Linear.Quaternion:$ccosh from linear-1.19.1.3

Percentage Accurate: 88.7% → 99.8%

Time: 12.8s

Alternatives: 21

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-/.f64 N/A

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

   2. clear-num N/A

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

   3. lift-*.f64 N/A

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

   4. associate-/r* N/A

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

   5. clear-num N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 99.9

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

4. Applied rewrites 99.9%

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

Alternative 2: 74.6% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sinh y) (sin x)) x)))
   (if (<= t_0 (- INFINITY))
     (/ (* (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) 1.0)))))

C

double code(double x, double y) {
	double t_0 = (sinh(y) * sin(x)) / x;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (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) / 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(sinh(y) * sin(x)) / x)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(sinh(y) * Float64(x * Float64(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 = Float64(sinh(y) / 1.0);
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 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

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

   8. Applied rewrites 32.8%

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

   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-/.f64 N/A

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

      2. clear-num N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. clear-num N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 84.6%

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

4. Final simplification 80.8%

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

Alternative 3: 74.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y \cdot \sin x}{x}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\frac{\sinh y \cdot \left(x \cdot \left(x \cdot \left(x \cdot -0.16666666666666666\right)\right)\right)}{x}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-25}:\\ \;\;\;\;\frac{y}{\frac{x}{\sin x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sinh y}{1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sinh y) (sin x)) x)))
   (if (<= t_0 (- INFINITY))
     (/ (* (sinh y) (* x (* x (* x -0.16666666666666666)))) x)
     (if (<= t_0 2e-25) (/ y (/ x (sin x))) (/ (sinh y) 1.0)))))

C

double code(double x, double y) {
	double t_0 = (sinh(y) * sin(x)) / x;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (sinh(y) * (x * (x * (x * -0.16666666666666666)))) / x;
	} else if (t_0 <= 2e-25) {
		tmp = y / (x / sin(x));
	} else {
		tmp = sinh(y) / 1.0;
	}
	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) / 1.0;
	}
	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) / 1.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(sinh(y) * sin(x)) / x)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(sinh(y) * Float64(x * Float64(x * Float64(x * -0.16666666666666666)))) / x);
	elseif (t_0 <= 2e-25)
		tmp = Float64(y / Float64(x / sin(x)));
	else
		tmp = Float64(sinh(y) / 1.0);
	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) / 1.0;
	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[(x * N[(x * N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[t$95$0, 2e-25], N[(y / N[(x / N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sinh[y], $MachinePrecision] / 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 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

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

   8. Applied rewrites 32.8%

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

   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

   7. Applied rewrites 99.4%

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

   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-/.f64 N/A

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

      2. clear-num N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. clear-num N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 85.3%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -\infty:\\ \;\;\;\;\frac{\sinh y \cdot \left(x \cdot \left(x \cdot \left(x \cdot -0.16666666666666666\right)\right)\right)}{x}\\ \mathbf{elif}\;\frac{\sinh y \cdot \sin x}{x} \leq 2 \cdot 10^{-25}:\\ \;\;\;\;\frac{y}{\frac{x}{\sin x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sinh y}{1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y \cdot \sin x}{x}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), x\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)}{x}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-25}:\\ \;\;\;\;\frac{y}{\frac{x}{\sin x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sinh y}{1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sinh y) (sin x)) x)))
   (if (<= t_0 (- INFINITY))
     (/
      (*
       (fma
        (* x x)
        (*
         x
         (fma
          (* x x)
          (fma (* x x) -0.0001984126984126984 0.008333333333333333)
          -0.16666666666666666))
        x)
       (fma
        (fma y (* y 0.008333333333333333) 0.16666666666666666)
        (* y (* y y))
        y))
      x)
     (if (<= t_0 2e-25) (/ y (/ x (sin x))) (/ (sinh y) 1.0)))))

C

double code(double x, double y) {
	double t_0 = (sinh(y) * sin(x)) / x;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (fma((x * x), (x * fma((x * x), fma((x * x), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666)), x) * fma(fma(y, (y * 0.008333333333333333), 0.16666666666666666), (y * (y * y)), y)) / x;
	} else if (t_0 <= 2e-25) {
		tmp = y / (x / sin(x));
	} else {
		tmp = sinh(y) / 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(sinh(y) * sin(x)) / x)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(fma(Float64(x * x), Float64(x * fma(Float64(x * x), fma(Float64(x * x), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666)), x) * fma(fma(y, Float64(y * 0.008333333333333333), 0.16666666666666666), Float64(y * Float64(y * y)), y)) / x);
	elseif (t_0 <= 2e-25)
		tmp = Float64(y / Float64(x / sin(x)));
	else
		tmp = Float64(sinh(y) / 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sinh[y], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[(x * x), $MachinePrecision] * N[(x * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] * N[(N[(y * N[(y * 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[t$95$0, 2e-25], N[(y / N[(x / N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sinh[y], $MachinePrecision] / 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 75.2

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

   5. Applied rewrites 75.2%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-*l* N/A

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

      4. *-lft-identity N/A

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

      5. lower-fma.f64 N/A

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

   8. Applied rewrites 61.2%

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

   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

   7. Applied rewrites 99.4%

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

   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-/.f64 N/A

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

      2. clear-num N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. clear-num N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 85.3%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -\infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), x\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)}{x}\\ \mathbf{elif}\;\frac{\sinh y \cdot \sin x}{x} \leq 2 \cdot 10^{-25}:\\ \;\;\;\;\frac{y}{\frac{x}{\sin x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sinh y}{1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y \cdot \sin x}{x}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), x\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)}{x}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-25}:\\ \;\;\;\;y \cdot \frac{\sin x}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sinh y}{1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sinh y) (sin x)) x)))
   (if (<= t_0 (- INFINITY))
     (/
      (*
       (fma
        (* x x)
        (*
         x
         (fma
          (* x x)
          (fma (* x x) -0.0001984126984126984 0.008333333333333333)
          -0.16666666666666666))
        x)
       (fma
        (fma y (* y 0.008333333333333333) 0.16666666666666666)
        (* y (* y y))
        y))
      x)
     (if (<= t_0 2e-25) (* y (/ (sin x) x)) (/ (sinh y) 1.0)))))

C

double code(double x, double y) {
	double t_0 = (sinh(y) * sin(x)) / x;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (fma((x * x), (x * fma((x * x), fma((x * x), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666)), x) * fma(fma(y, (y * 0.008333333333333333), 0.16666666666666666), (y * (y * y)), y)) / x;
	} else if (t_0 <= 2e-25) {
		tmp = y * (sin(x) / x);
	} else {
		tmp = sinh(y) / 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(sinh(y) * sin(x)) / x)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(fma(Float64(x * x), Float64(x * fma(Float64(x * x), fma(Float64(x * x), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666)), x) * fma(fma(y, Float64(y * 0.008333333333333333), 0.16666666666666666), Float64(y * Float64(y * y)), y)) / x);
	elseif (t_0 <= 2e-25)
		tmp = Float64(y * Float64(sin(x) / x));
	else
		tmp = Float64(sinh(y) / 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sinh[y], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[(x * x), $MachinePrecision] * N[(x * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] * N[(N[(y * N[(y * 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[t$95$0, 2e-25], N[(y * N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[Sinh[y], $MachinePrecision] / 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 75.2

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

   5. Applied rewrites 75.2%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-*l* N/A

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

      4. *-lft-identity N/A

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

      5. lower-fma.f64 N/A

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

   8. Applied rewrites 61.2%

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

   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-/.f64 N/A

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

      2. clear-num N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. clear-num N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 85.3%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -\infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), x\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)}{x}\\ \mathbf{elif}\;\frac{\sinh y \cdot \sin x}{x} \leq 2 \cdot 10^{-25}:\\ \;\;\;\;y \cdot \frac{\sin x}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sinh y}{1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 58.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y \cdot \sin x}{x}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-235}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, -0.16666666666666666 \cdot \left(x \cdot x\right), x\right) \cdot \mathsf{fma}\left(y \cdot y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y\right)}{x}\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(\mathsf{fma}\left(y, \left(y \cdot y\right) \cdot 0.16666666666666666, y\right) \cdot \left(y \cdot \mathsf{fma}\left(0.16666666666666666, y \cdot y, -1\right)\right)\right) \cdot \frac{1}{y \cdot \mathsf{fma}\left(y, y \cdot 0.16666666666666666, -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, y \cdot \left(y \cdot 0.16666666666666666\right), y\right)}{x} \cdot \mathsf{fma}\left(x \cdot x, x \cdot \mathsf{fma}\left(x \cdot x, 0.008333333333333333, -0.16666666666666666\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sinh y) (sin x)) x)))
   (if (<= t_0 -5e-235)
     (/
      (*
       (fma x (* -0.16666666666666666 (* x x)) x)
       (fma
        (* y y)
        (*
         y
         (fma
          (* y y)
          (fma (* y y) 0.0001984126984126984 0.008333333333333333)
          0.16666666666666666))
        y))
      x)
     (if (<= t_0 0.0)
       (*
        (*
         (fma y (* (* y y) 0.16666666666666666) y)
         (* y (fma 0.16666666666666666 (* y y) -1.0)))
        (/ 1.0 (* y (fma y (* y 0.16666666666666666) -1.0))))
       (*
        (/ (fma y (* y (* y 0.16666666666666666)) y) x)
        (fma
         (* x x)
         (* x (fma (* x x) 0.008333333333333333 -0.16666666666666666))
         x))))))

C

double code(double x, double y) {
	double t_0 = (sinh(y) * sin(x)) / x;
	double tmp;
	if (t_0 <= -5e-235) {
		tmp = (fma(x, (-0.16666666666666666 * (x * x)), x) * fma((y * y), (y * fma((y * y), fma((y * y), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666)), y)) / x;
	} else if (t_0 <= 0.0) {
		tmp = (fma(y, ((y * y) * 0.16666666666666666), y) * (y * fma(0.16666666666666666, (y * y), -1.0))) * (1.0 / (y * fma(y, (y * 0.16666666666666666), -1.0)));
	} else {
		tmp = (fma(y, (y * (y * 0.16666666666666666)), y) / x) * fma((x * x), (x * fma((x * x), 0.008333333333333333, -0.16666666666666666)), x);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(sinh(y) * sin(x)) / x)
	tmp = 0.0
	if (t_0 <= -5e-235)
		tmp = Float64(Float64(fma(x, Float64(-0.16666666666666666 * Float64(x * x)), x) * fma(Float64(y * y), Float64(y * fma(Float64(y * y), fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666)), y)) / x);
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(fma(y, Float64(Float64(y * y) * 0.16666666666666666), y) * Float64(y * fma(0.16666666666666666, Float64(y * y), -1.0))) * Float64(1.0 / Float64(y * fma(y, Float64(y * 0.16666666666666666), -1.0))));
	else
		tmp = Float64(Float64(fma(y, Float64(y * Float64(y * 0.16666666666666666)), y) / x) * fma(Float64(x * x), Float64(x * fma(Float64(x * x), 0.008333333333333333, -0.16666666666666666)), x));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sinh[y], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-235], N[(N[(N[(x * N[(-0.16666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(y * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(N[(y * N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] + y), $MachinePrecision] * N[(y * N[(0.16666666666666666 * N[(y * y), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(y * N[(y * N[(y * 0.16666666666666666), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y * N[(y * N[(y * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] / x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * N[(N[(x * x), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. +-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 60.0

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

   8. Applied rewrites 60.0%

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

   if -4.9999999999999998e-235 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 0.0

   1. Initial program 62.7%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*r/ N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. associate-*r* N/A

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

      11. associate-/l* N/A

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

      12. distribute-rgt-out N/A

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

      13. *-rgt-identity N/A

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

      14. distribute-lft-in N/A

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

   5. Applied rewrites 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

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

   8. Applied rewrites 44.9%

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

   10. Applied rewrites 25.4%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 49.6%

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

   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 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 82.2%

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

   7. Applied rewrites 90.8%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 72.3%

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

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -5 \cdot 10^{-235}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, -0.16666666666666666 \cdot \left(x \cdot x\right), x\right) \cdot \mathsf{fma}\left(y \cdot y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y\right)}{x}\\ \mathbf{elif}\;\frac{\sinh y \cdot \sin x}{x} \leq 0:\\ \;\;\;\;\left(\mathsf{fma}\left(y, \left(y \cdot y\right) \cdot 0.16666666666666666, y\right) \cdot \left(y \cdot \mathsf{fma}\left(0.16666666666666666, y \cdot y, -1\right)\right)\right) \cdot \frac{1}{y \cdot \mathsf{fma}\left(y, y \cdot 0.16666666666666666, -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, y \cdot \left(y \cdot 0.16666666666666666\right), y\right)}{x} \cdot \mathsf{fma}\left(x \cdot x, x \cdot \mathsf{fma}\left(x \cdot x, 0.008333333333333333, -0.16666666666666666\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 57.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y \cdot \sin x}{x}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-235}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right) \cdot \mathsf{fma}\left(x, x \cdot \left(x \cdot -0.16666666666666666\right), x\right)}{x}\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(\mathsf{fma}\left(y, \left(y \cdot y\right) \cdot 0.16666666666666666, y\right) \cdot \left(y \cdot \mathsf{fma}\left(0.16666666666666666, y \cdot y, -1\right)\right)\right) \cdot \frac{1}{y \cdot \mathsf{fma}\left(y, y \cdot 0.16666666666666666, -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, y \cdot \left(y \cdot 0.16666666666666666\right), y\right)}{x} \cdot \mathsf{fma}\left(x \cdot x, x \cdot \mathsf{fma}\left(x \cdot x, 0.008333333333333333, -0.16666666666666666\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sinh y) (sin x)) x)))
   (if (<= t_0 -5e-235)
     (/
      (*
       (fma
        (fma y (* y 0.008333333333333333) 0.16666666666666666)
        (* y (* y y))
        y)
       (fma x (* x (* x -0.16666666666666666)) x))
      x)
     (if (<= t_0 0.0)
       (*
        (*
         (fma y (* (* y y) 0.16666666666666666) y)
         (* y (fma 0.16666666666666666 (* y y) -1.0)))
        (/ 1.0 (* y (fma y (* y 0.16666666666666666) -1.0))))
       (*
        (/ (fma y (* y (* y 0.16666666666666666)) y) x)
        (fma
         (* x x)
         (* x (fma (* x x) 0.008333333333333333 -0.16666666666666666))
         x))))))

C

double code(double x, double y) {
	double t_0 = (sinh(y) * sin(x)) / x;
	double tmp;
	if (t_0 <= -5e-235) {
		tmp = (fma(fma(y, (y * 0.008333333333333333), 0.16666666666666666), (y * (y * y)), y) * fma(x, (x * (x * -0.16666666666666666)), x)) / x;
	} else if (t_0 <= 0.0) {
		tmp = (fma(y, ((y * y) * 0.16666666666666666), y) * (y * fma(0.16666666666666666, (y * y), -1.0))) * (1.0 / (y * fma(y, (y * 0.16666666666666666), -1.0)));
	} else {
		tmp = (fma(y, (y * (y * 0.16666666666666666)), y) / x) * fma((x * x), (x * fma((x * x), 0.008333333333333333, -0.16666666666666666)), x);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(sinh(y) * sin(x)) / x)
	tmp = 0.0
	if (t_0 <= -5e-235)
		tmp = Float64(Float64(fma(fma(y, Float64(y * 0.008333333333333333), 0.16666666666666666), Float64(y * Float64(y * y)), y) * fma(x, Float64(x * Float64(x * -0.16666666666666666)), x)) / x);
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(fma(y, Float64(Float64(y * y) * 0.16666666666666666), y) * Float64(y * fma(0.16666666666666666, Float64(y * y), -1.0))) * Float64(1.0 / Float64(y * fma(y, Float64(y * 0.16666666666666666), -1.0))));
	else
		tmp = Float64(Float64(fma(y, Float64(y * Float64(y * 0.16666666666666666)), y) / x) * fma(Float64(x * x), Float64(x * fma(Float64(x * x), 0.008333333333333333, -0.16666666666666666)), x));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sinh[y], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-235], N[(N[(N[(N[(y * N[(y * 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] * N[(x * N[(x * N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(N[(y * N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] + y), $MachinePrecision] * N[(y * N[(0.16666666666666666 * N[(y * y), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(y * N[(y * N[(y * 0.16666666666666666), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y * N[(y * N[(y * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] / x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * N[(N[(x * x), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 85.1

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

   5. Applied rewrites 85.1%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 58.9

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

   8. Applied rewrites 58.9%

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

   if -4.9999999999999998e-235 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 0.0

   1. Initial program 62.7%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*r/ N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. associate-*r* N/A

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

      11. associate-/l* N/A

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

      12. distribute-rgt-out N/A

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

      13. *-rgt-identity N/A

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

      14. distribute-lft-in N/A

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

   5. Applied rewrites 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

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

   8. Applied rewrites 44.9%

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

   10. Applied rewrites 25.4%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 49.6%

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

   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 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 82.2%

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

   7. Applied rewrites 90.8%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 72.3%

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

4. Final simplification 61.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -5 \cdot 10^{-235}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right) \cdot \mathsf{fma}\left(x, x \cdot \left(x \cdot -0.16666666666666666\right), x\right)}{x}\\ \mathbf{elif}\;\frac{\sinh y \cdot \sin x}{x} \leq 0:\\ \;\;\;\;\left(\mathsf{fma}\left(y, \left(y \cdot y\right) \cdot 0.16666666666666666, y\right) \cdot \left(y \cdot \mathsf{fma}\left(0.16666666666666666, y \cdot y, -1\right)\right)\right) \cdot \frac{1}{y \cdot \mathsf{fma}\left(y, y \cdot 0.16666666666666666, -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, y \cdot \left(y \cdot 0.16666666666666666\right), y\right)}{x} \cdot \mathsf{fma}\left(x \cdot x, x \cdot \mathsf{fma}\left(x \cdot x, 0.008333333333333333, -0.16666666666666666\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 61.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -5 \cdot 10^{-235}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, -0.16666666666666666 \cdot \left(x \cdot x\right), x\right) \cdot \mathsf{fma}\left(y \cdot y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sinh y}{1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (sinh y) (sin x)) x) -5e-235)
   (/
    (*
     (fma x (* -0.16666666666666666 (* x x)) x)
     (fma
      (* y y)
      (*
       y
       (fma
        (* y y)
        (fma (* y y) 0.0001984126984126984 0.008333333333333333)
        0.16666666666666666))
      y))
    x)
   (/ (sinh y) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (((sinh(y) * sin(x)) / x) <= -5e-235) {
		tmp = (fma(x, (-0.16666666666666666 * (x * x)), x) * fma((y * y), (y * fma((y * y), fma((y * y), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666)), y)) / x;
	} else {
		tmp = sinh(y) / 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(sinh(y) * sin(x)) / x) <= -5e-235)
		tmp = Float64(Float64(fma(x, Float64(-0.16666666666666666 * Float64(x * x)), x) * fma(Float64(y * y), Float64(y * fma(Float64(y * y), fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666)), y)) / x);
	else
		tmp = Float64(sinh(y) / 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[Sinh[y], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], -5e-235], N[(N[(N[(x * N[(-0.16666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(y * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[Sinh[y], $MachinePrecision] / 1.0), $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

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. +-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 60.0

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

   8. Applied rewrites 60.0%

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

   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. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. clear-num N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 64.2%

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

4. Final simplification 62.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -5 \cdot 10^{-235}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, -0.16666666666666666 \cdot \left(x \cdot x\right), x\right) \cdot \mathsf{fma}\left(y \cdot y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sinh y}{1}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 60.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq 5 \cdot 10^{-283}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot y, y \cdot \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), y\right)}{x} \cdot \mathsf{fma}\left(x \cdot -0.16666666666666666, x \cdot x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, y \cdot \left(y \cdot 0.16666666666666666\right), y\right)}{x} \cdot \mathsf{fma}\left(x \cdot x, x \cdot \mathsf{fma}\left(x \cdot x, 0.008333333333333333, -0.16666666666666666\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (sinh y) (sin x)) x) 5e-283)
   (*
    (/
     (fma
      (* y y)
      (* y (fma y (* y 0.008333333333333333) 0.16666666666666666))
      y)
     x)
    (fma (* x -0.16666666666666666) (* x x) x))
   (*
    (/ (fma y (* y (* y 0.16666666666666666)) y) x)
    (fma
     (* x x)
     (* x (fma (* x x) 0.008333333333333333 -0.16666666666666666))
     x))))

C

double code(double x, double y) {
	double tmp;
	if (((sinh(y) * sin(x)) / x) <= 5e-283) {
		tmp = (fma((y * y), (y * fma(y, (y * 0.008333333333333333), 0.16666666666666666)), y) / x) * fma((x * -0.16666666666666666), (x * x), x);
	} else {
		tmp = (fma(y, (y * (y * 0.16666666666666666)), y) / x) * fma((x * x), (x * fma((x * x), 0.008333333333333333, -0.16666666666666666)), x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(sinh(y) * sin(x)) / x) <= 5e-283)
		tmp = Float64(Float64(fma(Float64(y * y), Float64(y * fma(y, Float64(y * 0.008333333333333333), 0.16666666666666666)), y) / x) * fma(Float64(x * -0.16666666666666666), Float64(x * x), x));
	else
		tmp = Float64(Float64(fma(y, Float64(y * Float64(y * 0.16666666666666666)), y) / x) * fma(Float64(x * x), Float64(x * fma(Float64(x * x), 0.008333333333333333, -0.16666666666666666)), x));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[Sinh[y], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], 5e-283], N[(N[(N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * N[(y * 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] / x), $MachinePrecision] * N[(N[(x * -0.16666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y * N[(y * N[(y * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] / x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * N[(N[(x * x), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 5.0000000000000001e-283

   1. Initial program 85.4%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 76.4

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

   5. Applied rewrites 76.4%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 38.5

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

   8. Applied rewrites 38.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   10. Applied rewrites 55.7%

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

   if 5.0000000000000001e-283 < (/.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 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 82.0%

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

   7. Applied rewrites 90.7%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 73.0%

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

4. Final simplification 62.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq 5 \cdot 10^{-283}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot y, y \cdot \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), y\right)}{x} \cdot \mathsf{fma}\left(x \cdot -0.16666666666666666, x \cdot x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, y \cdot \left(y \cdot 0.16666666666666666\right), y\right)}{x} \cdot \mathsf{fma}\left(x \cdot x, x \cdot \mathsf{fma}\left(x \cdot x, 0.008333333333333333, -0.16666666666666666\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 41.5% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 86.1

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

   5. Applied rewrites 86.1%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 56.8

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

   8. Applied rewrites 56.8%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 18.2%

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

   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. 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 88.7%

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

   7. Applied rewrites 94.1%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 63.2%

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

4. Final simplification 44.7%

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

Alternative 11: 39.3% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (sinh y) (sin x)) x) -1e-291)
   (/
    (*
     (fma
      (fma y (* y 0.008333333333333333) 0.16666666666666666)
      (* y (* y y))
      y)
     (* x (* -0.16666666666666666 (* x x))))
    x)
   (*
    (fma (* y y) (* y 0.16666666666666666) y)
    (fma
     x
     (* x (fma 0.008333333333333333 (* x x) -0.16666666666666666))
     1.0))))

C

double code(double x, double y) {
	double tmp;
	if (((sinh(y) * sin(x)) / x) <= -1e-291) {
		tmp = (fma(fma(y, (y * 0.008333333333333333), 0.16666666666666666), (y * (y * y)), y) * (x * (-0.16666666666666666 * (x * x)))) / x;
	} else {
		tmp = fma((y * y), (y * 0.16666666666666666), y) * fma(x, (x * fma(0.008333333333333333, (x * x), -0.16666666666666666)), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(sinh(y) * sin(x)) / x) <= -1e-291)
		tmp = Float64(Float64(fma(fma(y, Float64(y * 0.008333333333333333), 0.16666666666666666), Float64(y * Float64(y * y)), y) * Float64(x * Float64(-0.16666666666666666 * Float64(x * x)))) / x);
	else
		tmp = Float64(fma(Float64(y * y), Float64(y * 0.16666666666666666), y) * fma(x, Float64(x * fma(0.008333333333333333, Float64(x * x), -0.16666666666666666)), 1.0));
	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[(N[(N[(y * N[(y * 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] * N[(x * N[(-0.16666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(y * y), $MachinePrecision] * N[(y * 0.16666666666666666), $MachinePrecision] + y), $MachinePrecision] * N[(x * N[(x * N[(0.008333333333333333 * N[(x * x), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $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 y around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 86.1

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

   5. Applied rewrites 86.1%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 56.8

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

   8. Applied rewrites 56.8%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 18.2%

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

   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. 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 88.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 57.4%

      \[\leadsto \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right), 1\right) \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, y \cdot 0.16666666666666666, y\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 41.3%

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

Alternative 12: 56.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq 5 \cdot 10^{-283}:\\ \;\;\;\;\mathsf{fma}\left(x, -0.16666666666666666 \cdot \left(x \cdot x\right), x\right) \cdot \frac{\mathsf{fma}\left(y, y \cdot \left(y \cdot 0.16666666666666666\right), y\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, y \cdot 0.16666666666666666, y\right) \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (sinh y) (sin x)) x) 5e-283)
   (*
    (fma x (* -0.16666666666666666 (* x x)) x)
    (/ (fma y (* y (* y 0.16666666666666666)) y) x))
   (*
    (fma (* y y) (* y 0.16666666666666666) y)
    (fma
     x
     (* x (fma 0.008333333333333333 (* x x) -0.16666666666666666))
     1.0))))

C

double code(double x, double y) {
	double tmp;
	if (((sinh(y) * sin(x)) / x) <= 5e-283) {
		tmp = fma(x, (-0.16666666666666666 * (x * x)), x) * (fma(y, (y * (y * 0.16666666666666666)), y) / x);
	} else {
		tmp = fma((y * y), (y * 0.16666666666666666), y) * fma(x, (x * fma(0.008333333333333333, (x * x), -0.16666666666666666)), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(sinh(y) * sin(x)) / x) <= 5e-283)
		tmp = Float64(fma(x, Float64(-0.16666666666666666 * Float64(x * x)), x) * Float64(fma(y, Float64(y * Float64(y * 0.16666666666666666)), y) / x));
	else
		tmp = Float64(fma(Float64(y * y), Float64(y * 0.16666666666666666), y) * fma(x, Float64(x * fma(0.008333333333333333, Float64(x * x), -0.16666666666666666)), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[Sinh[y], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], 5e-283], N[(N[(x * N[(-0.16666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] * N[(N[(y * N[(y * N[(y * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y * y), $MachinePrecision] * N[(y * 0.16666666666666666), $MachinePrecision] + y), $MachinePrecision] * N[(x * N[(x * N[(0.008333333333333333 * N[(x * x), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 5.0000000000000001e-283

   1. Initial program 85.4%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*r/ N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. associate-*r* N/A

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

      11. associate-/l* N/A

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

      12. distribute-rgt-out N/A

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

      13. *-rgt-identity N/A

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

      14. distribute-lft-in N/A

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

   5. Applied rewrites 86.2%

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

   7. Applied rewrites 89.7%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 54.5%

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

   if 5.0000000000000001e-283 < (/.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 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 82.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 64.3%

      \[\leadsto \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right), 1\right) \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, y \cdot 0.16666666666666666, y\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq 5 \cdot 10^{-283}:\\ \;\;\;\;\mathsf{fma}\left(x, -0.16666666666666666 \cdot \left(x \cdot x\right), x\right) \cdot \frac{\mathsf{fma}\left(y, y \cdot \left(y \cdot 0.16666666666666666\right), y\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, y \cdot 0.16666666666666666, y\right) \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 46.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq 5 \cdot 10^{-236}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, y \cdot 0.16666666666666666, y\right) \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (sinh y) (sin x)) x) 5e-236)
   (*
    y
    (fma
     (* x x)
     (fma
      (* x x)
      (fma (* x x) -0.0001984126984126984 0.008333333333333333)
      -0.16666666666666666)
     1.0))
   (*
    (fma (* y y) (* y 0.16666666666666666) y)
    (fma
     x
     (* x (fma 0.008333333333333333 (* x x) -0.16666666666666666))
     1.0))))

C

double code(double x, double y) {
	double tmp;
	if (((sinh(y) * sin(x)) / x) <= 5e-236) {
		tmp = y * fma((x * x), fma((x * x), fma((x * x), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666), 1.0);
	} else {
		tmp = fma((y * y), (y * 0.16666666666666666), y) * fma(x, (x * fma(0.008333333333333333, (x * x), -0.16666666666666666)), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(sinh(y) * sin(x)) / x) <= 5e-236)
		tmp = Float64(y * fma(Float64(x * x), fma(Float64(x * x), fma(Float64(x * x), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666), 1.0));
	else
		tmp = Float64(fma(Float64(y * y), Float64(y * 0.16666666666666666), y) * fma(x, Float64(x * fma(0.008333333333333333, Float64(x * x), -0.16666666666666666)), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[Sinh[y], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], 5e-236], N[(y * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y * y), $MachinePrecision] * N[(y * 0.16666666666666666), $MachinePrecision] + y), $MachinePrecision] * N[(x * N[(x * N[(0.008333333333333333 * N[(x * x), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 4.9999999999999998e-236

   1. Initial program 85.9%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 66.2

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

   5. Applied rewrites 66.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 41.0%

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

   if 4.9999999999999998e-236 < (/.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 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 80.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 67.4%

      \[\leadsto \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right), 1\right) \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, y \cdot 0.16666666666666666, y\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq 5 \cdot 10^{-236}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, y \cdot 0.16666666666666666, y\right) \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 46.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq 4 \cdot 10^{-152}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, y \cdot 0.16666666666666666, y\right) \cdot \mathsf{fma}\left(x, x \cdot \left(\left(x \cdot x\right) \cdot 0.008333333333333333\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (sinh y) (sin x)) x) 4e-152)
   (*
    y
    (fma
     (* x x)
     (fma
      (* x x)
      (fma (* x x) -0.0001984126984126984 0.008333333333333333)
      -0.16666666666666666)
     1.0))
   (*
    (fma (* y y) (* y 0.16666666666666666) y)
    (fma x (* x (* (* x x) 0.008333333333333333)) 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (((sinh(y) * sin(x)) / x) <= 4e-152) {
		tmp = y * fma((x * x), fma((x * x), fma((x * x), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666), 1.0);
	} else {
		tmp = fma((y * y), (y * 0.16666666666666666), y) * fma(x, (x * ((x * x) * 0.008333333333333333)), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(sinh(y) * sin(x)) / x) <= 4e-152)
		tmp = Float64(y * fma(Float64(x * x), fma(Float64(x * x), fma(Float64(x * x), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666), 1.0));
	else
		tmp = Float64(fma(Float64(y * y), Float64(y * 0.16666666666666666), y) * fma(x, Float64(x * Float64(Float64(x * x) * 0.008333333333333333)), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[Sinh[y], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], 4e-152], N[(y * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y * y), $MachinePrecision] * N[(y * 0.16666666666666666), $MachinePrecision] + y), $MachinePrecision] * N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 4.00000000000000026e-152

   1. Initial program 86.2%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y \cdot \sin x}{x}} \]
   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 67.0

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

   5. Applied rewrites 67.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 40.7%

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

   if 4.00000000000000026e-152 < (/.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 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 79.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 58.9%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 69.3%

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

   12. Taylor expanded in x around inf

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

   14. Applied rewrites 69.3%

       \[\leadsto \mathsf{fma}\left(x, x \cdot \left(\left(x \cdot x\right) \cdot 0.008333333333333333\right), 1\right) \cdot \mathsf{fma}\left(y \cdot y, y \cdot 0.16666666666666666, y\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq 4 \cdot 10^{-152}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, y \cdot 0.16666666666666666, y\right) \cdot \mathsf{fma}\left(x, x \cdot \left(\left(x \cdot x\right) \cdot 0.008333333333333333\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 45.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -2 \cdot 10^{-180}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \left(y \cdot y\right) \cdot 0.16666666666666666, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (sinh y) (sin x)) x) -2e-180)
   (*
    y
    (fma
     (* x x)
     (fma
      (* x x)
      (fma (* x x) -0.0001984126984126984 0.008333333333333333)
      -0.16666666666666666)
     1.0))
   (fma y (* (* y y) 0.16666666666666666) y)))

C

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

Julia

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

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[Sinh[y], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], -2e-180], N[(y * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y \cdot \sin x}{x}} \]
   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 35.3

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

   5. Applied rewrites 35.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 40.4%

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

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

   1. Initial program 86.1%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*r/ N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. associate-*r* N/A

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

      11. associate-/l* N/A

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

      12. distribute-rgt-out N/A

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

      13. *-rgt-identity N/A

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

      14. distribute-lft-in N/A

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

   5. Applied rewrites 89.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 53.7%

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

4. Final simplification 49.3%

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

Alternative 16: 38.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[Sinh[y], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], -5e-235], N[(N[(N[(y * y), $MachinePrecision] * N[(y * 0.16666666666666666), $MachinePrecision] + y), $MachinePrecision] * N[(-0.16666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   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 77.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 54.1%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 17.4%

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

   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

      \[\leadsto \color{blue}{y \cdot \left(\frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{\sin x}{x}\right)} \]
   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 89.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 54.8%

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

4. Final simplification 40.6%

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

Alternative 17: 37.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[Sinh[y], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], -5e-235], N[(y * N[(-0.16666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y \cdot \sin x}{x}} \]
   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 42.4

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

   5. Applied rewrites 42.4%

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

   7. Applied rewrites 54.9%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 38.4%

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

   11. Taylor expanded in x around inf

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

   13. Applied rewrites 16.6%

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

   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

      \[\leadsto \color{blue}{y \cdot \left(\frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{\sin x}{x}\right)} \]
   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 89.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 54.8%

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

4. Final simplification 40.3%

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

Alternative 18: 26.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -5 \cdot 10^{-235}:\\ \;\;\;\;y \cdot \left(-0.16666666666666666 \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (sinh y) (sin x)) x) -5e-235)
   (* y (* -0.16666666666666666 (* x x)))
   (* y 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (((sinh(y) * sin(x)) / x) <= -5e-235) {
		tmp = y * (-0.16666666666666666 * (x * x));
	} else {
		tmp = y * 1.0;
	}
	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 = y * ((-0.16666666666666666d0) * (x * x))
    else
        tmp = y * 1.0d0
    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 = y * (-0.16666666666666666 * (x * x));
	} else {
		tmp = y * 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((math.sinh(y) * math.sin(x)) / x) <= -5e-235:
		tmp = y * (-0.16666666666666666 * (x * x))
	else:
		tmp = y * 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(sinh(y) * sin(x)) / x) <= -5e-235)
		tmp = Float64(y * Float64(-0.16666666666666666 * Float64(x * x)));
	else
		tmp = Float64(y * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((sinh(y) * sin(x)) / x) <= -5e-235)
		tmp = y * (-0.16666666666666666 * (x * x));
	else
		tmp = y * 1.0;
	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[(y * N[(-0.16666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * 1.0), $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 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 42.4

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

   5. Applied rewrites 42.4%

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

   7. Applied rewrites 54.9%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 38.4%

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

   11. Taylor expanded in x around inf

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

   13. Applied rewrites 16.6%

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

   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

      \[\leadsto y \cdot 1 \]
   7. Step-by-step derivation

   8. Applied rewrites 31.5%

      \[\leadsto y \cdot 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 25.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -5 \cdot 10^{-235}:\\ \;\;\;\;y \cdot \left(-0.16666666666666666 \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 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-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 99.8

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

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 20: 36.3% accurate, 12.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(y, -0.16666666666666666 \cdot \left(x \cdot x\right), y\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (fma y (* -0.16666666666666666 (* x x)) y))

C

double code(double x, double y) {
	return fma(y, (-0.16666666666666666 * (x * x)), y);
}

Julia

function code(x, y)
	return fma(y, Float64(-0.16666666666666666 * Float64(x * x)), y)
end

Wolfram

code[x_, y_] := N[(y * N[(-0.16666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision]

Derivation

1. Initial program 90.8%

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

3. Taylor expanded in y around 0

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

   1. 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. Step-by-step derivation

7. Applied rewrites 64.9%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 35.6%

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

11. Final simplification 35.6%

    \[\leadsto \mathsf{fma}\left(y, -0.16666666666666666 \cdot \left(x \cdot x\right), y\right) \]
12. Add Preprocessing

Alternative 21: 28.1% accurate, 36.2× speedup

Math

\[\begin{array}{l} \\ y \cdot 1 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* y 1.0))

C

double code(double x, double y) {
	return y * 1.0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = y * 1.0d0
end function

Java

public static double code(double x, double y) {
	return y * 1.0;
}

Python

def code(x, y):
	return y * 1.0

Julia

function code(x, y)
	return Float64(y * 1.0)
end

MATLAB

function tmp = code(x, y)
	tmp = y * 1.0;
end

Wolfram

code[x_, y_] := N[(y * 1.0), $MachinePrecision]

Derivation

1. Initial program 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

   \[\leadsto y \cdot 1 \]
7. Step-by-step derivation

8. Applied rewrites 28.9%

   \[\leadsto y \cdot 1 \]
9. Add Preprocessing

Developer Target 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 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))