   11. Applied rewrites 64.4%

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

   if 2.70000000000000018e86 < 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 82.7%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)} \cdot \mathsf{fma}\left(y \cdot y, y \cdot \frac{1}{6}, y\right) \]
   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 16.4

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

   8. Applied rewrites 16.4%

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

   9. Taylor expanded in y around inf

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

       1. lower-*.f64 N/A

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

       2. cube-mult N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 15.8

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

   11. Applied rewrites 15.8%

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

   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. cube-mult N/A

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

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 55.0

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

   14. Applied rewrites 55.0%

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

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.7 \cdot 10^{+86}:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, x \cdot x, 1\right), y \cdot \left(y \cdot \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right)\right), \left(x \cdot x\right) \cdot -0.16666666666666666\right), y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(y \cdot y\right) \cdot 0.16666666666666666\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 62.6% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y \cdot y\right) \cdot 0.16666666666666666\\ \mathbf{if}\;x \leq 3.1:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right), \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), t\_0\right), y\right)\\ \mathbf{elif}\;x \leq 2.05 \cdot 10^{+108}:\\ \;\;\;\;\left(y \cdot \left(y \cdot y\right)\right) \cdot \mathsf{fma}\left(x \cdot x, -0.027777777777777776, 0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* y y) 0.16666666666666666)))
   (if (<= x 3.1)
     (fma
      y
      (fma
       (* (* y y) (* y y))
       (fma (* y y) 0.0001984126984126984 0.008333333333333333)
       t_0)
      y)
     (if (<= x 2.05e+108)
       (*
        (* y (* y y))
        (fma (* x x) -0.027777777777777776 0.16666666666666666))
       (* y t_0)))))

C

double code(double x, double y) {
	double t_0 = (y * y) * 0.16666666666666666;
	double tmp;
	if (x <= 3.1) {
		tmp = fma(y, fma(((y * y) * (y * y)), fma((y * y), 0.0001984126984126984, 0.008333333333333333), t_0), y);
	} else if (x <= 2.05e+108) {
		tmp = (y * (y * y)) * fma((x * x), -0.027777777777777776, 0.16666666666666666);
	} else {
		tmp = y * t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(y * y) * 0.16666666666666666)
	tmp = 0.0
	if (x <= 3.1)
		tmp = fma(y, fma(Float64(Float64(y * y) * Float64(y * y)), fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333), t_0), y);
	elseif (x <= 2.05e+108)
		tmp = Float64(Float64(y * Float64(y * y)) * fma(Float64(x * x), -0.027777777777777776, 0.16666666666666666));
	else
		tmp = Float64(y * t_0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]}, If[LessEqual[x, 3.1], N[(y * N[(N[(N[(y * y), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + t$95$0), $MachinePrecision] + y), $MachinePrecision], If[LessEqual[x, 2.05e+108], N[(N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.027777777777777776 + 0.16666666666666666), $MachinePrecision]), $MachinePrecision], N[(y * t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < 3.10000000000000009

   1. Initial program 87.8%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 64.8

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

   5. Applied rewrites 64.8%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 58.8%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

          \[\leadsto y \cdot \color{blue}{\left(\left(\frac{1}{6} \cdot {y}^{2} + {y}^{4} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) + 1\right)} \]

       2. distribute-lft-in N/A

          \[\leadsto \color{blue}{y \cdot \left(\frac{1}{6} \cdot {y}^{2} + {y}^{4} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) + y \cdot 1} \]

       3. *-rgt-identity N/A

          \[\leadsto y \cdot \left(\frac{1}{6} \cdot {y}^{2} + {y}^{4} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) + \color{blue}{y} \]

       4. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{1}{6} \cdot {y}^{2} + {y}^{4} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right), y\right)} \]

   11. Applied rewrites 68.3%

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

   if 3.10000000000000009 < x < 2.05e108

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*r/ N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. associate-*r* N/A

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

      11. associate-/l* N/A

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

      12. distribute-rgt-out N/A

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

      13. *-rgt-identity N/A

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

      14. distribute-lft-in N/A

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

   5. Applied rewrites 91.7%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)} \cdot \mathsf{fma}\left(y \cdot y, y \cdot \frac{1}{6}, y\right) \]
   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 14.2

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

   8. Applied rewrites 14.2%

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

   9. Taylor expanded in y around inf

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

       1. lower-*.f64 N/A

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

       2. cube-mult N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 15.1

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

   11. Applied rewrites 15.1%

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

   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. associate-*r* N/A

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

       2. distribute-rgt-out N/A

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

       3. lower-*.f64 N/A

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

       4. cube-mult N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 N/A

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

       9. *-commutative N/A

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

       10. lower-fma.f64 N/A

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

       11. unpow2 N/A

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

       12. lower-*.f64 20.3

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

   14. Applied rewrites 20.3%

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

   if 2.05e108 < 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 84.2%

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

   6. Taylor expanded in 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 14.4

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

   8. Applied rewrites 14.4%

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

   9. Taylor expanded in y around inf

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

       1. lower-*.f64 N/A

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

       2. cube-mult N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 13.7

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

   11. Applied rewrites 13.7%

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

   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. cube-mult N/A

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

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 58.3

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

   14. Applied rewrites 58.3%

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

4. Final simplification 62.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.1:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right), \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), \left(y \cdot y\right) \cdot 0.16666666666666666\right), y\right)\\ \mathbf{elif}\;x \leq 2.05 \cdot 10^{+108}:\\ \;\;\;\;\left(y \cdot \left(y \cdot y\right)\right) \cdot \mathsf{fma}\left(x \cdot x, -0.027777777777777776, 0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(y \cdot y\right) \cdot 0.16666666666666666\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 60.7% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.1:\\ \;\;\;\;\mathsf{fma}\left(y, y \cdot \left(y \cdot \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right)\right), y\right)\\ \mathbf{elif}\;x \leq 2.05 \cdot 10^{+108}:\\ \;\;\;\;\left(y \cdot \left(y \cdot y\right)\right) \cdot \mathsf{fma}\left(x \cdot x, -0.027777777777777776, 0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(y \cdot y\right) \cdot 0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 3.1)
   (fma y (* y (* y (fma y (* y 0.008333333333333333) 0.16666666666666666))) y)
   (if (<= x 2.05e+108)
     (* (* y (* y y)) (fma (* x x) -0.027777777777777776 0.16666666666666666))
     (* y (* (* y y) 0.16666666666666666)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 3.1) {
		tmp = fma(y, (y * (y * fma(y, (y * 0.008333333333333333), 0.16666666666666666))), y);
	} else if (x <= 2.05e+108) {
		tmp = (y * (y * y)) * fma((x * x), -0.027777777777777776, 0.16666666666666666);
	} else {
		tmp = y * ((y * y) * 0.16666666666666666);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 3.1)
		tmp = fma(y, Float64(y * Float64(y * fma(y, Float64(y * 0.008333333333333333), 0.16666666666666666))), y);
	elseif (x <= 2.05e+108)
		tmp = Float64(Float64(y * Float64(y * y)) * fma(Float64(x * x), -0.027777777777777776, 0.16666666666666666));
	else
		tmp = Float64(y * Float64(Float64(y * y) * 0.16666666666666666));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 3.1], N[(y * N[(y * N[(y * N[(y * N[(y * 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision], If[LessEqual[x, 2.05e+108], N[(N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.027777777777777776 + 0.16666666666666666), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 3.10000000000000009

   1. Initial program 87.8%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 62.7

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.008333333333333333, -0.16666666666666666\right), x \cdot \color{blue}{\left(x \cdot x\right)}, x\right) \cdot \sinh y}{x} \]

   5. Applied rewrites 62.7%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.008333333333333333, -0.16666666666666666\right), x \cdot \left(x \cdot x\right), x\right)} \cdot \sinh y}{x} \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left(1 + \left({x}^{2} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) + {y}^{2} \cdot \left(\frac{1}{120} \cdot \frac{{y}^{2} \cdot \left(x + {x}^{3} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right)}{x} + \frac{1}{6} \cdot \frac{x + {x}^{3} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)}{x}\right)\right)\right)} \]

   8. Applied rewrites 68.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.008333333333333333, -0.16666666666666666\right), \left(\mathsf{fma}\left(\mathsf{fma}\left(x, x \cdot 0.008333333333333333, -0.16666666666666666\right), x \cdot \left(x \cdot x\right), x\right) \cdot \frac{y \cdot y}{x}\right) \cdot \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right)\right), y\right)} \]

   9. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{{y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)}, y\right) \]
   10. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(y, \color{blue}{\left(y \cdot y\right)} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right), y\right) \]

       2. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(y, \color{blue}{y \cdot \left(y \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)}, y\right) \]

       3. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, \color{blue}{y \cdot \left(y \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)}, y\right) \]

       4. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, y \cdot \color{blue}{\left(y \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)}, y\right) \]

       5. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(y, y \cdot \left(y \cdot \color{blue}{\left(\frac{1}{120} \cdot {y}^{2} + \frac{1}{6}\right)}\right), y\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(y, y \cdot \left(y \cdot \left(\color{blue}{{y}^{2} \cdot \frac{1}{120}} + \frac{1}{6}\right)\right), y\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(y, y \cdot \left(y \cdot \left(\color{blue}{\left(y \cdot y\right)} \cdot \frac{1}{120} + \frac{1}{6}\right)\right), y\right) \]

       8. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(y, y \cdot \left(y \cdot \left(\color{blue}{y \cdot \left(y \cdot \frac{1}{120}\right)} + \frac{1}{6}\right)\right), y\right) \]

       9. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, y \cdot \left(y \cdot \color{blue}{\mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right)}\right), y\right) \]

       10. lower-*.f64 66.8

           \[\leadsto \mathsf{fma}\left(y, y \cdot \left(y \cdot \mathsf{fma}\left(y, \color{blue}{y \cdot 0.008333333333333333}, 0.16666666666666666\right)\right), y\right) \]

   11. Applied rewrites 66.8%

       \[\leadsto \mathsf{fma}\left(y, \color{blue}{y \cdot \left(y \cdot \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right)\right)}, y\right) \]

   if 3.10000000000000009 < x < 2.05e108

   1. Initial program 99.8%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left(\frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{\sin x}{x}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto y \cdot \color{blue}{\left(\frac{\sin x}{x} + \frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x}\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{y \cdot \frac{\sin x}{x} + y \cdot \left(\frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x}\right)} \]

      3. associate-*r/ N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + y \cdot \color{blue}{\frac{\frac{1}{6} \cdot \left({y}^{2} \cdot \sin x\right)}{x}} \]

      4. *-commutative N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + y \cdot \frac{\frac{1}{6} \cdot \color{blue}{\left(\sin x \cdot {y}^{2}\right)}}{x} \]

      5. associate-*r* N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + y \cdot \frac{\color{blue}{\left(\frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}}}{x} \]

      6. associate-*r/ N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + \color{blue}{\frac{y \cdot \left(\left(\frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}\right)}{x}} \]

      7. associate-*r* N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + \frac{y \cdot \color{blue}{\left(\frac{1}{6} \cdot \left(\sin x \cdot {y}^{2}\right)\right)}}{x} \]

      8. *-commutative N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + \frac{y \cdot \left(\frac{1}{6} \cdot \color{blue}{\left({y}^{2} \cdot \sin x\right)}\right)}{x} \]

      9. associate-*r* N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + \frac{y \cdot \color{blue}{\left(\left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \sin x\right)}}{x} \]

      10. associate-*r* N/A

          \[\leadsto y \cdot \frac{\sin x}{x} + \frac{\color{blue}{\left(y \cdot \left(\frac{1}{6} \cdot {y}^{2}\right)\right) \cdot \sin x}}{x} \]

      11. associate-/l* N/A

          \[\leadsto y \cdot \frac{\sin x}{x} + \color{blue}{\left(y \cdot \left(\frac{1}{6} \cdot {y}^{2}\right)\right) \cdot \frac{\sin x}{x}} \]

      12. distribute-rgt-out N/A

          \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot \left(y + y \cdot \left(\frac{1}{6} \cdot {y}^{2}\right)\right)} \]

      13. *-rgt-identity N/A

          \[\leadsto \frac{\sin x}{x} \cdot \left(\color{blue}{y \cdot 1} + y \cdot \left(\frac{1}{6} \cdot {y}^{2}\right)\right) \]

      14. distribute-lft-in N/A

          \[\leadsto \frac{\sin x}{x} \cdot \color{blue}{\left(y \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)} \]

   5. Applied rewrites 91.7%

      \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot \mathsf{fma}\left(y \cdot y, y \cdot 0.16666666666666666, y\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)} \cdot \mathsf{fma}\left(y \cdot y, y \cdot \frac{1}{6}, y\right) \]
   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 14.2

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right) \cdot \mathsf{fma}\left(y \cdot y, y \cdot 0.16666666666666666, y\right) \]

   8. Applied rewrites 14.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)} \cdot \mathsf{fma}\left(y \cdot y, y \cdot 0.16666666666666666, y\right) \]

   9. Taylor expanded in y around inf

      \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{-1}{5040}, \frac{1}{120}\right), \frac{-1}{6}\right), 1\right) \cdot \color{blue}{\left(\frac{1}{6} \cdot {y}^{3}\right)} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{-1}{5040}, \frac{1}{120}\right), \frac{-1}{6}\right), 1\right) \cdot \color{blue}{\left(\frac{1}{6} \cdot {y}^{3}\right)} \]

       2. cube-mult N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{-1}{5040}, \frac{1}{120}\right), \frac{-1}{6}\right), 1\right) \cdot \left(\frac{1}{6} \cdot \color{blue}{\left(y \cdot \left(y \cdot y\right)\right)}\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{-1}{5040}, \frac{1}{120}\right), \frac{-1}{6}\right), 1\right) \cdot \left(\frac{1}{6} \cdot \left(y \cdot \color{blue}{{y}^{2}}\right)\right) \]

       4. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{-1}{5040}, \frac{1}{120}\right), \frac{-1}{6}\right), 1\right) \cdot \left(\frac{1}{6} \cdot \color{blue}{\left(y \cdot {y}^{2}\right)}\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{-1}{5040}, \frac{1}{120}\right), \frac{-1}{6}\right), 1\right) \cdot \left(\frac{1}{6} \cdot \left(y \cdot \color{blue}{\left(y \cdot y\right)}\right)\right) \]

       6. lower-*.f64 15.1

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right) \cdot \left(0.16666666666666666 \cdot \left(y \cdot \color{blue}{\left(y \cdot y\right)}\right)\right) \]

   11. Applied rewrites 15.1%

       \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right) \cdot \color{blue}{\left(0.16666666666666666 \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)} \]

   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. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{-1}{36} \cdot {x}^{2}\right) \cdot {y}^{3}} + \frac{1}{6} \cdot {y}^{3} \]

       2. distribute-rgt-out N/A

          \[\leadsto \color{blue}{{y}^{3} \cdot \left(\frac{-1}{36} \cdot {x}^{2} + \frac{1}{6}\right)} \]

       3. lower-*.f64 N/A

          \[\leadsto \color{blue}{{y}^{3} \cdot \left(\frac{-1}{36} \cdot {x}^{2} + \frac{1}{6}\right)} \]

       4. cube-mult N/A

          \[\leadsto \color{blue}{\left(y \cdot \left(y \cdot y\right)\right)} \cdot \left(\frac{-1}{36} \cdot {x}^{2} + \frac{1}{6}\right) \]

       5. unpow2 N/A

          \[\leadsto \left(y \cdot \color{blue}{{y}^{2}}\right) \cdot \left(\frac{-1}{36} \cdot {x}^{2} + \frac{1}{6}\right) \]

       6. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(y \cdot {y}^{2}\right)} \cdot \left(\frac{-1}{36} \cdot {x}^{2} + \frac{1}{6}\right) \]

       7. unpow2 N/A

          \[\leadsto \left(y \cdot \color{blue}{\left(y \cdot y\right)}\right) \cdot \left(\frac{-1}{36} \cdot {x}^{2} + \frac{1}{6}\right) \]

       8. lower-*.f64 N/A

          \[\leadsto \left(y \cdot \color{blue}{\left(y \cdot y\right)}\right) \cdot \left(\frac{-1}{36} \cdot {x}^{2} + \frac{1}{6}\right) \]

       9. *-commutative N/A

          \[\leadsto \left(y \cdot \left(y \cdot y\right)\right) \cdot \left(\color{blue}{{x}^{2} \cdot \frac{-1}{36}} + \frac{1}{6}\right) \]

       10. lower-fma.f64 N/A

           \[\leadsto \left(y \cdot \left(y \cdot y\right)\right) \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{-1}{36}, \frac{1}{6}\right)} \]

       11. unpow2 N/A

           \[\leadsto \left(y \cdot \left(y \cdot y\right)\right) \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{36}, \frac{1}{6}\right) \]

       12. lower-*.f64 20.3

           \[\leadsto \left(y \cdot \left(y \cdot y\right)\right) \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, -0.027777777777777776, 0.16666666666666666\right) \]

   14. Applied rewrites 20.3%

       \[\leadsto \color{blue}{\left(y \cdot \left(y \cdot y\right)\right) \cdot \mathsf{fma}\left(x \cdot x, -0.027777777777777776, 0.16666666666666666\right)} \]

   if 2.05e108 < 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 84.2%

      \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot \mathsf{fma}\left(y \cdot y, y \cdot 0.16666666666666666, y\right)} \]

   6. Taylor expanded in 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 14.4

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right) \cdot \mathsf{fma}\left(y \cdot y, y \cdot 0.16666666666666666, y\right) \]

   8. Applied rewrites 14.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)} \cdot \mathsf{fma}\left(y \cdot y, y \cdot 0.16666666666666666, y\right) \]

   9. Taylor expanded in y around inf

      \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{-1}{5040}, \frac{1}{120}\right), \frac{-1}{6}\right), 1\right) \cdot \color{blue}{\left(\frac{1}{6} \cdot {y}^{3}\right)} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{-1}{5040}, \frac{1}{120}\right), \frac{-1}{6}\right), 1\right) \cdot \color{blue}{\left(\frac{1}{6} \cdot {y}^{3}\right)} \]

       2. cube-mult N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{-1}{5040}, \frac{1}{120}\right), \frac{-1}{6}\right), 1\right) \cdot \left(\frac{1}{6} \cdot \color{blue}{\left(y \cdot \left(y \cdot y\right)\right)}\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{-1}{5040}, \frac{1}{120}\right), \frac{-1}{6}\right), 1\right) \cdot \left(\frac{1}{6} \cdot \left(y \cdot \color{blue}{{y}^{2}}\right)\right) \]

       4. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{-1}{5040}, \frac{1}{120}\right), \frac{-1}{6}\right), 1\right) \cdot \left(\frac{1}{6} \cdot \color{blue}{\left(y \cdot {y}^{2}\right)}\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{-1}{5040}, \frac{1}{120}\right), \frac{-1}{6}\right), 1\right) \cdot \left(\frac{1}{6} \cdot \left(y \cdot \color{blue}{\left(y \cdot y\right)}\right)\right) \]

       6. lower-*.f64 13.7

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right) \cdot \left(0.16666666666666666 \cdot \left(y \cdot \color{blue}{\left(y \cdot y\right)}\right)\right) \]

   11. Applied rewrites 13.7%

       \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right) \cdot \color{blue}{\left(0.16666666666666666 \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)} \]

   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. cube-mult N/A

          \[\leadsto \color{blue}{\left(y \cdot \left(y \cdot y\right)\right)} \cdot \frac{1}{6} \]

       3. unpow2 N/A

          \[\leadsto \left(y \cdot \color{blue}{{y}^{2}}\right) \cdot \frac{1}{6} \]

       4. associate-*l* N/A

          \[\leadsto \color{blue}{y \cdot \left({y}^{2} \cdot \frac{1}{6}\right)} \]

       5. *-commutative N/A

          \[\leadsto y \cdot \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right)} \]

       6. lower-*.f64 N/A

          \[\leadsto \color{blue}{y \cdot \left(\frac{1}{6} \cdot {y}^{2}\right)} \]

       7. lower-*.f64 N/A

          \[\leadsto y \cdot \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right)} \]

       8. unpow2 N/A

          \[\leadsto y \cdot \left(\frac{1}{6} \cdot \color{blue}{\left(y \cdot y\right)}\right) \]

       9. lower-*.f64 58.3

          \[\leadsto y \cdot \left(0.16666666666666666 \cdot \color{blue}{\left(y \cdot y\right)}\right) \]

   14. Applied rewrites 58.3%

       \[\leadsto \color{blue}{y \cdot \left(0.16666666666666666 \cdot \left(y \cdot y\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 61.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.1:\\ \;\;\;\;\mathsf{fma}\left(y, y \cdot \left(y \cdot \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right)\right), y\right)\\ \mathbf{elif}\;x \leq 2.05 \cdot 10^{+108}:\\ \;\;\;\;\left(y \cdot \left(y \cdot y\right)\right) \cdot \mathsf{fma}\left(x \cdot x, -0.027777777777777776, 0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(y \cdot y\right) \cdot 0.16666666666666666\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 56.7% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot y\right)\\ \mathbf{if}\;x \leq 3.1:\\ \;\;\;\;\mathsf{fma}\left(0.16666666666666666, t\_0, y\right)\\ \mathbf{elif}\;x \leq 2.05 \cdot 10^{+108}:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(x \cdot x, -0.027777777777777776, 0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(y \cdot y\right) \cdot 0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y y))))
   (if (<= x 3.1)
     (fma 0.16666666666666666 t_0 y)
     (if (<= x 2.05e+108)
       (* t_0 (fma (* x x) -0.027777777777777776 0.16666666666666666))
       (* y (* (* y y) 0.16666666666666666))))))

C

double code(double x, double y) {
	double t_0 = y * (y * y);
	double tmp;
	if (x <= 3.1) {
		tmp = fma(0.16666666666666666, t_0, y);
	} else if (x <= 2.05e+108) {
		tmp = t_0 * fma((x * x), -0.027777777777777776, 0.16666666666666666);
	} else {
		tmp = y * ((y * y) * 0.16666666666666666);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(y * Float64(y * y))
	tmp = 0.0
	if (x <= 3.1)
		tmp = fma(0.16666666666666666, t_0, y);
	elseif (x <= 2.05e+108)
		tmp = Float64(t_0 * fma(Float64(x * x), -0.027777777777777776, 0.16666666666666666));
	else
		tmp = Float64(y * Float64(Float64(y * y) * 0.16666666666666666));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 3.1], N[(0.16666666666666666 * t$95$0 + y), $MachinePrecision], If[LessEqual[x, 2.05e+108], N[(t$95$0 * N[(N[(x * x), $MachinePrecision] * -0.027777777777777776 + 0.16666666666666666), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < 3.10000000000000009

   1. Initial program 87.8%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left(\frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{\sin x}{x}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto y \cdot \color{blue}{\left(\frac{\sin x}{x} + \frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x}\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{y \cdot \frac{\sin x}{x} + y \cdot \left(\frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x}\right)} \]

      3. associate-*r/ N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + y \cdot \color{blue}{\frac{\frac{1}{6} \cdot \left({y}^{2} \cdot \sin x\right)}{x}} \]

      4. *-commutative N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + y \cdot \frac{\frac{1}{6} \cdot \color{blue}{\left(\sin x \cdot {y}^{2}\right)}}{x} \]

      5. associate-*r* N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + y \cdot \frac{\color{blue}{\left(\frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}}}{x} \]

      6. associate-*r/ N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + \color{blue}{\frac{y \cdot \left(\left(\frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}\right)}{x}} \]

      7. associate-*r* N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + \frac{y \cdot \color{blue}{\left(\frac{1}{6} \cdot \left(\sin x \cdot {y}^{2}\right)\right)}}{x} \]

      8. *-commutative N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + \frac{y \cdot \left(\frac{1}{6} \cdot \color{blue}{\left({y}^{2} \cdot \sin x\right)}\right)}{x} \]

      9. associate-*r* N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + \frac{y \cdot \color{blue}{\left(\left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \sin x\right)}}{x} \]

      10. associate-*r* N/A

          \[\leadsto y \cdot \frac{\sin x}{x} + \frac{\color{blue}{\left(y \cdot \left(\frac{1}{6} \cdot {y}^{2}\right)\right) \cdot \sin x}}{x} \]

      11. associate-/l* N/A

          \[\leadsto y \cdot \frac{\sin x}{x} + \color{blue}{\left(y \cdot \left(\frac{1}{6} \cdot {y}^{2}\right)\right) \cdot \frac{\sin x}{x}} \]

      12. distribute-rgt-out N/A

          \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot \left(y + y \cdot \left(\frac{1}{6} \cdot {y}^{2}\right)\right)} \]

      13. *-rgt-identity N/A

          \[\leadsto \frac{\sin x}{x} \cdot \left(\color{blue}{y \cdot 1} + y \cdot \left(\frac{1}{6} \cdot {y}^{2}\right)\right) \]

      14. distribute-lft-in N/A

          \[\leadsto \frac{\sin x}{x} \cdot \color{blue}{\left(y \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)} \]

   5. Applied rewrites 83.8%

      \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot \mathsf{fma}\left(y \cdot y, y \cdot 0.16666666666666666, y\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{y + \frac{1}{6} \cdot {y}^{3}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{6} \cdot {y}^{3} + y} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{6}, {y}^{3}, y\right)} \]

      3. cube-mult N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{6}, \color{blue}{y \cdot \left(y \cdot y\right)}, y\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{6}, y \cdot \color{blue}{{y}^{2}}, y\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{6}, \color{blue}{y \cdot {y}^{2}}, y\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{6}, y \cdot \color{blue}{\left(y \cdot y\right)}, y\right) \]

      7. lower-*.f64 62.0

         \[\leadsto \mathsf{fma}\left(0.16666666666666666, y \cdot \color{blue}{\left(y \cdot y\right)}, y\right) \]

   8. Applied rewrites 62.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.16666666666666666, y \cdot \left(y \cdot y\right), y\right)} \]

   if 3.10000000000000009 < x < 2.05e108

   1. Initial program 99.8%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left(\frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{\sin x}{x}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto y \cdot \color{blue}{\left(\frac{\sin x}{x} + \frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x}\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{y \cdot \frac{\sin x}{x} + y \cdot \left(\frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x}\right)} \]

      3. associate-*r/ N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + y \cdot \color{blue}{\frac{\frac{1}{6} \cdot \left({y}^{2} \cdot \sin x\right)}{x}} \]

      4. *-commutative N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + y \cdot \frac{\frac{1}{6} \cdot \color{blue}{\left(\sin x \cdot {y}^{2}\right)}}{x} \]

      5. associate-*r* N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + y \cdot \frac{\color{blue}{\left(\frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}}}{x} \]

      6. associate-*r/ N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + \color{blue}{\frac{y \cdot \left(\left(\frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}\right)}{x}} \]

      7. associate-*r* N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + \frac{y \cdot \color{blue}{\left(\frac{1}{6} \cdot \left(\sin x \cdot {y}^{2}\right)\right)}}{x} \]

      8. *-commutative N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + \frac{y \cdot \left(\frac{1}{6} \cdot \color{blue}{\left({y}^{2} \cdot \sin x\right)}\right)}{x} \]

      9. associate-*r* N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + \frac{y \cdot \color{blue}{\left(\left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \sin x\right)}}{x} \]

      10. associate-*r* N/A

          \[\leadsto y \cdot \frac{\sin x}{x} + \frac{\color{blue}{\left(y \cdot \left(\frac{1}{6} \cdot {y}^{2}\right)\right) \cdot \sin x}}{x} \]

      11. associate-/l* N/A

          \[\leadsto y \cdot \frac{\sin x}{x} + \color{blue}{\left(y \cdot \left(\frac{1}{6} \cdot {y}^{2}\right)\right) \cdot \frac{\sin x}{x}} \]

      12. distribute-rgt-out N/A

          \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot \left(y + y \cdot \left(\frac{1}{6} \cdot {y}^{2}\right)\right)} \]

      13. *-rgt-identity N/A

          \[\leadsto \frac{\sin x}{x} \cdot \left(\color{blue}{y \cdot 1} + y \cdot \left(\frac{1}{6} \cdot {y}^{2}\right)\right) \]

      14. distribute-lft-in N/A

          \[\leadsto \frac{\sin x}{x} \cdot \color{blue}{\left(y \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)} \]

   5. Applied rewrites 91.7%

      \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot \mathsf{fma}\left(y \cdot y, y \cdot 0.16666666666666666, y\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)} \cdot \mathsf{fma}\left(y \cdot y, y \cdot \frac{1}{6}, y\right) \]
   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 14.2

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right) \cdot \mathsf{fma}\left(y \cdot y, y \cdot 0.16666666666666666, y\right) \]

   8. Applied rewrites 14.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)} \cdot \mathsf{fma}\left(y \cdot y, y \cdot 0.16666666666666666, y\right) \]

   9. Taylor expanded in y around inf

      \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{-1}{5040}, \frac{1}{120}\right), \frac{-1}{6}\right), 1\right) \cdot \color{blue}{\left(\frac{1}{6} \cdot {y}^{3}\right)} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{-1}{5040}, \frac{1}{120}\right), \frac{-1}{6}\right), 1\right) \cdot \color{blue}{\left(\frac{1}{6} \cdot {y}^{3}\right)} \]

       2. cube-mult N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{-1}{5040}, \frac{1}{120}\right), \frac{-1}{6}\right), 1\right) \cdot \left(\frac{1}{6} \cdot \color{blue}{\left(y \cdot \left(y \cdot y\right)\right)}\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{-1}{5040}, \frac{1}{120}\right), \frac{-1}{6}\right), 1\right) \cdot \left(\frac{1}{6} \cdot \left(y \cdot \color{blue}{{y}^{2}}\right)\right) \]

       4. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{-1}{5040}, \frac{1}{120}\right), \frac{-1}{6}\right), 1\right) \cdot \left(\frac{1}{6} \cdot \color{blue}{\left(y \cdot {y}^{2}\right)}\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{-1}{5040}, \frac{1}{120}\right), \frac{-1}{6}\right), 1\right) \cdot \left(\frac{1}{6} \cdot \left(y \cdot \color{blue}{\left(y \cdot y\right)}\right)\right) \]

       6. lower-*.f64 15.1

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right) \cdot \left(0.16666666666666666 \cdot \left(y \cdot \color{blue}{\left(y \cdot y\right)}\right)\right) \]

   11. Applied rewrites 15.1%

       \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right) \cdot \color{blue}{\left(0.16666666666666666 \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)} \]

   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. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{-1}{36} \cdot {x}^{2}\right) \cdot {y}^{3}} + \frac{1}{6} \cdot {y}^{3} \]

       2. distribute-rgt-out N/A

          \[\leadsto \color{blue}{{y}^{3} \cdot \left(\frac{-1}{36} \cdot {x}^{2} + \frac{1}{6}\right)} \]

       3. lower-*.f64 N/A

          \[\leadsto \color{blue}{{y}^{3} \cdot \left(\frac{-1}{36} \cdot {x}^{2} + \frac{1}{6}\right)} \]

       4. cube-mult N/A

          \[\leadsto \color{blue}{\left(y \cdot \left(y \cdot y\right)\right)} \cdot \left(\frac{-1}{36} \cdot {x}^{2} + \frac{1}{6}\right) \]

       5. unpow2 N/A

          \[\leadsto \left(y \cdot \color{blue}{{y}^{2}}\right) \cdot \left(\frac{-1}{36} \cdot {x}^{2} + \frac{1}{6}\right) \]

       6. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(y \cdot {y}^{2}\right)} \cdot \left(\frac{-1}{36} \cdot {x}^{2} + \frac{1}{6}\right) \]

       7. unpow2 N/A

          \[\leadsto \left(y \cdot \color{blue}{\left(y \cdot y\right)}\right) \cdot \left(\frac{-1}{36} \cdot {x}^{2} + \frac{1}{6}\right) \]

       8. lower-*.f64 N/A

          \[\leadsto \left(y \cdot \color{blue}{\left(y \cdot y\right)}\right) \cdot \left(\frac{-1}{36} \cdot {x}^{2} + \frac{1}{6}\right) \]

       9. *-commutative N/A

          \[\leadsto \left(y \cdot \left(y \cdot y\right)\right) \cdot \left(\color{blue}{{x}^{2} \cdot \frac{-1}{36}} + \frac{1}{6}\right) \]

       10. lower-fma.f64 N/A

           \[\leadsto \left(y \cdot \left(y \cdot y\right)\right) \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{-1}{36}, \frac{1}{6}\right)} \]

       11. unpow2 N/A

           \[\leadsto \left(y \cdot \left(y \cdot y\right)\right) \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{36}, \frac{1}{6}\right) \]

       12. lower-*.f64 20.3

           \[\leadsto \left(y \cdot \left(y \cdot y\right)\right) \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, -0.027777777777777776, 0.16666666666666666\right) \]

   14. Applied rewrites 20.3%

       \[\leadsto \color{blue}{\left(y \cdot \left(y \cdot y\right)\right) \cdot \mathsf{fma}\left(x \cdot x, -0.027777777777777776, 0.16666666666666666\right)} \]

   if 2.05e108 < 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 84.2%

      \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot \mathsf{fma}\left(y \cdot y, y \cdot 0.16666666666666666, y\right)} \]

   6. Taylor expanded in 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 14.4

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right) \cdot \mathsf{fma}\left(y \cdot y, y \cdot 0.16666666666666666, y\right) \]

   8. Applied rewrites 14.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)} \cdot \mathsf{fma}\left(y \cdot y, y \cdot 0.16666666666666666, y\right) \]

   9. Taylor expanded in y around inf

      \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{-1}{5040}, \frac{1}{120}\right), \frac{-1}{6}\right), 1\right) \cdot \color{blue}{\left(\frac{1}{6} \cdot {y}^{3}\right)} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{-1}{5040}, \frac{1}{120}\right), \frac{-1}{6}\right), 1\right) \cdot \color{blue}{\left(\frac{1}{6} \cdot {y}^{3}\right)} \]

       2. cube-mult N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{-1}{5040}, \frac{1}{120}\right), \frac{-1}{6}\right), 1\right) \cdot \left(\frac{1}{6} \cdot \color{blue}{\left(y \cdot \left(y \cdot y\right)\right)}\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{-1}{5040}, \frac{1}{120}\right), \frac{-1}{6}\right), 1\right) \cdot \left(\frac{1}{6} \cdot \left(y \cdot \color{blue}{{y}^{2}}\right)\right) \]

       4. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{-1}{5040}, \frac{1}{120}\right), \frac{-1}{6}\right), 1\right) \cdot \left(\frac{1}{6} \cdot \color{blue}{\left(y \cdot {y}^{2}\right)}\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{-1}{5040}, \frac{1}{120}\right), \frac{-1}{6}\right), 1\right) \cdot \left(\frac{1}{6} \cdot \left(y \cdot \color{blue}{\left(y \cdot y\right)}\right)\right) \]

       6. lower-*.f64 13.7

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right) \cdot \left(0.16666666666666666 \cdot \left(y \cdot \color{blue}{\left(y \cdot y\right)}\right)\right) \]

   11. Applied rewrites 13.7%

       \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right) \cdot \color{blue}{\left(0.16666666666666666 \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)} \]

   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. cube-mult N/A

          \[\leadsto \color{blue}{\left(y \cdot \left(y \cdot y\right)\right)} \cdot \frac{1}{6} \]

       3. unpow2 N/A

          \[\leadsto \left(y \cdot \color{blue}{{y}^{2}}\right) \cdot \frac{1}{6} \]

       4. associate-*l* N/A

          \[\leadsto \color{blue}{y \cdot \left({y}^{2} \cdot \frac{1}{6}\right)} \]

       5. *-commutative N/A

          \[\leadsto y \cdot \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right)} \]

       6. lower-*.f64 N/A

          \[\leadsto \color{blue}{y \cdot \left(\frac{1}{6} \cdot {y}^{2}\right)} \]

       7. lower-*.f64 N/A

          \[\leadsto y \cdot \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right)} \]

       8. unpow2 N/A

          \[\leadsto y \cdot \left(\frac{1}{6} \cdot \color{blue}{\left(y \cdot y\right)}\right) \]

       9. lower-*.f64 58.3

          \[\leadsto y \cdot \left(0.16666666666666666 \cdot \color{blue}{\left(y \cdot y\right)}\right) \]

   14. Applied rewrites 58.3%

       \[\leadsto \color{blue}{y \cdot \left(0.16666666666666666 \cdot \left(y \cdot y\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 57.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.1:\\ \;\;\;\;\mathsf{fma}\left(0.16666666666666666, y \cdot \left(y \cdot y\right), y\right)\\ \mathbf{elif}\;x \leq 2.05 \cdot 10^{+108}:\\ \;\;\;\;\left(y \cdot \left(y \cdot y\right)\right) \cdot \mathsf{fma}\left(x \cdot x, -0.027777777777777776, 0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(y \cdot y\right) \cdot 0.16666666666666666\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 56.6% accurate, 9.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.85 \cdot 10^{+35}:\\ \;\;\;\;\mathsf{fma}\left(0.16666666666666666, y \cdot \left(y \cdot y\right), y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(y \cdot y\right) \cdot 0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1.85e+35)
   (fma 0.16666666666666666 (* y (* y y)) y)
   (* y (* (* y y) 0.16666666666666666))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.85e+35) {
		tmp = fma(0.16666666666666666, (y * (y * y)), y);
	} else {
		tmp = y * ((y * y) * 0.16666666666666666);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.85e+35)
		tmp = fma(0.16666666666666666, Float64(y * Float64(y * y)), y);
	else
		tmp = Float64(y * Float64(Float64(y * y) * 0.16666666666666666));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.85e+35], N[(0.16666666666666666 * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision], N[(y * N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.85e35

   1. Initial program 88.2%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left(\frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{\sin x}{x}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto y \cdot \color{blue}{\left(\frac{\sin x}{x} + \frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x}\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{y \cdot \frac{\sin x}{x} + y \cdot \left(\frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x}\right)} \]

      3. associate-*r/ N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + y \cdot \color{blue}{\frac{\frac{1}{6} \cdot \left({y}^{2} \cdot \sin x\right)}{x}} \]

      4. *-commutative N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + y \cdot \frac{\frac{1}{6} \cdot \color{blue}{\left(\sin x \cdot {y}^{2}\right)}}{x} \]

      5. associate-*r* N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + y \cdot \frac{\color{blue}{\left(\frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}}}{x} \]

      6. associate-*r/ N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + \color{blue}{\frac{y \cdot \left(\left(\frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}\right)}{x}} \]

      7. associate-*r* N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + \frac{y \cdot \color{blue}{\left(\frac{1}{6} \cdot \left(\sin x \cdot {y}^{2}\right)\right)}}{x} \]

      8. *-commutative N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + \frac{y \cdot \left(\frac{1}{6} \cdot \color{blue}{\left({y}^{2} \cdot \sin x\right)}\right)}{x} \]

      9. associate-*r* N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + \frac{y \cdot \color{blue}{\left(\left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \sin x\right)}}{x} \]

      10. associate-*r* N/A

          \[\leadsto y \cdot \frac{\sin x}{x} + \frac{\color{blue}{\left(y \cdot \left(\frac{1}{6} \cdot {y}^{2}\right)\right) \cdot \sin x}}{x} \]

      11. associate-/l* N/A

          \[\leadsto y \cdot \frac{\sin x}{x} + \color{blue}{\left(y \cdot \left(\frac{1}{6} \cdot {y}^{2}\right)\right) \cdot \frac{\sin x}{x}} \]

      12. distribute-rgt-out N/A

          \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot \left(y + y \cdot \left(\frac{1}{6} \cdot {y}^{2}\right)\right)} \]

      13. *-rgt-identity N/A

          \[\leadsto \frac{\sin x}{x} \cdot \left(\color{blue}{y \cdot 1} + y \cdot \left(\frac{1}{6} \cdot {y}^{2}\right)\right) \]

      14. distribute-lft-in N/A

          \[\leadsto \frac{\sin x}{x} \cdot \color{blue}{\left(y \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)} \]

   5. Applied rewrites 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. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{6}, {y}^{3}, y\right)} \]

      3. cube-mult N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{6}, \color{blue}{y \cdot \left(y \cdot y\right)}, y\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{6}, y \cdot \color{blue}{{y}^{2}}, y\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{6}, \color{blue}{y \cdot {y}^{2}}, y\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{6}, y \cdot \color{blue}{\left(y \cdot y\right)}, y\right) \]

      7. lower-*.f64 60.3

         \[\leadsto \mathsf{fma}\left(0.16666666666666666, y \cdot \color{blue}{\left(y \cdot y\right)}, y\right) \]

   8. Applied rewrites 60.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.16666666666666666, y \cdot \left(y \cdot y\right), y\right)} \]

   if 1.85e35 < x

   1. Initial program 99.9%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left(\frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{\sin x}{x}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto y \cdot \color{blue}{\left(\frac{\sin x}{x} + \frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x}\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{y \cdot \frac{\sin x}{x} + y \cdot \left(\frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x}\right)} \]

      3. associate-*r/ N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + y \cdot \color{blue}{\frac{\frac{1}{6} \cdot \left({y}^{2} \cdot \sin x\right)}{x}} \]

      4. *-commutative N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + y \cdot \frac{\frac{1}{6} \cdot \color{blue}{\left(\sin x \cdot {y}^{2}\right)}}{x} \]

      5. associate-*r* N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + y \cdot \frac{\color{blue}{\left(\frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}}}{x} \]

      6. associate-*r/ N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + \color{blue}{\frac{y \cdot \left(\left(\frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}\right)}{x}} \]

      7. associate-*r* N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + \frac{y \cdot \color{blue}{\left(\frac{1}{6} \cdot \left(\sin x \cdot {y}^{2}\right)\right)}}{x} \]

      8. *-commutative N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + \frac{y \cdot \left(\frac{1}{6} \cdot \color{blue}{\left({y}^{2} \cdot \sin x\right)}\right)}{x} \]

      9. associate-*r* N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + \frac{y \cdot \color{blue}{\left(\left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \sin x\right)}}{x} \]

      10. associate-*r* N/A

          \[\leadsto y \cdot \frac{\sin x}{x} + \frac{\color{blue}{\left(y \cdot \left(\frac{1}{6} \cdot {y}^{2}\right)\right) \cdot \sin x}}{x} \]

      11. associate-/l* N/A

          \[\leadsto y \cdot \frac{\sin x}{x} + \color{blue}{\left(y \cdot \left(\frac{1}{6} \cdot {y}^{2}\right)\right) \cdot \frac{\sin x}{x}} \]

      12. distribute-rgt-out N/A

          \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot \left(y + y \cdot \left(\frac{1}{6} \cdot {y}^{2}\right)\right)} \]

      13. *-rgt-identity N/A

          \[\leadsto \frac{\sin x}{x} \cdot \left(\color{blue}{y \cdot 1} + y \cdot \left(\frac{1}{6} \cdot {y}^{2}\right)\right) \]

      14. distribute-lft-in N/A

          \[\leadsto \frac{\sin x}{x} \cdot \color{blue}{\left(y \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)} \]

   5. Applied rewrites 85.9%

      \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot \mathsf{fma}\left(y \cdot y, y \cdot 0.16666666666666666, y\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)} \cdot \mathsf{fma}\left(y \cdot y, y \cdot \frac{1}{6}, y\right) \]
   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 15.4

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right) \cdot \mathsf{fma}\left(y \cdot y, y \cdot 0.16666666666666666, y\right) \]

   8. Applied rewrites 15.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)} \cdot \mathsf{fma}\left(y \cdot y, y \cdot 0.16666666666666666, y\right) \]

   9. Taylor expanded in y around inf

      \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{-1}{5040}, \frac{1}{120}\right), \frac{-1}{6}\right), 1\right) \cdot \color{blue}{\left(\frac{1}{6} \cdot {y}^{3}\right)} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{-1}{5040}, \frac{1}{120}\right), \frac{-1}{6}\right), 1\right) \cdot \color{blue}{\left(\frac{1}{6} \cdot {y}^{3}\right)} \]

       2. cube-mult N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{-1}{5040}, \frac{1}{120}\right), \frac{-1}{6}\right), 1\right) \cdot \left(\frac{1}{6} \cdot \color{blue}{\left(y \cdot \left(y \cdot y\right)\right)}\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{-1}{5040}, \frac{1}{120}\right), \frac{-1}{6}\right), 1\right) \cdot \left(\frac{1}{6} \cdot \left(y \cdot \color{blue}{{y}^{2}}\right)\right) \]

       4. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{-1}{5040}, \frac{1}{120}\right), \frac{-1}{6}\right), 1\right) \cdot \left(\frac{1}{6} \cdot \color{blue}{\left(y \cdot {y}^{2}\right)}\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{-1}{5040}, \frac{1}{120}\right), \frac{-1}{6}\right), 1\right) \cdot \left(\frac{1}{6} \cdot \left(y \cdot \color{blue}{\left(y \cdot y\right)}\right)\right) \]

       6. lower-*.f64 15.2

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right) \cdot \left(0.16666666666666666 \cdot \left(y \cdot \color{blue}{\left(y \cdot y\right)}\right)\right) \]

   11. Applied rewrites 15.2%

       \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right) \cdot \color{blue}{\left(0.16666666666666666 \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)} \]

   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. cube-mult N/A

          \[\leadsto \color{blue}{\left(y \cdot \left(y \cdot y\right)\right)} \cdot \frac{1}{6} \]

       3. unpow2 N/A

          \[\leadsto \left(y \cdot \color{blue}{{y}^{2}}\right) \cdot \frac{1}{6} \]

       4. associate-*l* N/A

          \[\leadsto \color{blue}{y \cdot \left({y}^{2} \cdot \frac{1}{6}\right)} \]

       5. *-commutative N/A

          \[\leadsto y \cdot \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right)} \]

       6. lower-*.f64 N/A

          \[\leadsto \color{blue}{y \cdot \left(\frac{1}{6} \cdot {y}^{2}\right)} \]

       7. lower-*.f64 N/A

          \[\leadsto y \cdot \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right)} \]

       8. unpow2 N/A

          \[\leadsto y \cdot \left(\frac{1}{6} \cdot \color{blue}{\left(y \cdot y\right)}\right) \]

       9. lower-*.f64 45.2

          \[\leadsto y \cdot \left(0.16666666666666666 \cdot \color{blue}{\left(y \cdot y\right)}\right) \]

   14. Applied rewrites 45.2%

       \[\leadsto \color{blue}{y \cdot \left(0.16666666666666666 \cdot \left(y \cdot y\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.85 \cdot 10^{+35}:\\ \;\;\;\;\mathsf{fma}\left(0.16666666666666666, y \cdot \left(y \cdot y\right), y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(y \cdot y\right) \cdot 0.16666666666666666\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 28.1% accurate, 217.0× speedup

Math

\[\begin{array}{l} \\ y \end{array} \]

FPCore

(FPCore (x y) :precision binary64 y)

C

double code(double x, double y) {
	return y;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = y
end function

Java

public static double code(double x, double y) {
	return y;
}

Python

def code(x, y):
	return y

Julia

function code(x, y)
	return y
end

MATLAB

function tmp = code(x, y)
	tmp = y;
end

Wolfram

code[x_, y_] := y

Derivation

1. Initial program 90.8%

   \[\frac{\sin x \cdot \sinh y}{x} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\frac{y \cdot \sin x}{x}} \]
4. Step-by-step derivation

   1. associate-/l* N/A

      \[\leadsto \color{blue}{y \cdot \frac{\sin x}{x}} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{y \cdot \frac{\sin x}{x}} \]

   3. lower-/.f64 N/A

      \[\leadsto y \cdot \color{blue}{\frac{\sin x}{x}} \]

   4. lower-sin.f64 53.6

      \[\leadsto y \cdot \frac{\color{blue}{\sin x}}{x} \]

5. Applied rewrites 53.6%

   \[\leadsto \color{blue}{y \cdot \frac{\sin x}{x}} \]

6. Taylor expanded in x around 0

   \[\leadsto y \cdot \color{blue}{1} \]
7. Step-by-step derivation

8. Applied rewrites 28.9%

   \[\leadsto y \cdot \color{blue}{1} \]
9. Step-by-step derivation

   1. *-rgt-identity 28.9

      \[\leadsto \color{blue}{y} \]

10. Applied rewrites 28.9%

    \[\leadsto \color{blue}{y} \]
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 2024219 
(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))