Linear.Quaternion:$ccosh from linear-1.19.1.3

Percentage Accurate: 88.6% → 99.9%

Time: 13.5s

Alternatives: 24

Speedup: 1.0×

• 49.7% 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: 88.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.7%

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

   1. lift-sin.f64 N/A

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

   2. lift-sinh.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-*l/ N/A

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

   5. associate-/r/ N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 100.0

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

4. Applied egg-rr 100.0%

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

Alternative 2: 86.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y \cdot \sin x}{x}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\frac{\sinh y \cdot \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot -0.16666666666666666, x\right)}{x}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\frac{\sin x}{x} \cdot \mathsf{fma}\left(y \cdot y, y \cdot 0.16666666666666666, y\right)\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sinh y) (sin x)) x)))
   (if (<= t_0 (- INFINITY))
     (/ (* (sinh y) (fma x (* (* x x) -0.16666666666666666) x)) x)
     (if (<= t_0 2e-8)
       (* (/ (sin x) x) (fma (* y y) (* y 0.16666666666666666) y))
       (sinh y)))))

C

double code(double x, double y) {
	double t_0 = (sinh(y) * sin(x)) / x;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (sinh(y) * fma(x, ((x * x) * -0.16666666666666666), x)) / x;
	} else if (t_0 <= 2e-8) {
		tmp = (sin(x) / x) * fma((y * y), (y * 0.16666666666666666), y);
	} else {
		tmp = sinh(y);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(sinh(y) * sin(x)) / x)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(sinh(y) * fma(x, Float64(Float64(x * x) * -0.16666666666666666), x)) / x);
	elseif (t_0 <= 2e-8)
		tmp = Float64(Float64(sin(x) / x) * fma(Float64(y * y), Float64(y * 0.16666666666666666), y));
	else
		tmp = sinh(y);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sinh[y], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[Sinh[y], $MachinePrecision] * N[(x * N[(N[(x * x), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[t$95$0, 2e-8], N[(N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(y * 0.16666666666666666), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision], N[Sinh[y], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 72.5

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

   5. Simplified 72.5%

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

   if -inf.0 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 2e-8

   1. Initial program 74.1%

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

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

   if 2e-8 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 100.0%

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

      1. lift-sin.f64 N/A

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

      2. lift-sinh.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l/ N/A

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

      5. associate-/r/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 100.0

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

   4. Applied egg-rr 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. Simplified 73.0%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -\infty:\\ \;\;\;\;\frac{\sinh y \cdot \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot -0.16666666666666666, x\right)}{x}\\ \mathbf{elif}\;\frac{\sinh y \cdot \sin x}{x} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\frac{\sin x}{x} \cdot \mathsf{fma}\left(y \cdot y, y \cdot 0.16666666666666666, y\right)\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y \cdot \sin x}{x}\\ t_1 := \mathsf{fma}\left(y \cdot y, y \cdot 0.16666666666666666, y\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;t\_1 \cdot \mathsf{fma}\left(x \cdot x, x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot -0.0001984126984126984\right)\right), 1\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\frac{\sin x}{x} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sinh y) (sin x)) x))
        (t_1 (fma (* y y) (* y 0.16666666666666666) y)))
   (if (<= t_0 (- INFINITY))
     (* t_1 (fma (* x x) (* x (* x (* (* x x) -0.0001984126984126984))) 1.0))
     (if (<= t_0 2e-8) (* (/ (sin x) x) t_1) (sinh y)))))

C

double code(double x, double y) {
	double t_0 = (sinh(y) * sin(x)) / x;
	double t_1 = fma((y * y), (y * 0.16666666666666666), y);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = t_1 * fma((x * x), (x * (x * ((x * x) * -0.0001984126984126984))), 1.0);
	} else if (t_0 <= 2e-8) {
		tmp = (sin(x) / x) * t_1;
	} else {
		tmp = sinh(y);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(sinh(y) * sin(x)) / x)
	t_1 = fma(Float64(y * y), Float64(y * 0.16666666666666666), y)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(t_1 * fma(Float64(x * x), Float64(x * Float64(x * Float64(Float64(x * x) * -0.0001984126984126984))), 1.0));
	elseif (t_0 <= 2e-8)
		tmp = Float64(Float64(sin(x) / x) * t_1);
	else
		tmp = sinh(y);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sinh[y], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, Block[{t$95$1 = N[(N[(y * y), $MachinePrecision] * N[(y * 0.16666666666666666), $MachinePrecision] + y), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(t$95$1 * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * -0.0001984126984126984), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e-8], N[(N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision] * t$95$1), $MachinePrecision], N[Sinh[y], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in 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. Simplified 64.2%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 55.8

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

   8. Simplified 55.8%

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

   9. Taylor expanded in x around inf

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

       1. metadata-eval N/A

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

       2. pow-sqr N/A

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

       3. associate-*l* N/A

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

       4. unpow2 N/A

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

       5. associate-*r* N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 N/A

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

       12. unpow2 N/A

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

       13. lower-*.f64 55.8

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

   11. Simplified 55.8%

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

   if -inf.0 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 2e-8

   1. Initial program 74.1%

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

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

   if 2e-8 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 100.0%

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

      1. lift-sin.f64 N/A

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

      2. lift-sinh.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l/ N/A

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

      5. associate-/r/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 100.0

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

   4. Applied egg-rr 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. Simplified 73.0%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, y \cdot 0.16666666666666666, y\right) \cdot \mathsf{fma}\left(x \cdot x, x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot -0.0001984126984126984\right)\right), 1\right)\\ \mathbf{elif}\;\frac{\sinh y \cdot \sin x}{x} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\frac{\sin x}{x} \cdot \mathsf{fma}\left(y \cdot y, y \cdot 0.16666666666666666, y\right)\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y \cdot \sin x}{x}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, y \cdot 0.16666666666666666, y\right) \cdot \mathsf{fma}\left(x \cdot x, x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot -0.0001984126984126984\right)\right), 1\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\frac{y}{\frac{x}{\sin x}}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sinh y) (sin x)) x)))
   (if (<= t_0 (- INFINITY))
     (*
      (fma (* y y) (* y 0.16666666666666666) y)
      (fma (* x x) (* x (* x (* (* x x) -0.0001984126984126984))) 1.0))
     (if (<= t_0 2e-8) (/ y (/ x (sin x))) (sinh y)))))

C

double code(double x, double y) {
	double t_0 = (sinh(y) * sin(x)) / x;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma((y * y), (y * 0.16666666666666666), y) * fma((x * x), (x * (x * ((x * x) * -0.0001984126984126984))), 1.0);
	} else if (t_0 <= 2e-8) {
		tmp = y / (x / sin(x));
	} else {
		tmp = sinh(y);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(sinh(y) * sin(x)) / x)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(fma(Float64(y * y), Float64(y * 0.16666666666666666), y) * fma(Float64(x * x), Float64(x * Float64(x * Float64(Float64(x * x) * -0.0001984126984126984))), 1.0));
	elseif (t_0 <= 2e-8)
		tmp = Float64(y / Float64(x / sin(x)));
	else
		tmp = sinh(y);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sinh[y], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(y * y), $MachinePrecision] * N[(y * 0.16666666666666666), $MachinePrecision] + y), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * -0.0001984126984126984), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e-8], N[(y / N[(x / N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sinh[y], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in 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. Simplified 64.2%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 55.8

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

   8. Simplified 55.8%

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

   9. Taylor expanded in x around inf

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

       1. metadata-eval N/A

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

       2. pow-sqr N/A

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

       3. associate-*l* N/A

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

       4. unpow2 N/A

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

       5. associate-*r* N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 N/A

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

       12. unpow2 N/A

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

       13. lower-*.f64 55.8

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

   11. Simplified 55.8%

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

   if -inf.0 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 2e-8

   1. Initial program 74.1%

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

   3. Taylor expanded in y around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sin.f64 99.3

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

   5. Simplified 99.3%

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

      1. lift-sin.f64 N/A

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

      2. div-inv N/A

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

      3. div-inv N/A

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

      4. clear-num N/A

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

      5. lift-/.f64 N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 99.4

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

   7. Applied egg-rr 99.4%

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

   if 2e-8 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 100.0%

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

      1. lift-sin.f64 N/A

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

      2. lift-sinh.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l/ N/A

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

      5. associate-/r/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 100.0

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

   4. Applied egg-rr 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. Simplified 73.0%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, y \cdot 0.16666666666666666, y\right) \cdot \mathsf{fma}\left(x \cdot x, x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot -0.0001984126984126984\right)\right), 1\right)\\ \mathbf{elif}\;\frac{\sinh y \cdot \sin x}{x} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\frac{y}{\frac{x}{\sin x}}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 82.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y \cdot \sin x}{x}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, y \cdot 0.16666666666666666, y\right) \cdot \mathsf{fma}\left(x \cdot x, x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot -0.0001984126984126984\right)\right), 1\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-8}:\\ \;\;\;\;y \cdot \frac{\sin x}{x}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sinh y) (sin x)) x)))
   (if (<= t_0 (- INFINITY))
     (*
      (fma (* y y) (* y 0.16666666666666666) y)
      (fma (* x x) (* x (* x (* (* x x) -0.0001984126984126984))) 1.0))
     (if (<= t_0 2e-8) (* y (/ (sin x) x)) (sinh y)))))

C

double code(double x, double y) {
	double t_0 = (sinh(y) * sin(x)) / x;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma((y * y), (y * 0.16666666666666666), y) * fma((x * x), (x * (x * ((x * x) * -0.0001984126984126984))), 1.0);
	} else if (t_0 <= 2e-8) {
		tmp = y * (sin(x) / x);
	} else {
		tmp = sinh(y);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(sinh(y) * sin(x)) / x)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(fma(Float64(y * y), Float64(y * 0.16666666666666666), y) * fma(Float64(x * x), Float64(x * Float64(x * Float64(Float64(x * x) * -0.0001984126984126984))), 1.0));
	elseif (t_0 <= 2e-8)
		tmp = Float64(y * Float64(sin(x) / x));
	else
		tmp = sinh(y);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sinh[y], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(y * y), $MachinePrecision] * N[(y * 0.16666666666666666), $MachinePrecision] + y), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * -0.0001984126984126984), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e-8], N[(y * N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[Sinh[y], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in 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. Simplified 64.2%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 55.8

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

   8. Simplified 55.8%

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

   9. Taylor expanded in x around inf

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

       1. metadata-eval N/A

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

       2. pow-sqr N/A

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

       3. associate-*l* N/A

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

       4. unpow2 N/A

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

       5. associate-*r* N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 N/A

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

       12. unpow2 N/A

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

       13. lower-*.f64 55.8

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

   11. Simplified 55.8%

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

   if -inf.0 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 2e-8

   1. Initial program 74.1%

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

   3. Taylor expanded in y around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sin.f64 99.3

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

   5. Simplified 99.3%

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

   if 2e-8 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 100.0%

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

      1. lift-sin.f64 N/A

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

      2. lift-sinh.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l/ N/A

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

      5. associate-/r/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 100.0

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

   4. Applied egg-rr 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. Simplified 73.0%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, y \cdot 0.16666666666666666, y\right) \cdot \mathsf{fma}\left(x \cdot x, x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot -0.0001984126984126984\right)\right), 1\right)\\ \mathbf{elif}\;\frac{\sinh y \cdot \sin x}{x} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;y \cdot \frac{\sin x}{x}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 55.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(0.16666666666666666, y \cdot \left(y \cdot y\right), y\right)\\ \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -2 \cdot 10^{-154}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \left(x \cdot \left(x \cdot t\_0\right)\right) \cdot \mathsf{fma}\left(x \cdot x, -0.0001984126984126984, 0.008333333333333333\right), t\_0 \cdot \mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma 0.16666666666666666 (* y (* y y)) y)))
   (if (<= (/ (* (sinh y) (sin x)) x) -2e-154)
     (fma
      (* x x)
      (*
       (* x (* x t_0))
       (fma (* x x) -0.0001984126984126984 0.008333333333333333))
      (* t_0 (fma x (* x -0.16666666666666666) 1.0)))
     (sinh y))))

C

double code(double x, double y) {
	double t_0 = fma(0.16666666666666666, (y * (y * y)), y);
	double tmp;
	if (((sinh(y) * sin(x)) / x) <= -2e-154) {
		tmp = fma((x * x), ((x * (x * t_0)) * fma((x * x), -0.0001984126984126984, 0.008333333333333333)), (t_0 * fma(x, (x * -0.16666666666666666), 1.0)));
	} else {
		tmp = sinh(y);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(0.16666666666666666, Float64(y * Float64(y * y)), y)
	tmp = 0.0
	if (Float64(Float64(sinh(y) * sin(x)) / x) <= -2e-154)
		tmp = fma(Float64(x * x), Float64(Float64(x * Float64(x * t_0)) * fma(Float64(x * x), -0.0001984126984126984, 0.008333333333333333)), Float64(t_0 * fma(x, Float64(x * -0.16666666666666666), 1.0)));
	else
		tmp = sinh(y);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(0.16666666666666666 * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sinh[y], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], -2e-154], N[(N[(x * x), $MachinePrecision] * N[(N[(x * N[(x * t$95$0), $MachinePrecision]), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * N[(x * N[(x * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sinh[y], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -1.9999999999999999e-154

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*r/ N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. associate-*r* N/A

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

      11. associate-/l* N/A

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

      12. distribute-rgt-out N/A

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

      13. *-rgt-identity N/A

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

      14. distribute-lft-in N/A

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

   5. Simplified 69.0%

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

   6. Taylor expanded in x around 0

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

   8. Simplified 38.3%

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

   if -1.9999999999999999e-154 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 82.8%

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

      1. lift-sin.f64 N/A

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

      2. lift-sinh.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l/ N/A

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

      5. associate-/r/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 100.0

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

   4. Applied egg-rr 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. Simplified 59.5%

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

4. Final simplification 54.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -2 \cdot 10^{-154}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \left(x \cdot \left(x \cdot \mathsf{fma}\left(0.16666666666666666, y \cdot \left(y \cdot y\right), y\right)\right)\right) \cdot \mathsf{fma}\left(x \cdot x, -0.0001984126984126984, 0.008333333333333333\right), \mathsf{fma}\left(0.16666666666666666, y \cdot \left(y \cdot y\right), y\right) \cdot \mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 53.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(0.16666666666666666, y \cdot \left(y \cdot y\right), y\right)\\ \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq 10^{-36}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \left(x \cdot \left(x \cdot t\_0\right)\right) \cdot \mathsf{fma}\left(x \cdot x, -0.0001984126984126984, 0.008333333333333333\right), t\_0 \cdot \mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.008333333333333333, -0.16666666666666666\right), x \cdot \left(x \cdot x\right), x\right) \cdot \mathsf{fma}\left(y \cdot y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma 0.16666666666666666 (* y (* y y)) y)))
   (if (<= (/ (* (sinh y) (sin x)) x) 1e-36)
     (fma
      (* x x)
      (*
       (* x (* x t_0))
       (fma (* x x) -0.0001984126984126984 0.008333333333333333))
      (* t_0 (fma x (* x -0.16666666666666666) 1.0)))
     (/
      (*
       (fma
        (fma (* x x) 0.008333333333333333 -0.16666666666666666)
        (* x (* x x))
        x)
       (fma
        (* y y)
        (*
         y
         (fma
          (* y y)
          (fma (* y y) 0.0001984126984126984 0.008333333333333333)
          0.16666666666666666))
        y))
      x))))

C

double code(double x, double y) {
	double t_0 = fma(0.16666666666666666, (y * (y * y)), y);
	double tmp;
	if (((sinh(y) * sin(x)) / x) <= 1e-36) {
		tmp = fma((x * x), ((x * (x * t_0)) * fma((x * x), -0.0001984126984126984, 0.008333333333333333)), (t_0 * fma(x, (x * -0.16666666666666666), 1.0)));
	} else {
		tmp = (fma(fma((x * x), 0.008333333333333333, -0.16666666666666666), (x * (x * x)), x) * fma((y * y), (y * fma((y * y), fma((y * y), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666)), y)) / x;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(0.16666666666666666, Float64(y * Float64(y * y)), y)
	tmp = 0.0
	if (Float64(Float64(sinh(y) * sin(x)) / x) <= 1e-36)
		tmp = fma(Float64(x * x), Float64(Float64(x * Float64(x * t_0)) * fma(Float64(x * x), -0.0001984126984126984, 0.008333333333333333)), Float64(t_0 * fma(x, Float64(x * -0.16666666666666666), 1.0)));
	else
		tmp = Float64(Float64(fma(fma(Float64(x * x), 0.008333333333333333, -0.16666666666666666), Float64(x * Float64(x * x)), x) * fma(Float64(y * y), Float64(y * fma(Float64(y * y), fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666)), y)) / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(0.16666666666666666 * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sinh[y], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], 1e-36], N[(N[(x * x), $MachinePrecision] * N[(N[(x * N[(x * t$95$0), $MachinePrecision]), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * N[(x * N[(x * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(x * x), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision] * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(y * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 9.9999999999999994e-37

   1. Initial program 80.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*r/ N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. associate-*r* N/A

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

      11. associate-/l* N/A

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

      12. distribute-rgt-out N/A

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

      13. *-rgt-identity N/A

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

      14. distribute-lft-in N/A

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

   5. Simplified 89.6%

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

   6. Taylor expanded in x around 0

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

   8. Simplified 44.5%

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

   if 9.9999999999999994e-37 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. sub-neg N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 74.7

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

   5. Simplified 74.7%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-*l* N/A

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

      4. *-lft-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 66.5

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

   8. Simplified 66.5%

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

4. Final simplification 51.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq 10^{-36}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \left(x \cdot \left(x \cdot \mathsf{fma}\left(0.16666666666666666, y \cdot \left(y \cdot y\right), y\right)\right)\right) \cdot \mathsf{fma}\left(x \cdot x, -0.0001984126984126984, 0.008333333333333333\right), \mathsf{fma}\left(0.16666666666666666, y \cdot \left(y \cdot y\right), y\right) \cdot \mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.008333333333333333, -0.16666666666666666\right), x \cdot \left(x \cdot x\right), x\right) \cdot \mathsf{fma}\left(y \cdot y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 56.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq 10^{-36}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, y \cdot 0.16666666666666666, y\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.008333333333333333, -0.16666666666666666\right), x \cdot \left(x \cdot x\right), x\right) \cdot \mathsf{fma}\left(y \cdot y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (sinh y) (sin x)) x) 1e-36)
   (*
    (fma (* y y) (* y 0.16666666666666666) y)
    (fma
     (* x x)
     (fma
      (* x x)
      (fma (* x x) -0.0001984126984126984 0.008333333333333333)
      -0.16666666666666666)
     1.0))
   (/
    (*
     (fma
      (fma (* x x) 0.008333333333333333 -0.16666666666666666)
      (* x (* x x))
      x)
     (fma
      (* y y)
      (*
       y
       (fma
        (* y y)
        (fma (* y y) 0.0001984126984126984 0.008333333333333333)
        0.16666666666666666))
      y))
    x)))

C

double code(double x, double y) {
	double tmp;
	if (((sinh(y) * sin(x)) / x) <= 1e-36) {
		tmp = fma((y * y), (y * 0.16666666666666666), y) * fma((x * x), fma((x * x), fma((x * x), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666), 1.0);
	} else {
		tmp = (fma(fma((x * x), 0.008333333333333333, -0.16666666666666666), (x * (x * x)), x) * fma((y * y), (y * fma((y * y), fma((y * y), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666)), y)) / x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(sinh(y) * sin(x)) / x) <= 1e-36)
		tmp = Float64(fma(Float64(y * y), Float64(y * 0.16666666666666666), y) * fma(Float64(x * x), fma(Float64(x * x), fma(Float64(x * x), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666), 1.0));
	else
		tmp = Float64(Float64(fma(fma(Float64(x * x), 0.008333333333333333, -0.16666666666666666), Float64(x * 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);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[Sinh[y], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], 1e-36], N[(N[(N[(y * y), $MachinePrecision] * N[(y * 0.16666666666666666), $MachinePrecision] + y), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(x * x), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision] * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(y * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 9.9999999999999994e-37

   1. Initial program 80.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*r/ N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. associate-*r* N/A

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

      11. associate-/l* N/A

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

      12. distribute-rgt-out N/A

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

      13. *-rgt-identity N/A

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

      14. distribute-lft-in N/A

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

   5. Simplified 89.6%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 50.2

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

   8. Simplified 50.2%

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

   if 9.9999999999999994e-37 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. sub-neg N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 74.7

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

   5. Simplified 74.7%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-*l* N/A

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

      4. *-lft-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 66.5

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

   8. Simplified 66.5%

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

4. Final simplification 55.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq 10^{-36}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, y \cdot 0.16666666666666666, y\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.008333333333333333, -0.16666666666666666\right), x \cdot \left(x \cdot x\right), x\right) \cdot \mathsf{fma}\left(y \cdot y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 56.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq 10^{-36}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, y \cdot 0.16666666666666666, y\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.008333333333333333, -0.16666666666666666\right), x \cdot \left(x \cdot x\right), x\right) \cdot \mathsf{fma}\left(y \cdot y, y \cdot \mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), y\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (sinh y) (sin x)) x) 1e-36)
   (*
    (fma (* y y) (* y 0.16666666666666666) y)
    (fma
     (* x x)
     (fma
      (* x x)
      (fma (* x x) -0.0001984126984126984 0.008333333333333333)
      -0.16666666666666666)
     1.0))
   (/
    (*
     (fma
      (fma (* x x) 0.008333333333333333 -0.16666666666666666)
      (* x (* x x))
      x)
     (fma
      (* y y)
      (* y (fma (* y y) 0.008333333333333333 0.16666666666666666))
      y))
    x)))

C

double code(double x, double y) {
	double tmp;
	if (((sinh(y) * sin(x)) / x) <= 1e-36) {
		tmp = fma((y * y), (y * 0.16666666666666666), y) * fma((x * x), fma((x * x), fma((x * x), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666), 1.0);
	} else {
		tmp = (fma(fma((x * x), 0.008333333333333333, -0.16666666666666666), (x * (x * x)), x) * fma((y * y), (y * fma((y * y), 0.008333333333333333, 0.16666666666666666)), y)) / x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(sinh(y) * sin(x)) / x) <= 1e-36)
		tmp = Float64(fma(Float64(y * y), Float64(y * 0.16666666666666666), y) * fma(Float64(x * x), fma(Float64(x * x), fma(Float64(x * x), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666), 1.0));
	else
		tmp = Float64(Float64(fma(fma(Float64(x * x), 0.008333333333333333, -0.16666666666666666), Float64(x * Float64(x * x)), x) * fma(Float64(y * y), Float64(y * fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666)), y)) / 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-36], N[(N[(N[(y * y), $MachinePrecision] * N[(y * 0.16666666666666666), $MachinePrecision] + y), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(x * x), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision] * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(y * N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 9.9999999999999994e-37

   1. Initial program 80.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*r/ N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. associate-*r* N/A

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

      11. associate-/l* N/A

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

      12. distribute-rgt-out N/A

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

      13. *-rgt-identity N/A

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

      14. distribute-lft-in N/A

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

   5. Simplified 89.6%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 50.2

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

   8. Simplified 50.2%

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

   if 9.9999999999999994e-37 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. sub-neg N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 74.7

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

   5. Simplified 74.7%

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

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 65.2

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

   8. Simplified 65.2%

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

4. Final simplification 54.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq 10^{-36}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, y \cdot 0.16666666666666666, y\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.008333333333333333, -0.16666666666666666\right), x \cdot \left(x \cdot x\right), x\right) \cdot \mathsf{fma}\left(y \cdot y, y \cdot \mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), y\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 54.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq 10^{-36}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, y \cdot 0.16666666666666666, y\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(\mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right) \cdot \mathsf{fma}\left(x \cdot x, x \cdot \mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right), x\right)\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (sinh y) (sin x)) x) 1e-36)
   (*
    (fma (* y y) (* y 0.16666666666666666) y)
    (fma
     (* x x)
     (fma
      (* x x)
      (fma (* x x) -0.0001984126984126984 0.008333333333333333)
      -0.16666666666666666)
     1.0))
   (/
    (*
     y
     (*
      (fma 0.16666666666666666 (* y y) 1.0)
      (fma
       (* x x)
       (* x (fma 0.008333333333333333 (* x x) -0.16666666666666666))
       x)))
    x)))

C

double code(double x, double y) {
	double tmp;
	if (((sinh(y) * sin(x)) / x) <= 1e-36) {
		tmp = fma((y * y), (y * 0.16666666666666666), y) * fma((x * x), fma((x * x), fma((x * x), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666), 1.0);
	} else {
		tmp = (y * (fma(0.16666666666666666, (y * y), 1.0) * fma((x * x), (x * fma(0.008333333333333333, (x * x), -0.16666666666666666)), x))) / x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(sinh(y) * sin(x)) / x) <= 1e-36)
		tmp = Float64(fma(Float64(y * y), Float64(y * 0.16666666666666666), y) * fma(Float64(x * x), fma(Float64(x * x), fma(Float64(x * x), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666), 1.0));
	else
		tmp = Float64(Float64(y * Float64(fma(0.16666666666666666, Float64(y * y), 1.0) * fma(Float64(x * x), Float64(x * fma(0.008333333333333333, Float64(x * x), -0.16666666666666666)), x))) / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[Sinh[y], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], 1e-36], N[(N[(N[(y * y), $MachinePrecision] * N[(y * 0.16666666666666666), $MachinePrecision] + y), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(y * N[(N[(0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * N[(0.008333333333333333 * N[(x * x), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 9.9999999999999994e-37

   1. Initial program 80.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*r/ N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. associate-*r* N/A

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

      11. associate-/l* N/A

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

      12. distribute-rgt-out N/A

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

      13. *-rgt-identity N/A

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

      14. distribute-lft-in N/A

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

   5. Simplified 89.6%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 50.2

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

   8. Simplified 50.2%

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

   if 9.9999999999999994e-37 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. sub-neg N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 74.7

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

   5. Simplified 74.7%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

   8. Simplified 60.4%

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

4. Final simplification 53.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq 10^{-36}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, y \cdot 0.16666666666666666, y\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(\mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right) \cdot \mathsf{fma}\left(x \cdot x, x \cdot \mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right), x\right)\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 53.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq 1.5 \cdot 10^{-172}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, y \cdot 0.16666666666666666, y\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \mathsf{fma}\left(0.16666666666666666, y \cdot \left(y \cdot y\right), y\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (sinh y) (sin x)) x) 1.5e-172)
   (*
    (fma (* y y) (* y 0.16666666666666666) y)
    (fma
     (* x x)
     (fma
      (* x x)
      (fma (* x x) -0.0001984126984126984 0.008333333333333333)
      -0.16666666666666666)
     1.0))
   (/ (* x (fma 0.16666666666666666 (* y (* y y)) y)) x)))

C

double code(double x, double y) {
	double tmp;
	if (((sinh(y) * sin(x)) / x) <= 1.5e-172) {
		tmp = fma((y * y), (y * 0.16666666666666666), y) * fma((x * x), fma((x * x), fma((x * x), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666), 1.0);
	} else {
		tmp = (x * fma(0.16666666666666666, (y * (y * y)), y)) / x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(sinh(y) * sin(x)) / x) <= 1.5e-172)
		tmp = Float64(fma(Float64(y * y), Float64(y * 0.16666666666666666), y) * fma(Float64(x * x), fma(Float64(x * x), fma(Float64(x * x), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666), 1.0));
	else
		tmp = Float64(Float64(x * fma(0.16666666666666666, Float64(y * Float64(y * y)), y)) / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[Sinh[y], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], 1.5e-172], N[(N[(N[(y * y), $MachinePrecision] * N[(y * 0.16666666666666666), $MachinePrecision] + y), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(0.16666666666666666 * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 1.49999999999999992e-172

   1. Initial program 79.6%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left(\frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{\sin x}{x}\right)} \]
   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. Simplified 88.8%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 48.3

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

   8. Simplified 48.3%

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

   if 1.49999999999999992e-172 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 99.7%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-lft-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 75.6

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

   5. Simplified 75.6%

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

   6. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. cube-mult N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 61.2

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

   8. Simplified 61.2%

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

4. Final simplification 52.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq 1.5 \cdot 10^{-172}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, y \cdot 0.16666666666666666, y\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \mathsf{fma}\left(0.16666666666666666, y \cdot \left(y \cdot y\right), y\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 53.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq 1.5 \cdot 10^{-172}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, y \cdot 0.16666666666666666, y\right) \cdot \mathsf{fma}\left(x \cdot x, x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot -0.0001984126984126984\right)\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \mathsf{fma}\left(0.16666666666666666, y \cdot \left(y \cdot y\right), y\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (sinh y) (sin x)) x) 1.5e-172)
   (*
    (fma (* y y) (* y 0.16666666666666666) y)
    (fma (* x x) (* x (* x (* (* x x) -0.0001984126984126984))) 1.0))
   (/ (* x (fma 0.16666666666666666 (* y (* y y)) y)) x)))

C

double code(double x, double y) {
	double tmp;
	if (((sinh(y) * sin(x)) / x) <= 1.5e-172) {
		tmp = fma((y * y), (y * 0.16666666666666666), y) * fma((x * x), (x * (x * ((x * x) * -0.0001984126984126984))), 1.0);
	} else {
		tmp = (x * fma(0.16666666666666666, (y * (y * y)), y)) / x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(sinh(y) * sin(x)) / x) <= 1.5e-172)
		tmp = Float64(fma(Float64(y * y), Float64(y * 0.16666666666666666), y) * fma(Float64(x * x), Float64(x * Float64(x * Float64(Float64(x * x) * -0.0001984126984126984))), 1.0));
	else
		tmp = Float64(Float64(x * fma(0.16666666666666666, Float64(y * Float64(y * y)), y)) / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[Sinh[y], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], 1.5e-172], N[(N[(N[(y * y), $MachinePrecision] * N[(y * 0.16666666666666666), $MachinePrecision] + y), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * -0.0001984126984126984), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(0.16666666666666666 * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 1.49999999999999992e-172

   1. Initial program 79.6%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left(\frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{\sin x}{x}\right)} \]
   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. Simplified 88.8%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 48.3

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

   8. Simplified 48.3%

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

   9. Taylor expanded in x around inf

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

       1. metadata-eval N/A

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

       2. pow-sqr N/A

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

       3. associate-*l* N/A

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

       4. unpow2 N/A

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

       5. associate-*r* N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 N/A

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

       12. unpow2 N/A

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

       13. lower-*.f64 48.0

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

   11. Simplified 48.0%

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

   if 1.49999999999999992e-172 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 99.7%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-lft-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 75.6

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

   5. Simplified 75.6%

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

   6. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. cube-mult N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 61.2

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

   8. Simplified 61.2%

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

4. Final simplification 52.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq 1.5 \cdot 10^{-172}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, y \cdot 0.16666666666666666, y\right) \cdot \mathsf{fma}\left(x \cdot x, x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot -0.0001984126984126984\right)\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \mathsf{fma}\left(0.16666666666666666, y \cdot \left(y \cdot y\right), y\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 53.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(0.16666666666666666, y \cdot \left(y \cdot y\right), y\right)\\ \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq 1.5 \cdot 10^{-172}:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot t\_0}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma 0.16666666666666666 (* y (* y y)) y)))
   (if (<= (/ (* (sinh y) (sin x)) x) 1.5e-172)
     (* t_0 (fma x (* x -0.16666666666666666) 1.0))
     (/ (* x t_0) x))))

C

double code(double x, double y) {
	double t_0 = fma(0.16666666666666666, (y * (y * y)), y);
	double tmp;
	if (((sinh(y) * sin(x)) / x) <= 1.5e-172) {
		tmp = t_0 * fma(x, (x * -0.16666666666666666), 1.0);
	} else {
		tmp = (x * t_0) / x;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(0.16666666666666666, Float64(y * Float64(y * y)), y)
	tmp = 0.0
	if (Float64(Float64(sinh(y) * sin(x)) / x) <= 1.5e-172)
		tmp = Float64(t_0 * fma(x, Float64(x * -0.16666666666666666), 1.0));
	else
		tmp = Float64(Float64(x * t_0) / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(0.16666666666666666 * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sinh[y], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], 1.5e-172], N[(t$95$0 * N[(x * N[(x * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x * t$95$0), $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 1.49999999999999992e-172

   1. Initial program 79.6%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left(\frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{\sin x}{x}\right)} \]
   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. Simplified 88.8%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt1-in N/A

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

      5. +-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. cube-mult N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 48.4

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

   8. Simplified 48.4%

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

   if 1.49999999999999992e-172 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 99.7%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-lft-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 75.6

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

   5. Simplified 75.6%

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

   6. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. cube-mult N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 61.2

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

   8. Simplified 61.2%

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

4. Final simplification 53.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq 1.5 \cdot 10^{-172}:\\ \;\;\;\;\mathsf{fma}\left(0.16666666666666666, y \cdot \left(y \cdot y\right), y\right) \cdot \mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \mathsf{fma}\left(0.16666666666666666, y \cdot \left(y \cdot y\right), y\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 53.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(0.16666666666666666, y \cdot \left(y \cdot y\right), y\right)\\ \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -2 \cdot 10^{-252}:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma 0.16666666666666666 (* y (* y y)) y)))
   (if (<= (/ (* (sinh y) (sin x)) x) -2e-252)
     (* t_0 (fma x (* x -0.16666666666666666) 1.0))
     t_0)))

C

double code(double x, double y) {
	double t_0 = fma(0.16666666666666666, (y * (y * y)), y);
	double tmp;
	if (((sinh(y) * sin(x)) / x) <= -2e-252) {
		tmp = t_0 * fma(x, (x * -0.16666666666666666), 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(0.16666666666666666, Float64(y * Float64(y * y)), y)
	tmp = 0.0
	if (Float64(Float64(sinh(y) * sin(x)) / x) <= -2e-252)
		tmp = Float64(t_0 * fma(x, Float64(x * -0.16666666666666666), 1.0));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(0.16666666666666666 * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sinh[y], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], -2e-252], N[(t$95$0 * N[(x * N[(x * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -1.99999999999999989e-252

   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. Simplified 73.8%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt1-in N/A

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

      5. +-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. cube-mult N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 52.3

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

   8. Simplified 52.3%

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

   if -1.99999999999999989e-252 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 81.8%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left(\frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{\sin x}{x}\right)} \]
   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. Simplified 88.2%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. cube-mult N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 51.8

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

   8. Simplified 51.8%

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

4. Final simplification 52.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -2 \cdot 10^{-252}:\\ \;\;\;\;\mathsf{fma}\left(0.16666666666666666, y \cdot \left(y \cdot y\right), y\right) \cdot \mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.16666666666666666, y \cdot \left(y \cdot y\right), y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 53.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -2 \cdot 10^{-252}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, x \cdot x, 1\right) \cdot \mathsf{fma}\left(y, \left(y \cdot y\right) \cdot 0.16666666666666666, y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.16666666666666666, y \cdot \left(y \cdot y\right), y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (sinh y) (sin x)) x) -2e-252)
   (*
    (fma -0.16666666666666666 (* x x) 1.0)
    (fma y (* (* y y) 0.16666666666666666) y))
   (fma 0.16666666666666666 (* y (* y y)) y)))

C

double code(double x, double y) {
	double tmp;
	if (((sinh(y) * sin(x)) / x) <= -2e-252) {
		tmp = fma(-0.16666666666666666, (x * x), 1.0) * fma(y, ((y * y) * 0.16666666666666666), y);
	} else {
		tmp = fma(0.16666666666666666, (y * (y * y)), y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(sinh(y) * sin(x)) / x) <= -2e-252)
		tmp = Float64(fma(-0.16666666666666666, Float64(x * x), 1.0) * fma(y, Float64(Float64(y * y) * 0.16666666666666666), y));
	else
		tmp = fma(0.16666666666666666, Float64(y * Float64(y * y)), y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[Sinh[y], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], -2e-252], N[(N[(-0.16666666666666666 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(y * N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision], N[(0.16666666666666666 * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -1.99999999999999989e-252

   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. Simplified 73.8%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 52.5

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

   8. Simplified 52.5%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. associate-+r+ N/A

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

       3. associate-*r* N/A

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

       4. distribute-rgt1-in N/A

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

       5. +-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. +-commutative N/A

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

       8. lower-fma.f64 N/A

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

       9. unpow2 N/A

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

       10. lower-*.f64 N/A

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

       11. +-commutative N/A

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

       12. unpow3 N/A

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

       13. unpow2 N/A

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

       14. associate-*r* N/A

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

       15. *-commutative N/A

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

       16. lower-fma.f64 N/A

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

       17. lower-*.f64 N/A

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

       18. unpow2 N/A

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

       19. lower-*.f64 52.3

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

   11. Simplified 52.3%

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

   if -1.99999999999999989e-252 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 81.8%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left(\frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{\sin x}{x}\right)} \]
   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. Simplified 88.2%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. cube-mult N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 51.8

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

   8. Simplified 51.8%

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

4. Final simplification 52.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -2 \cdot 10^{-252}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, x \cdot x, 1\right) \cdot \mathsf{fma}\left(y, \left(y \cdot y\right) \cdot 0.16666666666666666, y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.16666666666666666, y \cdot \left(y \cdot y\right), y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 38.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -2 \cdot 10^{-252}:\\ \;\;\;\;\frac{y \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot -0.16666666666666666\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.16666666666666666, y \cdot \left(y \cdot y\right), y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (sinh y) (sin x)) x) -2e-252)
   (/ (* y (* x (* (* x x) -0.16666666666666666))) x)
   (fma 0.16666666666666666 (* y (* y y)) y)))

C

double code(double x, double y) {
	double tmp;
	if (((sinh(y) * sin(x)) / x) <= -2e-252) {
		tmp = (y * (x * ((x * x) * -0.16666666666666666))) / x;
	} else {
		tmp = fma(0.16666666666666666, (y * (y * y)), y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(sinh(y) * sin(x)) / x) <= -2e-252)
		tmp = Float64(Float64(y * Float64(x * Float64(Float64(x * x) * -0.16666666666666666))) / x);
	else
		tmp = fma(0.16666666666666666, Float64(y * Float64(y * y)), y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[Sinh[y], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], -2e-252], N[(N[(y * N[(x * N[(N[(x * x), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(0.16666666666666666 * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -1.99999999999999989e-252

   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 29.2

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

   5. Simplified 29.2%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 26.3

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

   8. Simplified 26.3%

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

   9. Taylor expanded in x around inf

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

       1. unpow3 N/A

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

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 14.8

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

   11. Simplified 14.8%

       \[\leadsto y \cdot \frac{\color{blue}{x \cdot \left(-0.16666666666666666 \cdot \left(x \cdot x\right)\right)}}{x} \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-*r/ N/A

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

       5. lower-/.f64 N/A

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

       6. lower-*.f64 16.1

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

   13. Applied egg-rr 16.1%

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

   if -1.99999999999999989e-252 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 81.8%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left(\frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{\sin x}{x}\right)} \]
   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. Simplified 88.2%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. cube-mult N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 51.8

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

   8. Simplified 51.8%

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

4. Final simplification 42.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -2 \cdot 10^{-252}:\\ \;\;\;\;\frac{y \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot -0.16666666666666666\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.16666666666666666, y \cdot \left(y \cdot y\right), y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 38.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -2 \cdot 10^{-252}:\\ \;\;\;\;y \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{-0.16666666666666666}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.16666666666666666, y \cdot \left(y \cdot y\right), y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (sinh y) (sin x)) x) -2e-252)
   (* y (* (* x (* x x)) (/ -0.16666666666666666 x)))
   (fma 0.16666666666666666 (* y (* y y)) y)))

C

double code(double x, double y) {
	double tmp;
	if (((sinh(y) * sin(x)) / x) <= -2e-252) {
		tmp = y * ((x * (x * x)) * (-0.16666666666666666 / x));
	} else {
		tmp = fma(0.16666666666666666, (y * (y * y)), y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(sinh(y) * sin(x)) / x) <= -2e-252)
		tmp = Float64(y * Float64(Float64(x * Float64(x * x)) * Float64(-0.16666666666666666 / x)));
	else
		tmp = fma(0.16666666666666666, Float64(y * Float64(y * y)), y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[Sinh[y], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], -2e-252], N[(y * N[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(-0.16666666666666666 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.16666666666666666 * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -1.99999999999999989e-252

   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 29.2

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

   5. Simplified 29.2%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 26.3

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

   8. Simplified 26.3%

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

   9. Taylor expanded in x around inf

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

       1. unpow3 N/A

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

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 14.8

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

   11. Simplified 14.8%

       \[\leadsto y \cdot \frac{\color{blue}{x \cdot \left(-0.16666666666666666 \cdot \left(x \cdot x\right)\right)}}{x} \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. /-rgt-identity N/A

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

       4. div-inv N/A

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

       5. metadata-eval N/A

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

       6. *-commutative N/A

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

       7. lift-*.f64 N/A

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

       8. associate-*l* N/A

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

       9. *-commutative N/A

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

       10. lift-*.f64 N/A

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

       11. times-frac N/A

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

       12. lift-*.f64 N/A

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

       13. lift-*.f64 N/A

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

       14. cube-unmult N/A

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

       15. metadata-eval N/A

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

       16. cube-div N/A

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

       17. /-rgt-identity N/A

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

       18. cube-unmult N/A

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

       19. lift-*.f64 N/A

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

       20. lift-*.f64 N/A

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

       21. lower-*.f64 N/A

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

       22. lower-/.f64 14.8

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

   13. Applied egg-rr 14.8%

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

   if -1.99999999999999989e-252 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 81.8%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left(\frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{\sin x}{x}\right)} \]
   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. Simplified 88.2%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. cube-mult N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 51.8

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

   8. Simplified 51.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.16666666666666666, y \cdot \left(y \cdot y\right), y\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 41.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -2 \cdot 10^{-252}:\\ \;\;\;\;y \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{-0.16666666666666666}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.16666666666666666, y \cdot \left(y \cdot y\right), y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 38.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -2 \cdot 10^{-252}:\\ \;\;\;\;y \cdot \left(x \cdot \left(x \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.16666666666666666, y \cdot \left(y \cdot y\right), y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (sinh y) (sin x)) x) -2e-252)
   (* y (* x (* x -0.16666666666666666)))
   (fma 0.16666666666666666 (* y (* y y)) y)))

C

double code(double x, double y) {
	double tmp;
	if (((sinh(y) * sin(x)) / x) <= -2e-252) {
		tmp = y * (x * (x * -0.16666666666666666));
	} else {
		tmp = fma(0.16666666666666666, (y * (y * y)), y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(sinh(y) * sin(x)) / x) <= -2e-252)
		tmp = Float64(y * Float64(x * Float64(x * -0.16666666666666666)));
	else
		tmp = fma(0.16666666666666666, Float64(y * Float64(y * y)), y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[Sinh[y], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], -2e-252], N[(y * N[(x * N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.16666666666666666 * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -1.99999999999999989e-252

   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 29.2

         \[\leadsto y \cdot \frac{\color{blue}{\sin x}}{x} \]

   5. Simplified 29.2%

      \[\leadsto \color{blue}{y \cdot \frac{\sin x}{x}} \]

   6. Taylor expanded in x around 0

      \[\leadsto y \cdot \frac{\color{blue}{x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)}}{x} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto y \cdot \frac{x \cdot \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2} + 1\right)}}{x} \]

      2. distribute-lft-in N/A

         \[\leadsto y \cdot \frac{\color{blue}{x \cdot \left(\frac{-1}{6} \cdot {x}^{2}\right) + x \cdot 1}}{x} \]

      3. *-rgt-identity N/A

         \[\leadsto y \cdot \frac{x \cdot \left(\frac{-1}{6} \cdot {x}^{2}\right) + \color{blue}{x}}{x} \]

      4. lower-fma.f64 N/A

         \[\leadsto y \cdot \frac{\color{blue}{\mathsf{fma}\left(x, \frac{-1}{6} \cdot {x}^{2}, x\right)}}{x} \]

      5. *-commutative N/A

         \[\leadsto y \cdot \frac{\mathsf{fma}\left(x, \color{blue}{{x}^{2} \cdot \frac{-1}{6}}, x\right)}{x} \]

      6. lower-*.f64 N/A

         \[\leadsto y \cdot \frac{\mathsf{fma}\left(x, \color{blue}{{x}^{2} \cdot \frac{-1}{6}}, x\right)}{x} \]

      7. unpow2 N/A

         \[\leadsto y \cdot \frac{\mathsf{fma}\left(x, \color{blue}{\left(x \cdot x\right)} \cdot \frac{-1}{6}, x\right)}{x} \]

      8. lower-*.f64 26.3

         \[\leadsto y \cdot \frac{\mathsf{fma}\left(x, \color{blue}{\left(x \cdot x\right)} \cdot -0.16666666666666666, x\right)}{x} \]

   8. Simplified 26.3%

      \[\leadsto y \cdot \frac{\color{blue}{\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot -0.16666666666666666, x\right)}}{x} \]

   9. Taylor expanded in x around inf

      \[\leadsto y \cdot \frac{\color{blue}{\frac{-1}{6} \cdot {x}^{3}}}{x} \]
   10. Step-by-step derivation

       1. unpow3 N/A

          \[\leadsto y \cdot \frac{\frac{-1}{6} \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot x\right)}}{x} \]

       2. unpow2 N/A

          \[\leadsto y \cdot \frac{\frac{-1}{6} \cdot \left(\color{blue}{{x}^{2}} \cdot x\right)}{x} \]

       3. associate-*r* N/A

          \[\leadsto y \cdot \frac{\color{blue}{\left(\frac{-1}{6} \cdot {x}^{2}\right) \cdot x}}{x} \]

       4. *-commutative N/A

          \[\leadsto y \cdot \frac{\color{blue}{x \cdot \left(\frac{-1}{6} \cdot {x}^{2}\right)}}{x} \]

       5. lower-*.f64 N/A

          \[\leadsto y \cdot \frac{\color{blue}{x \cdot \left(\frac{-1}{6} \cdot {x}^{2}\right)}}{x} \]

       6. lower-*.f64 N/A

          \[\leadsto y \cdot \frac{x \cdot \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2}\right)}}{x} \]

       7. unpow2 N/A

          \[\leadsto y \cdot \frac{x \cdot \left(\frac{-1}{6} \cdot \color{blue}{\left(x \cdot x\right)}\right)}{x} \]

       8. lower-*.f64 14.8

          \[\leadsto y \cdot \frac{x \cdot \left(-0.16666666666666666 \cdot \color{blue}{\left(x \cdot x\right)}\right)}{x} \]

   11. Simplified 14.8%

       \[\leadsto y \cdot \frac{\color{blue}{x \cdot \left(-0.16666666666666666 \cdot \left(x \cdot x\right)\right)}}{x} \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto y \cdot \frac{x \cdot \left(\frac{-1}{6} \cdot \color{blue}{\left(x \cdot x\right)}\right)}{x} \]

       2. lift-*.f64 N/A

          \[\leadsto y \cdot \frac{x \cdot \color{blue}{\left(\frac{-1}{6} \cdot \left(x \cdot x\right)\right)}}{x} \]

       3. lift-*.f64 N/A

          \[\leadsto y \cdot \frac{\color{blue}{x \cdot \left(\frac{-1}{6} \cdot \left(x \cdot x\right)\right)}}{x} \]

       4. div-inv N/A

          \[\leadsto y \cdot \color{blue}{\left(\left(x \cdot \left(\frac{-1}{6} \cdot \left(x \cdot x\right)\right)\right) \cdot \frac{1}{x}\right)} \]

       5. lift-*.f64 N/A

          \[\leadsto y \cdot \left(\color{blue}{\left(x \cdot \left(\frac{-1}{6} \cdot \left(x \cdot x\right)\right)\right)} \cdot \frac{1}{x}\right) \]

       6. lift-*.f64 N/A

          \[\leadsto y \cdot \left(\left(x \cdot \color{blue}{\left(\frac{-1}{6} \cdot \left(x \cdot x\right)\right)}\right) \cdot \frac{1}{x}\right) \]

       7. associate-*r* N/A

          \[\leadsto y \cdot \left(\color{blue}{\left(\left(x \cdot \frac{-1}{6}\right) \cdot \left(x \cdot x\right)\right)} \cdot \frac{1}{x}\right) \]

       8. lift-/.f64 N/A

          \[\leadsto y \cdot \left(\left(\left(x \cdot \frac{-1}{6}\right) \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\frac{1}{x}}\right) \]

       9. associate-*l* N/A

          \[\leadsto y \cdot \color{blue}{\left(\left(x \cdot \frac{-1}{6}\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{x}\right)\right)} \]

       10. lift-*.f64 N/A

           \[\leadsto y \cdot \left(\left(x \cdot \frac{-1}{6}\right) \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{x}\right)\right) \]

       11. pow2 N/A

           \[\leadsto y \cdot \left(\left(x \cdot \frac{-1}{6}\right) \cdot \left(\color{blue}{{x}^{2}} \cdot \frac{1}{x}\right)\right) \]

       12. lift-/.f64 N/A

           \[\leadsto y \cdot \left(\left(x \cdot \frac{-1}{6}\right) \cdot \left({x}^{2} \cdot \color{blue}{\frac{1}{x}}\right)\right) \]

       13. inv-pow N/A

           \[\leadsto y \cdot \left(\left(x \cdot \frac{-1}{6}\right) \cdot \left({x}^{2} \cdot \color{blue}{{x}^{-1}}\right)\right) \]

       14. pow-prod-up N/A

           \[\leadsto y \cdot \left(\left(x \cdot \frac{-1}{6}\right) \cdot \color{blue}{{x}^{\left(2 + -1\right)}}\right) \]

       15. metadata-eval N/A

           \[\leadsto y \cdot \left(\left(x \cdot \frac{-1}{6}\right) \cdot {x}^{\color{blue}{1}}\right) \]

       16. unpow1 N/A

           \[\leadsto y \cdot \left(\left(x \cdot \frac{-1}{6}\right) \cdot \color{blue}{x}\right) \]

       17. lower-*.f64 N/A

           \[\leadsto y \cdot \color{blue}{\left(\left(x \cdot \frac{-1}{6}\right) \cdot x\right)} \]

       18. lower-*.f64 14.7

           \[\leadsto y \cdot \left(\color{blue}{\left(x \cdot -0.16666666666666666\right)} \cdot x\right) \]

   13. Applied egg-rr 14.7%

       \[\leadsto y \cdot \color{blue}{\left(\left(x \cdot -0.16666666666666666\right) \cdot x\right)} \]

   if -1.99999999999999989e-252 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 81.8%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left(\frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{\sin x}{x}\right)} \]
   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. Simplified 88.2%

      \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot \mathsf{fma}\left(y \cdot y, y \cdot 0.16666666666666666, y\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{y + \frac{1}{6} \cdot {y}^{3}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{6} \cdot {y}^{3} + y} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{6}, {y}^{3}, y\right)} \]

      3. cube-mult N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{6}, \color{blue}{y \cdot \left(y \cdot y\right)}, y\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{6}, y \cdot \color{blue}{{y}^{2}}, y\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{6}, \color{blue}{y \cdot {y}^{2}}, y\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{6}, y \cdot \color{blue}{\left(y \cdot y\right)}, y\right) \]

      7. lower-*.f64 51.8

         \[\leadsto \mathsf{fma}\left(0.16666666666666666, y \cdot \color{blue}{\left(y \cdot y\right)}, y\right) \]

   8. Simplified 51.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.16666666666666666, y \cdot \left(y \cdot y\right), y\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 41.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -2 \cdot 10^{-252}:\\ \;\;\;\;y \cdot \left(x \cdot \left(x \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.16666666666666666, y \cdot \left(y \cdot y\right), y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 35.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq 1.5 \cdot 10^{-172}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (sinh y) (sin x)) x) 1.5e-172)
   (* y (fma x (* x -0.16666666666666666) 1.0))
   (/ (* y x) x)))

C

double code(double x, double y) {
	double tmp;
	if (((sinh(y) * sin(x)) / x) <= 1.5e-172) {
		tmp = y * fma(x, (x * -0.16666666666666666), 1.0);
	} else {
		tmp = (y * x) / x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(sinh(y) * sin(x)) / x) <= 1.5e-172)
		tmp = Float64(y * fma(x, Float64(x * -0.16666666666666666), 1.0));
	else
		tmp = Float64(Float64(y * x) / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[Sinh[y], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], 1.5e-172], N[(y * N[(x * N[(x * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(y * x), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 1.49999999999999992e-172

   1. Initial program 79.6%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y \cdot \sin x}{x}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{y \cdot \frac{\sin x}{x}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{y \cdot \frac{\sin x}{x}} \]

      3. lower-/.f64 N/A

         \[\leadsto y \cdot \color{blue}{\frac{\sin x}{x}} \]

      4. lower-sin.f64 69.8

         \[\leadsto y \cdot \frac{\color{blue}{\sin x}}{x} \]

   5. Simplified 69.8%

      \[\leadsto \color{blue}{y \cdot \frac{\sin x}{x}} \]

   6. Taylor expanded in x around 0

      \[\leadsto y \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {x}^{2}\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto y \cdot \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2} + 1\right)} \]

      2. *-commutative N/A

         \[\leadsto y \cdot \left(\color{blue}{{x}^{2} \cdot \frac{-1}{6}} + 1\right) \]

      3. unpow2 N/A

         \[\leadsto y \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{-1}{6} + 1\right) \]

      4. associate-*l* N/A

         \[\leadsto y \cdot \left(\color{blue}{x \cdot \left(x \cdot \frac{-1}{6}\right)} + 1\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto y \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)} \]

      6. lower-*.f64 37.4

         \[\leadsto y \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot -0.16666666666666666}, 1\right) \]

   8. Simplified 37.4%

      \[\leadsto y \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right)} \]

   if 1.49999999999999992e-172 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 99.7%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{y \cdot \sin x}}{x} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{y \cdot \sin x}}{x} \]

      2. lower-sin.f64 22.0

         \[\leadsto \frac{y \cdot \color{blue}{\sin x}}{x} \]

   5. Simplified 22.0%

      \[\leadsto \frac{\color{blue}{y \cdot \sin x}}{x} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{x \cdot y}}{x} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot x}}{x} \]

      2. lower-*.f64 29.0

         \[\leadsto \frac{\color{blue}{y \cdot x}}{x} \]

   8. Simplified 29.0%

      \[\leadsto \frac{\color{blue}{y \cdot x}}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 34.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq 1.5 \cdot 10^{-172}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 26.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -2 \cdot 10^{-252}:\\ \;\;\;\;y \cdot \left(x \cdot \left(x \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (sinh y) (sin x)) x) -2e-252)
   (* y (* x (* x -0.16666666666666666)))
   y))

C

double code(double x, double y) {
	double tmp;
	if (((sinh(y) * sin(x)) / x) <= -2e-252) {
		tmp = y * (x * (x * -0.16666666666666666));
	} else {
		tmp = y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (((sinh(y) * sin(x)) / x) <= (-2d-252)) then
        tmp = y * (x * (x * (-0.16666666666666666d0)))
    else
        tmp = y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (((Math.sinh(y) * Math.sin(x)) / x) <= -2e-252) {
		tmp = y * (x * (x * -0.16666666666666666));
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((math.sinh(y) * math.sin(x)) / x) <= -2e-252:
		tmp = y * (x * (x * -0.16666666666666666))
	else:
		tmp = y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(sinh(y) * sin(x)) / x) <= -2e-252)
		tmp = Float64(y * Float64(x * Float64(x * -0.16666666666666666)));
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((sinh(y) * sin(x)) / x) <= -2e-252)
		tmp = y * (x * (x * -0.16666666666666666));
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[Sinh[y], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], -2e-252], N[(y * N[(x * N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], y]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -1.99999999999999989e-252

   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 29.2

         \[\leadsto y \cdot \frac{\color{blue}{\sin x}}{x} \]

   5. Simplified 29.2%

      \[\leadsto \color{blue}{y \cdot \frac{\sin x}{x}} \]

   6. Taylor expanded in x around 0

      \[\leadsto y \cdot \frac{\color{blue}{x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)}}{x} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto y \cdot \frac{x \cdot \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2} + 1\right)}}{x} \]

      2. distribute-lft-in N/A

         \[\leadsto y \cdot \frac{\color{blue}{x \cdot \left(\frac{-1}{6} \cdot {x}^{2}\right) + x \cdot 1}}{x} \]

      3. *-rgt-identity N/A

         \[\leadsto y \cdot \frac{x \cdot \left(\frac{-1}{6} \cdot {x}^{2}\right) + \color{blue}{x}}{x} \]

      4. lower-fma.f64 N/A

         \[\leadsto y \cdot \frac{\color{blue}{\mathsf{fma}\left(x, \frac{-1}{6} \cdot {x}^{2}, x\right)}}{x} \]

      5. *-commutative N/A

         \[\leadsto y \cdot \frac{\mathsf{fma}\left(x, \color{blue}{{x}^{2} \cdot \frac{-1}{6}}, x\right)}{x} \]

      6. lower-*.f64 N/A

         \[\leadsto y \cdot \frac{\mathsf{fma}\left(x, \color{blue}{{x}^{2} \cdot \frac{-1}{6}}, x\right)}{x} \]

      7. unpow2 N/A

         \[\leadsto y \cdot \frac{\mathsf{fma}\left(x, \color{blue}{\left(x \cdot x\right)} \cdot \frac{-1}{6}, x\right)}{x} \]

      8. lower-*.f64 26.3

         \[\leadsto y \cdot \frac{\mathsf{fma}\left(x, \color{blue}{\left(x \cdot x\right)} \cdot -0.16666666666666666, x\right)}{x} \]

   8. Simplified 26.3%

      \[\leadsto y \cdot \frac{\color{blue}{\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot -0.16666666666666666, x\right)}}{x} \]

   9. Taylor expanded in x around inf

      \[\leadsto y \cdot \frac{\color{blue}{\frac{-1}{6} \cdot {x}^{3}}}{x} \]
   10. Step-by-step derivation

       1. unpow3 N/A

          \[\leadsto y \cdot \frac{\frac{-1}{6} \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot x\right)}}{x} \]

       2. unpow2 N/A

          \[\leadsto y \cdot \frac{\frac{-1}{6} \cdot \left(\color{blue}{{x}^{2}} \cdot x\right)}{x} \]

       3. associate-*r* N/A

          \[\leadsto y \cdot \frac{\color{blue}{\left(\frac{-1}{6} \cdot {x}^{2}\right) \cdot x}}{x} \]

       4. *-commutative N/A

          \[\leadsto y \cdot \frac{\color{blue}{x \cdot \left(\frac{-1}{6} \cdot {x}^{2}\right)}}{x} \]

       5. lower-*.f64 N/A

          \[\leadsto y \cdot \frac{\color{blue}{x \cdot \left(\frac{-1}{6} \cdot {x}^{2}\right)}}{x} \]

       6. lower-*.f64 N/A

          \[\leadsto y \cdot \frac{x \cdot \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2}\right)}}{x} \]

       7. unpow2 N/A

          \[\leadsto y \cdot \frac{x \cdot \left(\frac{-1}{6} \cdot \color{blue}{\left(x \cdot x\right)}\right)}{x} \]

       8. lower-*.f64 14.8

          \[\leadsto y \cdot \frac{x \cdot \left(-0.16666666666666666 \cdot \color{blue}{\left(x \cdot x\right)}\right)}{x} \]

   11. Simplified 14.8%

       \[\leadsto y \cdot \frac{\color{blue}{x \cdot \left(-0.16666666666666666 \cdot \left(x \cdot x\right)\right)}}{x} \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto y \cdot \frac{x \cdot \left(\frac{-1}{6} \cdot \color{blue}{\left(x \cdot x\right)}\right)}{x} \]

       2. lift-*.f64 N/A

          \[\leadsto y \cdot \frac{x \cdot \color{blue}{\left(\frac{-1}{6} \cdot \left(x \cdot x\right)\right)}}{x} \]

       3. lift-*.f64 N/A

          \[\leadsto y \cdot \frac{\color{blue}{x \cdot \left(\frac{-1}{6} \cdot \left(x \cdot x\right)\right)}}{x} \]

       4. div-inv N/A

          \[\leadsto y \cdot \color{blue}{\left(\left(x \cdot \left(\frac{-1}{6} \cdot \left(x \cdot x\right)\right)\right) \cdot \frac{1}{x}\right)} \]

       5. lift-*.f64 N/A

          \[\leadsto y \cdot \left(\color{blue}{\left(x \cdot \left(\frac{-1}{6} \cdot \left(x \cdot x\right)\right)\right)} \cdot \frac{1}{x}\right) \]

       6. lift-*.f64 N/A

          \[\leadsto y \cdot \left(\left(x \cdot \color{blue}{\left(\frac{-1}{6} \cdot \left(x \cdot x\right)\right)}\right) \cdot \frac{1}{x}\right) \]

       7. associate-*r* N/A

          \[\leadsto y \cdot \left(\color{blue}{\left(\left(x \cdot \frac{-1}{6}\right) \cdot \left(x \cdot x\right)\right)} \cdot \frac{1}{x}\right) \]

       8. lift-/.f64 N/A

          \[\leadsto y \cdot \left(\left(\left(x \cdot \frac{-1}{6}\right) \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\frac{1}{x}}\right) \]

       9. associate-*l* N/A

          \[\leadsto y \cdot \color{blue}{\left(\left(x \cdot \frac{-1}{6}\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{x}\right)\right)} \]

       10. lift-*.f64 N/A

           \[\leadsto y \cdot \left(\left(x \cdot \frac{-1}{6}\right) \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{x}\right)\right) \]

       11. pow2 N/A

           \[\leadsto y \cdot \left(\left(x \cdot \frac{-1}{6}\right) \cdot \left(\color{blue}{{x}^{2}} \cdot \frac{1}{x}\right)\right) \]

       12. lift-/.f64 N/A

           \[\leadsto y \cdot \left(\left(x \cdot \frac{-1}{6}\right) \cdot \left({x}^{2} \cdot \color{blue}{\frac{1}{x}}\right)\right) \]

       13. inv-pow N/A

           \[\leadsto y \cdot \left(\left(x \cdot \frac{-1}{6}\right) \cdot \left({x}^{2} \cdot \color{blue}{{x}^{-1}}\right)\right) \]

       14. pow-prod-up N/A

           \[\leadsto y \cdot \left(\left(x \cdot \frac{-1}{6}\right) \cdot \color{blue}{{x}^{\left(2 + -1\right)}}\right) \]

       15. metadata-eval N/A

           \[\leadsto y \cdot \left(\left(x \cdot \frac{-1}{6}\right) \cdot {x}^{\color{blue}{1}}\right) \]

       16. unpow1 N/A

           \[\leadsto y \cdot \left(\left(x \cdot \frac{-1}{6}\right) \cdot \color{blue}{x}\right) \]

       17. lower-*.f64 N/A

           \[\leadsto y \cdot \color{blue}{\left(\left(x \cdot \frac{-1}{6}\right) \cdot x\right)} \]

       18. lower-*.f64 14.7

           \[\leadsto y \cdot \left(\color{blue}{\left(x \cdot -0.16666666666666666\right)} \cdot x\right) \]

   13. Applied egg-rr 14.7%

       \[\leadsto y \cdot \color{blue}{\left(\left(x \cdot -0.16666666666666666\right) \cdot x\right)} \]

   if -1.99999999999999989e-252 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   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 61.9

         \[\leadsto y \cdot \frac{\color{blue}{\sin x}}{x} \]

   5. Simplified 61.9%

      \[\leadsto \color{blue}{y \cdot \frac{\sin x}{x}} \]

   6. Taylor expanded in x around 0

      \[\leadsto y \cdot \color{blue}{1} \]
   7. Step-by-step derivation

   8. Simplified 33.6%

      \[\leadsto y \cdot \color{blue}{1} \]
   9. Step-by-step derivation

      1. *-rgt-identity 33.6

         \[\leadsto \color{blue}{y} \]

   10. Applied egg-rr 33.6%

       \[\leadsto \color{blue}{y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 28.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -2 \cdot 10^{-252}:\\ \;\;\;\;y \cdot \left(x \cdot \left(x \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 26.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -2 \cdot 10^{-252}:\\ \;\;\;\;y \cdot \left(\left(x \cdot x\right) \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (sinh y) (sin x)) x) -2e-252)
   (* y (* (* x x) -0.16666666666666666))
   y))

C

double code(double x, double y) {
	double tmp;
	if (((sinh(y) * sin(x)) / x) <= -2e-252) {
		tmp = y * ((x * x) * -0.16666666666666666);
	} else {
		tmp = y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (((sinh(y) * sin(x)) / x) <= (-2d-252)) then
        tmp = y * ((x * x) * (-0.16666666666666666d0))
    else
        tmp = y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (((Math.sinh(y) * Math.sin(x)) / x) <= -2e-252) {
		tmp = y * ((x * x) * -0.16666666666666666);
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((math.sinh(y) * math.sin(x)) / x) <= -2e-252:
		tmp = y * ((x * x) * -0.16666666666666666)
	else:
		tmp = y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(sinh(y) * sin(x)) / x) <= -2e-252)
		tmp = Float64(y * Float64(Float64(x * x) * -0.16666666666666666));
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((sinh(y) * sin(x)) / x) <= -2e-252)
		tmp = y * ((x * x) * -0.16666666666666666);
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[Sinh[y], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], -2e-252], N[(y * N[(N[(x * x), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision], y]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -1.99999999999999989e-252

   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 29.2

         \[\leadsto y \cdot \frac{\color{blue}{\sin x}}{x} \]

   5. Simplified 29.2%

      \[\leadsto \color{blue}{y \cdot \frac{\sin x}{x}} \]

   6. Taylor expanded in x around 0

      \[\leadsto y \cdot \frac{\color{blue}{x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)}}{x} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto y \cdot \frac{x \cdot \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2} + 1\right)}}{x} \]

      2. distribute-lft-in N/A

         \[\leadsto y \cdot \frac{\color{blue}{x \cdot \left(\frac{-1}{6} \cdot {x}^{2}\right) + x \cdot 1}}{x} \]

      3. *-rgt-identity N/A

         \[\leadsto y \cdot \frac{x \cdot \left(\frac{-1}{6} \cdot {x}^{2}\right) + \color{blue}{x}}{x} \]

      4. lower-fma.f64 N/A

         \[\leadsto y \cdot \frac{\color{blue}{\mathsf{fma}\left(x, \frac{-1}{6} \cdot {x}^{2}, x\right)}}{x} \]

      5. *-commutative N/A

         \[\leadsto y \cdot \frac{\mathsf{fma}\left(x, \color{blue}{{x}^{2} \cdot \frac{-1}{6}}, x\right)}{x} \]

      6. lower-*.f64 N/A

         \[\leadsto y \cdot \frac{\mathsf{fma}\left(x, \color{blue}{{x}^{2} \cdot \frac{-1}{6}}, x\right)}{x} \]

      7. unpow2 N/A

         \[\leadsto y \cdot \frac{\mathsf{fma}\left(x, \color{blue}{\left(x \cdot x\right)} \cdot \frac{-1}{6}, x\right)}{x} \]

      8. lower-*.f64 26.3

         \[\leadsto y \cdot \frac{\mathsf{fma}\left(x, \color{blue}{\left(x \cdot x\right)} \cdot -0.16666666666666666, x\right)}{x} \]

   8. Simplified 26.3%

      \[\leadsto y \cdot \frac{\color{blue}{\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot -0.16666666666666666, x\right)}}{x} \]

   9. Taylor expanded in x around inf

      \[\leadsto y \cdot \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2}\right)} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto y \cdot \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2}\right)} \]

       2. unpow2 N/A

          \[\leadsto y \cdot \left(\frac{-1}{6} \cdot \color{blue}{\left(x \cdot x\right)}\right) \]

       3. lower-*.f64 14.7

          \[\leadsto y \cdot \left(-0.16666666666666666 \cdot \color{blue}{\left(x \cdot x\right)}\right) \]

   11. Simplified 14.7%

       \[\leadsto y \cdot \color{blue}{\left(-0.16666666666666666 \cdot \left(x \cdot x\right)\right)} \]

   if -1.99999999999999989e-252 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   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 61.9

         \[\leadsto y \cdot \frac{\color{blue}{\sin x}}{x} \]

   5. Simplified 61.9%

      \[\leadsto \color{blue}{y \cdot \frac{\sin x}{x}} \]

   6. Taylor expanded in x around 0

      \[\leadsto y \cdot \color{blue}{1} \]
   7. Step-by-step derivation

   8. Simplified 33.6%

      \[\leadsto y \cdot \color{blue}{1} \]
   9. Step-by-step derivation

      1. *-rgt-identity 33.6

         \[\leadsto \color{blue}{y} \]

   10. Applied egg-rr 33.6%

       \[\leadsto \color{blue}{y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 28.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -2 \cdot 10^{-252}:\\ \;\;\;\;y \cdot \left(\left(x \cdot x\right) \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sin x \cdot \frac{\sinh y}{x} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* (sin x) (/ (sinh y) x)))

C

double code(double x, double y) {
	return sin(x) * (sinh(y) / x);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = sin(x) * (sinh(y) / x)
end function

Java

public static double code(double x, double y) {
	return Math.sin(x) * (Math.sinh(y) / x);
}

Python

def code(x, y):
	return math.sin(x) * (math.sinh(y) / x)

Julia

function code(x, y)
	return Float64(sin(x) * Float64(sinh(y) / x))
end

MATLAB

function tmp = code(x, y)
	tmp = sin(x) * (sinh(y) / x);
end

Wolfram

code[x_, y_] := N[(N[Sin[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 86.7%

   \[\frac{\sin x \cdot \sinh y}{x} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-sin.f64 N/A

      \[\leadsto \frac{\color{blue}{\sin x} \cdot \sinh y}{x} \]

   2. lift-sinh.f64 N/A

      \[\leadsto \frac{\sin x \cdot \color{blue}{\sinh y}}{x} \]

   3. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\sinh y \cdot \sin x}}{x} \]

   4. associate-*l/ N/A

      \[\leadsto \color{blue}{\frac{\sinh y}{x} \cdot \sin x} \]

   5. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{\sinh y}{x} \cdot \sin x} \]

   6. lower-/.f64 99.9

      \[\leadsto \color{blue}{\frac{\sinh y}{x}} \cdot \sin x \]

4. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{\frac{\sinh y}{x} \cdot \sin x} \]

5. Final simplification 99.9%

   \[\leadsto \sin x \cdot \frac{\sinh y}{x} \]
6. Add Preprocessing

Alternative 23: 36.4% accurate, 12.8× speedup

Math

\[\begin{array}{l} \\ y \cdot \mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* y (fma x (* x -0.16666666666666666) 1.0)))

C

double code(double x, double y) {
	return y * fma(x, (x * -0.16666666666666666), 1.0);
}

Julia

function code(x, y)
	return Float64(y * fma(x, Float64(x * -0.16666666666666666), 1.0))
end

Wolfram

code[x_, y_] := N[(y * N[(x * N[(x * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 86.7%

   \[\frac{\sin x \cdot \sinh y}{x} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\frac{y \cdot \sin x}{x}} \]
4. Step-by-step derivation

   1. associate-/l* N/A

      \[\leadsto \color{blue}{y \cdot \frac{\sin x}{x}} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{y \cdot \frac{\sin x}{x}} \]

   3. lower-/.f64 N/A

      \[\leadsto y \cdot \color{blue}{\frac{\sin x}{x}} \]

   4. lower-sin.f64 53.0

      \[\leadsto y \cdot \frac{\color{blue}{\sin x}}{x} \]

5. Simplified 53.0%

   \[\leadsto \color{blue}{y \cdot \frac{\sin x}{x}} \]

6. Taylor expanded in x around 0

   \[\leadsto y \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {x}^{2}\right)} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto y \cdot \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2} + 1\right)} \]

   2. *-commutative N/A

      \[\leadsto y \cdot \left(\color{blue}{{x}^{2} \cdot \frac{-1}{6}} + 1\right) \]

   3. unpow2 N/A

      \[\leadsto y \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{-1}{6} + 1\right) \]

   4. associate-*l* N/A

      \[\leadsto y \cdot \left(\color{blue}{x \cdot \left(x \cdot \frac{-1}{6}\right)} + 1\right) \]

   5. lower-fma.f64 N/A

      \[\leadsto y \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)} \]

   6. lower-*.f64 36.8

      \[\leadsto y \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot -0.16666666666666666}, 1\right) \]

8. Simplified 36.8%

   \[\leadsto y \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right)} \]
9. Add Preprocessing

Alternative 24: 28.1% accurate, 217.0× speedup

Math

\[\begin{array}{l} \\ y \end{array} \]

FPCore

(FPCore (x y) :precision binary64 y)

C

double code(double x, double y) {
	return y;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = y
end function

Java

public static double code(double x, double y) {
	return y;
}

Python

def code(x, y):
	return y

Julia

function code(x, y)
	return y
end

MATLAB

function tmp = code(x, y)
	tmp = y;
end

Wolfram

code[x_, y_] := y

Derivation

1. Initial program 86.7%

   \[\frac{\sin x \cdot \sinh y}{x} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\frac{y \cdot \sin x}{x}} \]
4. Step-by-step derivation

   1. associate-/l* N/A

      \[\leadsto \color{blue}{y \cdot \frac{\sin x}{x}} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{y \cdot \frac{\sin x}{x}} \]

   3. lower-/.f64 N/A

      \[\leadsto y \cdot \color{blue}{\frac{\sin x}{x}} \]

   4. lower-sin.f64 53.0

      \[\leadsto y \cdot \frac{\color{blue}{\sin x}}{x} \]

5. Simplified 53.0%

   \[\leadsto \color{blue}{y \cdot \frac{\sin x}{x}} \]

6. Taylor expanded in x around 0

   \[\leadsto y \cdot \color{blue}{1} \]
7. Step-by-step derivation

8. Simplified 28.2%

   \[\leadsto y \cdot \color{blue}{1} \]
9. Step-by-step derivation

   1. *-rgt-identity 28.2

      \[\leadsto \color{blue}{y} \]

10. Applied egg-rr 28.2%

    \[\leadsto \color{blue}{y} \]
11. Add Preprocessing

Developer Target 1: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sin x \cdot \frac{\sinh y}{x} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* (sin x) (/ (sinh y) x)))

C

double code(double x, double y) {
	return sin(x) * (sinh(y) / x);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = sin(x) * (sinh(y) / x)
end function

Java

public static double code(double x, double y) {
	return Math.sin(x) * (Math.sinh(y) / x);
}

Python

def code(x, y):
	return math.sin(x) * (math.sinh(y) / x)

Julia

function code(x, y)
	return Float64(sin(x) * Float64(sinh(y) / x))
end

MATLAB

function tmp = code(x, y)
	tmp = sin(x) * (sinh(y) / x);
end

Wolfram

code[x_, y_] := N[(N[Sin[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024208 
(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))