Linear.Quaternion:$ccos from linear-1.19.1.3

Percentage Accurate: 100.0% → 100.0%

Time: 15.7s

Alternatives: 21

Speedup: 1.0×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

Alternative 2: 74.8% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. *-lowering-*.f64 74.5

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

   5. Simplified 74.5%

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

   6. Taylor expanded in x around inf

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

      1. unpow3 N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 N/A

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 25.5

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

   8. Simplified 25.5%

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

   if -inf.0 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. +-commutative N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 98.6

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

   5. Simplified 98.6%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 79.4%

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

4. Final simplification 78.1%

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

Alternative 3: 84.6% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. +-commutative N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. +-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. *-lowering-*.f64 77.6

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

   5. Simplified 77.6%

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

   6. Taylor expanded in x around 0

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. sub-neg N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. metadata-eval N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. accelerator-lowering-fma.f64 N/A

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

      15. unpow2 N/A

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

      16. *-lowering-*.f64 62.4

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

   8. Simplified 62.4%

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

   if -inf.0 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 98.2

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

   5. Simplified 98.2%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 79.4%

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

4. Final simplification 85.9%

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

Alternative 4: 84.4% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. +-commutative N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. +-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. *-lowering-*.f64 77.6

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

   5. Simplified 77.6%

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

   6. Taylor expanded in x around 0

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. sub-neg N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. metadata-eval N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. accelerator-lowering-fma.f64 N/A

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

      15. unpow2 N/A

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

      16. *-lowering-*.f64 62.4

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

   8. Simplified 62.4%

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

   if -inf.0 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. sin-lowering-sin.f64 97.7

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

   5. Simplified 97.7%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 79.4%

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

4. Final simplification 85.6%

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

Alternative 5: 82.2% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. +-commutative N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. +-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. *-lowering-*.f64 77.6

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

   5. Simplified 77.6%

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

   6. Taylor expanded in x around 0

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. sub-neg N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. metadata-eval N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. accelerator-lowering-fma.f64 N/A

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

      15. unpow2 N/A

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

      16. *-lowering-*.f64 62.4

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

   8. Simplified 62.4%

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

   if -inf.0 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. sin-lowering-sin.f64 97.7

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

   5. Simplified 97.7%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. *-lowering-*.f64 74.6

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

   5. Simplified 74.6%

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

   6. Taylor expanded in y around 0

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. +-commutative N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 67.0

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

   8. Simplified 67.0%

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

   9. Taylor expanded in x around 0

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

   11. Simplified 73.3%

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

4. Final simplification 84.1%

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

Alternative 6: 89.0% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. +-commutative N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. +-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. *-lowering-*.f64 92.7

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

   5. Simplified 92.7%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 79.4%

      \[\leadsto \color{blue}{x} \cdot \frac{\sinh y}{y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 87.0% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. +-commutative N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 88.6

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

   5. Simplified 88.6%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 79.4%

      \[\leadsto \color{blue}{x} \cdot \frac{\sinh y}{y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 57.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin x \cdot \frac{\sinh y}{y} \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right) \cdot \left(x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (sin x) (/ (sinh y) y)) 2e-7)
   (*
    (fma x (* x -0.16666666666666666) 1.0)
    (*
     x
     (fma
      (* y y)
      (fma
       (* y y)
       (fma y (* y 0.0001984126984126984) 0.008333333333333333)
       0.16666666666666666)
      1.0)))
   (*
    x
    (/
     (*
      y
      (fma
       y
       (*
        y
        (fma
         (* y y)
         (fma (* y y) 0.0001984126984126984 0.008333333333333333)
         0.16666666666666666))
       1.0))
     y))))

C

double code(double x, double y) {
	double tmp;
	if ((sin(x) * (sinh(y) / y)) <= 2e-7) {
		tmp = fma(x, (x * -0.16666666666666666), 1.0) * (x * fma((y * y), fma((y * y), fma(y, (y * 0.0001984126984126984), 0.008333333333333333), 0.16666666666666666), 1.0));
	} else {
		tmp = x * ((y * fma(y, (y * fma((y * y), fma((y * y), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666)), 1.0)) / y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(sin(x) * Float64(sinh(y) / y)) <= 2e-7)
		tmp = Float64(fma(x, Float64(x * -0.16666666666666666), 1.0) * Float64(x * fma(Float64(y * y), fma(Float64(y * y), fma(y, Float64(y * 0.0001984126984126984), 0.008333333333333333), 0.16666666666666666), 1.0)));
	else
		tmp = Float64(x * Float64(Float64(y * fma(y, Float64(y * fma(Float64(y * y), fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666)), 1.0)) / y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Sin[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], 2e-7], N[(N[(x * N[(x * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * 0.0001984126984126984), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(y * N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < 1.9999999999999999e-7

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. *-lowering-*.f64 68.6

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

   5. Simplified 68.6%

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

   6. Taylor expanded in y around 0

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. +-commutative N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 66.4

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

   8. Simplified 66.4%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. clear-num N/A

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

      7. associate-/r* N/A

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

      8. *-inverses N/A

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

   10. Applied egg-rr 64.1%

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

   if 1.9999999999999999e-7 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. *-lowering-*.f64 51.4

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

   5. Simplified 51.4%

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

   6. Taylor expanded in y around 0

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. +-commutative N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 46.3

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

   8. Simplified 46.3%

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

   9. Taylor expanded in x around 0

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

   11. Simplified 51.3%

       \[\leadsto \color{blue}{x} \cdot \frac{y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)}{y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 47.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right)\\ \mathbf{if}\;\sin x \cdot \frac{\sinh y}{y} \leq -0.02:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(x \cdot \mathsf{fma}\left(y \cdot y, -0.16666666666666666 \cdot \mathsf{fma}\left(y, y \cdot t\_0, 0.16666666666666666\right), -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, t\_0, 0.16666666666666666\right), 1\right)}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma (* y y) 0.0001984126984126984 0.008333333333333333)))
   (if (<= (* (sin x) (/ (sinh y) y)) -0.02)
     (*
      (* x x)
      (*
       x
       (fma
        (* y y)
        (* -0.16666666666666666 (fma y (* y t_0) 0.16666666666666666))
        -0.16666666666666666)))
     (*
      x
      (/ (* y (fma y (* y (fma (* y y) t_0 0.16666666666666666)) 1.0)) y)))))

C

double code(double x, double y) {
	double t_0 = fma((y * y), 0.0001984126984126984, 0.008333333333333333);
	double tmp;
	if ((sin(x) * (sinh(y) / y)) <= -0.02) {
		tmp = (x * x) * (x * fma((y * y), (-0.16666666666666666 * fma(y, (y * t_0), 0.16666666666666666)), -0.16666666666666666));
	} else {
		tmp = x * ((y * fma(y, (y * fma((y * y), t_0, 0.16666666666666666)), 1.0)) / y);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333)
	tmp = 0.0
	if (Float64(sin(x) * Float64(sinh(y) / y)) <= -0.02)
		tmp = Float64(Float64(x * x) * Float64(x * fma(Float64(y * y), Float64(-0.16666666666666666 * fma(y, Float64(y * t_0), 0.16666666666666666)), -0.16666666666666666)));
	else
		tmp = Float64(x * Float64(Float64(y * fma(y, Float64(y * fma(Float64(y * y), t_0, 0.16666666666666666)), 1.0)) / y));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision]}, If[LessEqual[N[(N[Sin[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], -0.02], N[(N[(x * x), $MachinePrecision] * N[(x * N[(N[(y * y), $MachinePrecision] * N[(-0.16666666666666666 * N[(y * N[(y * t$95$0), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(y * N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * t$95$0 + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. *-lowering-*.f64 45.0

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

   5. Simplified 45.0%

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

   6. Taylor expanded in y around 0

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. +-commutative N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 41.9

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

   8. Simplified 41.9%

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

   9. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

       4. unpow3 N/A

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

       5. unpow2 N/A

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

       6. associate-*l* N/A

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

       7. *-lowering-*.f64 N/A

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

       8. unpow2 N/A

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

       9. *-lowering-*.f64 N/A

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

       10. *-lowering-*.f64 N/A

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

       11. +-commutative N/A

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

       12. distribute-rgt-in N/A

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

   11. Simplified 16.1%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. *-lowering-*.f64 72.3

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

   5. Simplified 72.3%

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

   6. Taylor expanded in y around 0

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. +-commutative N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 68.9

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

   8. Simplified 68.9%

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

   9. Taylor expanded in x around 0

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

   11. Simplified 71.7%

       \[\leadsto \color{blue}{x} \cdot \frac{y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)}{y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin x \cdot \frac{\sinh y}{y} \leq -0.02:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(x \cdot \mathsf{fma}\left(y \cdot y, -0.16666666666666666 \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 47.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right)\\ \mathbf{if}\;\sin x \cdot \frac{\sinh y}{y} \leq -0.02:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(x \cdot \mathsf{fma}\left(y \cdot y, -0.16666666666666666 \cdot t\_0, -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(y \cdot y, t\_0, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (fma
          y
          (* y (fma (* y y) 0.0001984126984126984 0.008333333333333333))
          0.16666666666666666)))
   (if (<= (* (sin x) (/ (sinh y) y)) -0.02)
     (*
      (* x x)
      (* x (fma (* y y) (* -0.16666666666666666 t_0) -0.16666666666666666)))
     (* x (fma (* y y) t_0 1.0)))))

C

double code(double x, double y) {
	double t_0 = fma(y, (y * fma((y * y), 0.0001984126984126984, 0.008333333333333333)), 0.16666666666666666);
	double tmp;
	if ((sin(x) * (sinh(y) / y)) <= -0.02) {
		tmp = (x * x) * (x * fma((y * y), (-0.16666666666666666 * t_0), -0.16666666666666666));
	} else {
		tmp = x * fma((y * y), t_0, 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(y, Float64(y * fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333)), 0.16666666666666666)
	tmp = 0.0
	if (Float64(sin(x) * Float64(sinh(y) / y)) <= -0.02)
		tmp = Float64(Float64(x * x) * Float64(x * fma(Float64(y * y), Float64(-0.16666666666666666 * t_0), -0.16666666666666666)));
	else
		tmp = Float64(x * fma(Float64(y * y), t_0, 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]}, If[LessEqual[N[(N[Sin[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], -0.02], N[(N[(x * x), $MachinePrecision] * N[(x * N[(N[(y * y), $MachinePrecision] * N[(-0.16666666666666666 * t$95$0), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(y * y), $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. *-lowering-*.f64 45.0

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

   5. Simplified 45.0%

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

   6. Taylor expanded in y around 0

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. +-commutative N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 41.9

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

   8. Simplified 41.9%

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

   9. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

       4. unpow3 N/A

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

       5. unpow2 N/A

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

       6. associate-*l* N/A

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

       7. *-lowering-*.f64 N/A

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

       8. unpow2 N/A

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

       9. *-lowering-*.f64 N/A

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

       10. *-lowering-*.f64 N/A

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

       11. +-commutative N/A

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

       12. distribute-rgt-in N/A

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

   11. Simplified 16.1%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. *-lowering-*.f64 72.3

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

   5. Simplified 72.3%

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

   6. Taylor expanded in y around 0

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. +-commutative N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 68.9

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

   8. Simplified 68.9%

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

   9. Taylor expanded in x around 0

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

       1. *-lowering-*.f64 N/A

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

       2. +-commutative N/A

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

       3. accelerator-lowering-fma.f64 N/A

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

       4. unpow2 N/A

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

       5. *-lowering-*.f64 N/A

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

       6. +-commutative N/A

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

       7. unpow2 N/A

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

       8. associate-*l* N/A

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

       9. accelerator-lowering-fma.f64 N/A

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

       10. *-lowering-*.f64 N/A

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

       11. +-commutative N/A

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

       12. *-commutative N/A

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

       13. accelerator-lowering-fma.f64 N/A

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

       14. unpow2 N/A

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

       15. *-lowering-*.f64 70.5

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

   11. Simplified 70.5%

       \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin x \cdot \frac{\sinh y}{y} \leq -0.02:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(x \cdot \mathsf{fma}\left(y \cdot y, -0.16666666666666666 \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 56.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin x \cdot \frac{\sinh y}{y} \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\left(x \cdot \mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right)\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (sin x) (/ (sinh y) y)) 2e-7)
   (*
    (* x (fma x (* x -0.16666666666666666) 1.0))
    (fma y (* y (fma y (* y 0.008333333333333333) 0.16666666666666666)) 1.0))
   (*
    x
    (fma
     (* y y)
     (fma
      y
      (* y (fma (* y y) 0.0001984126984126984 0.008333333333333333))
      0.16666666666666666)
     1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((sin(x) * (sinh(y) / y)) <= 2e-7) {
		tmp = (x * fma(x, (x * -0.16666666666666666), 1.0)) * fma(y, (y * fma(y, (y * 0.008333333333333333), 0.16666666666666666)), 1.0);
	} else {
		tmp = x * fma((y * y), fma(y, (y * fma((y * y), 0.0001984126984126984, 0.008333333333333333)), 0.16666666666666666), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(sin(x) * Float64(sinh(y) / y)) <= 2e-7)
		tmp = Float64(Float64(x * fma(x, Float64(x * -0.16666666666666666), 1.0)) * fma(y, Float64(y * fma(y, Float64(y * 0.008333333333333333), 0.16666666666666666)), 1.0));
	else
		tmp = Float64(x * fma(Float64(y * y), fma(y, Float64(y * fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333)), 0.16666666666666666), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Sin[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], 2e-7], N[(N[(x * N[(x * N[(x * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[(y * N[(y * N[(y * N[(y * 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < 1.9999999999999999e-7

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. *-lowering-*.f64 68.6

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

   5. Simplified 68.6%

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

   6. Taylor expanded in y around 0

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. +-commutative N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 66.4

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

   8. Simplified 66.4%

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

   9. Taylor expanded in y around 0

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

       1. +-commutative N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. accelerator-lowering-fma.f64 N/A

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

       5. *-lowering-*.f64 N/A

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

       6. +-commutative N/A

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

       7. *-commutative N/A

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

       8. unpow2 N/A

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

       9. associate-*l* N/A

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

       10. accelerator-lowering-fma.f64 N/A

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

       11. *-lowering-*.f64 60.5

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

   11. Simplified 60.5%

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

   if 1.9999999999999999e-7 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. *-lowering-*.f64 51.4

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

   5. Simplified 51.4%

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

   6. Taylor expanded in y around 0

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. +-commutative N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 46.3

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

   8. Simplified 46.3%

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

   9. Taylor expanded in x around 0

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

       1. *-lowering-*.f64 N/A

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

       2. +-commutative N/A

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

       3. accelerator-lowering-fma.f64 N/A

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

       4. unpow2 N/A

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

       5. *-lowering-*.f64 N/A

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

       6. +-commutative N/A

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

       7. unpow2 N/A

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

       8. associate-*l* N/A

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

       9. accelerator-lowering-fma.f64 N/A

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

       10. *-lowering-*.f64 N/A

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

       11. +-commutative N/A

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

       12. *-commutative N/A

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

       13. accelerator-lowering-fma.f64 N/A

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

       14. unpow2 N/A

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

       15. *-lowering-*.f64 49.2

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

   11. Simplified 49.2%

       \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 47.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin x \cdot \frac{\sinh y}{y} \leq -0.02:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, -0.001388888888888889, -0.027777777777777776\right), -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (sin x) (/ (sinh y) y)) -0.02)
   (*
    (* x x)
    (*
     x
     (fma
      (* y y)
      (fma (* y y) -0.001388888888888889 -0.027777777777777776)
      -0.16666666666666666)))
   (*
    x
    (fma
     (* y y)
     (fma
      y
      (* y (fma (* y y) 0.0001984126984126984 0.008333333333333333))
      0.16666666666666666)
     1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((sin(x) * (sinh(y) / y)) <= -0.02) {
		tmp = (x * x) * (x * fma((y * y), fma((y * y), -0.001388888888888889, -0.027777777777777776), -0.16666666666666666));
	} else {
		tmp = x * fma((y * y), fma(y, (y * fma((y * y), 0.0001984126984126984, 0.008333333333333333)), 0.16666666666666666), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(sin(x) * Float64(sinh(y) / y)) <= -0.02)
		tmp = Float64(Float64(x * x) * Float64(x * fma(Float64(y * y), fma(Float64(y * y), -0.001388888888888889, -0.027777777777777776), -0.16666666666666666)));
	else
		tmp = Float64(x * fma(Float64(y * y), fma(y, Float64(y * fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333)), 0.16666666666666666), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Sin[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], -0.02], N[(N[(x * x), $MachinePrecision] * N[(x * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * -0.001388888888888889 + -0.027777777777777776), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. *-lowering-*.f64 45.0

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

   5. Simplified 45.0%

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

   6. Taylor expanded in y around 0

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. +-commutative N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 41.9

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

   8. Simplified 41.9%

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

   9. Taylor expanded in y around 0

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

       1. accelerator-lowering-fma.f64 N/A

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

       2. +-commutative N/A

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

       3. *-commutative N/A

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

       4. accelerator-lowering-fma.f64 N/A

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

       5. unpow2 N/A

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

       6. *-lowering-*.f64 N/A

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

       7. *-lowering-*.f64 N/A

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

       8. unpow2 N/A

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

       9. *-lowering-*.f64 N/A

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

       10. +-commutative N/A

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

       11. *-commutative N/A

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

       12. *-commutative N/A

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

       13. associate-*r* N/A

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

   11. Simplified 26.5%

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

   12. Taylor expanded in x around inf

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

       1. unpow3 N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. *-lowering-*.f64 N/A

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

       5. unpow2 N/A

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

       6. *-lowering-*.f64 N/A

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

       7. *-lowering-*.f64 N/A

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

       8. sub-neg N/A

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

       9. *-commutative N/A

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

       10. associate-*l* N/A

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

       11. *-commutative N/A

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

       12. metadata-eval N/A

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

       13. accelerator-lowering-fma.f64 N/A

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

   14. Simplified 15.2%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. *-lowering-*.f64 72.3

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

   5. Simplified 72.3%

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

   6. Taylor expanded in y around 0

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. +-commutative N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 68.9

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

   8. Simplified 68.9%

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

   9. Taylor expanded in x around 0

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

       1. *-lowering-*.f64 N/A

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

       2. +-commutative N/A

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

       3. accelerator-lowering-fma.f64 N/A

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

       4. unpow2 N/A

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

       5. *-lowering-*.f64 N/A

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

       6. +-commutative N/A

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

       7. unpow2 N/A

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

       8. associate-*l* N/A

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

       9. accelerator-lowering-fma.f64 N/A

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

       10. *-lowering-*.f64 N/A

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

       11. +-commutative N/A

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

       12. *-commutative N/A

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

       13. accelerator-lowering-fma.f64 N/A

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

       14. unpow2 N/A

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

       15. *-lowering-*.f64 70.5

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

   11. Simplified 70.5%

       \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 42.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin x \cdot \frac{\sinh y}{y} \leq -0.02:\\ \;\;\;\;\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(y \cdot y, -0.027777777777777776, -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (sin x) (/ (sinh y) y)) -0.02)
   (* (* x (* x x)) (fma (* y y) -0.027777777777777776 -0.16666666666666666))
   (* x (fma 0.16666666666666666 (* y y) 1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((sin(x) * (sinh(y) / y)) <= -0.02) {
		tmp = (x * (x * x)) * fma((y * y), -0.027777777777777776, -0.16666666666666666);
	} else {
		tmp = x * fma(0.16666666666666666, (y * y), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(sin(x) * Float64(sinh(y) / y)) <= -0.02)
		tmp = Float64(Float64(x * Float64(x * x)) * fma(Float64(y * y), -0.027777777777777776, -0.16666666666666666));
	else
		tmp = Float64(x * fma(0.16666666666666666, Float64(y * y), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Sin[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], -0.02], N[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * -0.027777777777777776 + -0.16666666666666666), $MachinePrecision]), $MachinePrecision], N[(x * N[(0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. *-lowering-*.f64 45.0

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

   5. Simplified 45.0%

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

   6. Taylor expanded in x around inf

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

      1. unpow3 N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 N/A

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 16.0

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

   8. Simplified 16.0%

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

   9. Taylor expanded in y around 0

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. metadata-eval N/A

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

       4. associate-*r* N/A

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

       5. distribute-rgt-out N/A

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

       6. metadata-eval N/A

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

       7. distribute-lft-in N/A

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

       8. *-lowering-*.f64 N/A

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

       9. cube-mult N/A

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

       10. unpow2 N/A

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

       11. *-lowering-*.f64 N/A

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

       12. unpow2 N/A

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

       13. *-lowering-*.f64 N/A

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

       14. distribute-rgt-in N/A

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

       15. metadata-eval N/A

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

       16. +-commutative N/A

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

       17. *-commutative N/A

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

       18. associate-*l* N/A

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

       19. metadata-eval N/A

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

       20. accelerator-lowering-fma.f64 N/A

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

       21. unpow2 N/A

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

       22. *-lowering-*.f64 15.2

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

   11. Simplified 15.2%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 80.4

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

   5. Simplified 80.4%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

      3. +-commutative N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 62.7

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

   8. Simplified 62.7%

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

4. Final simplification 45.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin x \cdot \frac{\sinh y}{y} \leq -0.02:\\ \;\;\;\;\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(y \cdot y, -0.027777777777777776, -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 40.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin x \cdot \frac{\sinh y}{y} \leq -0.02:\\ \;\;\;\;x \cdot \left(x \cdot \left(x \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (sin x) (/ (sinh y) y)) -0.02)
   (* x (* x (* x -0.16666666666666666)))
   (* x (fma 0.16666666666666666 (* y y) 1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((sin(x) * (sinh(y) / y)) <= -0.02) {
		tmp = x * (x * (x * -0.16666666666666666));
	} else {
		tmp = x * fma(0.16666666666666666, (y * y), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(sin(x) * Float64(sinh(y) / y)) <= -0.02)
		tmp = Float64(x * Float64(x * Float64(x * -0.16666666666666666)));
	else
		tmp = Float64(x * fma(0.16666666666666666, Float64(y * y), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Sin[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], -0.02], N[(x * N[(x * N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. *-lowering-*.f64 45.0

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

   5. Simplified 45.0%

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

   6. Taylor expanded in x around inf

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

      1. unpow3 N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 N/A

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 16.0

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

   8. Simplified 16.0%

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

   9. Taylor expanded in y around 0

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

       1. unpow3 N/A

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

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

       5. *-lowering-*.f64 N/A

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

       6. *-commutative N/A

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

       7. *-lowering-*.f64 N/A

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

       8. unpow2 N/A

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

       9. *-lowering-*.f64 15.4

          \[\leadsto x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot -0.16666666666666666\right) \]

   11. Simplified 15.4%

       \[\leadsto \color{blue}{x \cdot \left(\left(x \cdot x\right) \cdot -0.16666666666666666\right)} \]
   12. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(x \cdot \frac{-1}{6}\right)\right)} \]

       2. *-commutative N/A

          \[\leadsto x \cdot \color{blue}{\left(\left(x \cdot \frac{-1}{6}\right) \cdot x\right)} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \color{blue}{\left(\left(x \cdot \frac{-1}{6}\right) \cdot x\right)} \]

       4. *-lowering-*.f64 15.4

          \[\leadsto x \cdot \left(\color{blue}{\left(x \cdot -0.16666666666666666\right)} \cdot x\right) \]

   13. Applied egg-rr 15.4%

       \[\leadsto x \cdot \color{blue}{\left(\left(x \cdot -0.16666666666666666\right) \cdot x\right)} \]

   if -0.0200000000000000004 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

      \[\sin x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \sin x \cdot \color{blue}{\left(1 + \frac{1}{6} \cdot {y}^{2}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \sin x \cdot \color{blue}{\left(\frac{1}{6} \cdot {y}^{2} + 1\right)} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \sin x \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{6}, {y}^{2}, 1\right)} \]

      3. unpow2 N/A

         \[\leadsto \sin x \cdot \mathsf{fma}\left(\frac{1}{6}, \color{blue}{y \cdot y}, 1\right) \]

      4. *-lowering-*.f64 80.4

         \[\leadsto \sin x \cdot \mathsf{fma}\left(0.16666666666666666, \color{blue}{y \cdot y}, 1\right) \]

   5. Simplified 80.4%

      \[\leadsto \sin x \cdot \color{blue}{\mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(1 + \frac{1}{6} \cdot {y}^{2}\right) \cdot x} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(1 + \frac{1}{6} \cdot {y}^{2}\right) \cdot x} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{6} \cdot {y}^{2} + 1\right)} \cdot x \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{6}, {y}^{2}, 1\right)} \cdot x \]

      5. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{6}, \color{blue}{y \cdot y}, 1\right) \cdot x \]

      6. *-lowering-*.f64 62.7

         \[\leadsto \mathsf{fma}\left(0.16666666666666666, \color{blue}{y \cdot y}, 1\right) \cdot x \]

   8. Simplified 62.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right) \cdot x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 45.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin x \cdot \frac{\sinh y}{y} \leq -0.02:\\ \;\;\;\;x \cdot \left(x \cdot \left(x \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 29.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin x \cdot \frac{\sinh y}{y} \leq -0.02:\\ \;\;\;\;x \cdot \left(x \cdot \left(x \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (sin x) (/ (sinh y) y)) -0.02)
   (* x (* x (* x -0.16666666666666666)))
   x))

C

double code(double x, double y) {
	double tmp;
	if ((sin(x) * (sinh(y) / y)) <= -0.02) {
		tmp = x * (x * (x * -0.16666666666666666));
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((sin(x) * (sinh(y) / y)) <= (-0.02d0)) then
        tmp = x * (x * (x * (-0.16666666666666666d0)))
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((Math.sin(x) * (Math.sinh(y) / y)) <= -0.02) {
		tmp = x * (x * (x * -0.16666666666666666));
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (math.sin(x) * (math.sinh(y) / y)) <= -0.02:
		tmp = x * (x * (x * -0.16666666666666666))
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(sin(x) * Float64(sinh(y) / y)) <= -0.02)
		tmp = Float64(x * Float64(x * Float64(x * -0.16666666666666666)));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((sin(x) * (sinh(y) / y)) <= -0.02)
		tmp = x * (x * (x * -0.16666666666666666));
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Sin[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], -0.02], N[(x * N[(x * N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < -0.0200000000000000004

   1. Initial program 100.0%

      \[\sin x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)} \cdot \frac{\sinh y}{y} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)} \cdot \frac{\sinh y}{y} \]

      2. +-commutative N/A

         \[\leadsto \left(x \cdot \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2} + 1\right)}\right) \cdot \frac{\sinh y}{y} \]

      3. *-commutative N/A

         \[\leadsto \left(x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{-1}{6}} + 1\right)\right) \cdot \frac{\sinh y}{y} \]

      4. unpow2 N/A

         \[\leadsto \left(x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{-1}{6} + 1\right)\right) \cdot \frac{\sinh y}{y} \]

      5. associate-*l* N/A

         \[\leadsto \left(x \cdot \left(\color{blue}{x \cdot \left(x \cdot \frac{-1}{6}\right)} + 1\right)\right) \cdot \frac{\sinh y}{y} \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(x \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)}\right) \cdot \frac{\sinh y}{y} \]

      7. *-lowering-*.f64 45.0

         \[\leadsto \left(x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot -0.16666666666666666}, 1\right)\right) \cdot \frac{\sinh y}{y} \]

   5. Simplified 45.0%

      \[\leadsto \color{blue}{\left(x \cdot \mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right)\right)} \cdot \frac{\sinh y}{y} \]

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\left(\frac{-1}{6} \cdot {x}^{3}\right)} \cdot \frac{\sinh y}{y} \]
   7. Step-by-step derivation

      1. unpow3 N/A

         \[\leadsto \left(\frac{-1}{6} \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot x\right)}\right) \cdot \frac{\sinh y}{y} \]

      2. unpow2 N/A

         \[\leadsto \left(\frac{-1}{6} \cdot \left(\color{blue}{{x}^{2}} \cdot x\right)\right) \cdot \frac{\sinh y}{y} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(\frac{-1}{6} \cdot {x}^{2}\right) \cdot x\right)} \cdot \frac{\sinh y}{y} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot \left(\frac{-1}{6} \cdot {x}^{2}\right)\right)} \cdot \frac{\sinh y}{y} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot \left(\frac{-1}{6} \cdot {x}^{2}\right)\right)} \cdot \frac{\sinh y}{y} \]

      6. *-commutative N/A

         \[\leadsto \left(x \cdot \color{blue}{\left({x}^{2} \cdot \frac{-1}{6}\right)}\right) \cdot \frac{\sinh y}{y} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \left(x \cdot \color{blue}{\left({x}^{2} \cdot \frac{-1}{6}\right)}\right) \cdot \frac{\sinh y}{y} \]

      8. unpow2 N/A

         \[\leadsto \left(x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{-1}{6}\right)\right) \cdot \frac{\sinh y}{y} \]

      9. *-lowering-*.f64 16.0

         \[\leadsto \left(x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot -0.16666666666666666\right)\right) \cdot \frac{\sinh y}{y} \]

   8. Simplified 16.0%

      \[\leadsto \color{blue}{\left(x \cdot \left(\left(x \cdot x\right) \cdot -0.16666666666666666\right)\right)} \cdot \frac{\sinh y}{y} \]

   9. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{-1}{6} \cdot {x}^{3}} \]
   10. Step-by-step derivation

       1. unpow3 N/A

          \[\leadsto \frac{-1}{6} \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot x\right)} \]

       2. unpow2 N/A

          \[\leadsto \frac{-1}{6} \cdot \left(\color{blue}{{x}^{2}} \cdot x\right) \]

       3. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2}\right) \cdot x} \]

       4. *-commutative N/A

          \[\leadsto \color{blue}{x \cdot \left(\frac{-1}{6} \cdot {x}^{2}\right)} \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{x \cdot \left(\frac{-1}{6} \cdot {x}^{2}\right)} \]

       6. *-commutative N/A

          \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \frac{-1}{6}\right)} \]

       7. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \frac{-1}{6}\right)} \]

       8. unpow2 N/A

          \[\leadsto x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{-1}{6}\right) \]

       9. *-lowering-*.f64 15.4

          \[\leadsto x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot -0.16666666666666666\right) \]

   11. Simplified 15.4%

       \[\leadsto \color{blue}{x \cdot \left(\left(x \cdot x\right) \cdot -0.16666666666666666\right)} \]
   12. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(x \cdot \frac{-1}{6}\right)\right)} \]

       2. *-commutative N/A

          \[\leadsto x \cdot \color{blue}{\left(\left(x \cdot \frac{-1}{6}\right) \cdot x\right)} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \color{blue}{\left(\left(x \cdot \frac{-1}{6}\right) \cdot x\right)} \]

       4. *-lowering-*.f64 15.4

          \[\leadsto x \cdot \left(\color{blue}{\left(x \cdot -0.16666666666666666\right)} \cdot x\right) \]

   13. Applied egg-rr 15.4%

       \[\leadsto x \cdot \color{blue}{\left(\left(x \cdot -0.16666666666666666\right) \cdot x\right)} \]

   if -0.0200000000000000004 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

      \[\sin x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\sin x} \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 60.6

         \[\leadsto \color{blue}{\sin x} \]

   5. Simplified 60.6%

      \[\leadsto \color{blue}{\sin x} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x} \]
   7. Step-by-step derivation

   8. Simplified 44.0%

      \[\leadsto \color{blue}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 33.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin x \cdot \frac{\sinh y}{y} \leq -0.02:\\ \;\;\;\;x \cdot \left(x \cdot \left(x \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 58.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right)\\ \mathbf{if}\;\sin x \leq 8 \cdot 10^{-54}:\\ \;\;\;\;\left(x \cdot \mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right)\right) \cdot \frac{y \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(t\_0, y \cdot \left(y \cdot y\right), y \cdot 0.16666666666666666\right), 1\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, t\_0, 0.16666666666666666\right), 1\right) \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.008333333333333333, -0.16666666666666666\right), 1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma y (* y 0.0001984126984126984) 0.008333333333333333)))
   (if (<= (sin x) 8e-54)
     (*
      (* x (fma x (* x -0.16666666666666666) 1.0))
      (/
       (* y (fma y (fma t_0 (* y (* y y)) (* y 0.16666666666666666)) 1.0))
       y))
     (*
      (fma (* y y) (fma (* y y) t_0 0.16666666666666666) 1.0)
      (*
       x
       (fma
        x
        (* x (fma x (* x 0.008333333333333333) -0.16666666666666666))
        1.0))))))

C

double code(double x, double y) {
	double t_0 = fma(y, (y * 0.0001984126984126984), 0.008333333333333333);
	double tmp;
	if (sin(x) <= 8e-54) {
		tmp = (x * fma(x, (x * -0.16666666666666666), 1.0)) * ((y * fma(y, fma(t_0, (y * (y * y)), (y * 0.16666666666666666)), 1.0)) / y);
	} else {
		tmp = fma((y * y), fma((y * y), t_0, 0.16666666666666666), 1.0) * (x * fma(x, (x * fma(x, (x * 0.008333333333333333), -0.16666666666666666)), 1.0));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(y, Float64(y * 0.0001984126984126984), 0.008333333333333333)
	tmp = 0.0
	if (sin(x) <= 8e-54)
		tmp = Float64(Float64(x * fma(x, Float64(x * -0.16666666666666666), 1.0)) * Float64(Float64(y * fma(y, fma(t_0, Float64(y * Float64(y * y)), Float64(y * 0.16666666666666666)), 1.0)) / y));
	else
		tmp = Float64(fma(Float64(y * y), fma(Float64(y * y), t_0, 0.16666666666666666), 1.0) * Float64(x * fma(x, Float64(x * fma(x, Float64(x * 0.008333333333333333), -0.16666666666666666)), 1.0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(y * 0.0001984126984126984), $MachinePrecision] + 0.008333333333333333), $MachinePrecision]}, If[LessEqual[N[Sin[x], $MachinePrecision], 8e-54], N[(N[(x * N[(x * N[(x * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(y * N[(y * N[(t$95$0 * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision] + N[(y * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * t$95$0 + 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x * N[(x * N[(x * N[(x * N[(x * 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 x) < 8.0000000000000002e-54

   1. Initial program 100.0%

      \[\sin x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)} \cdot \frac{\sinh y}{y} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)} \cdot \frac{\sinh y}{y} \]

      2. +-commutative N/A

         \[\leadsto \left(x \cdot \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2} + 1\right)}\right) \cdot \frac{\sinh y}{y} \]

      3. *-commutative N/A

         \[\leadsto \left(x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{-1}{6}} + 1\right)\right) \cdot \frac{\sinh y}{y} \]

      4. unpow2 N/A

         \[\leadsto \left(x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{-1}{6} + 1\right)\right) \cdot \frac{\sinh y}{y} \]

      5. associate-*l* N/A

         \[\leadsto \left(x \cdot \left(\color{blue}{x \cdot \left(x \cdot \frac{-1}{6}\right)} + 1\right)\right) \cdot \frac{\sinh y}{y} \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(x \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)}\right) \cdot \frac{\sinh y}{y} \]

      7. *-lowering-*.f64 70.8

         \[\leadsto \left(x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot -0.16666666666666666}, 1\right)\right) \cdot \frac{\sinh y}{y} \]

   5. Simplified 70.8%

      \[\leadsto \color{blue}{\left(x \cdot \mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right)\right)} \cdot \frac{\sinh y}{y} \]

   6. Taylor expanded in y around 0

      \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)\right) \cdot \frac{\color{blue}{y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)}}{y} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)\right) \cdot \frac{\color{blue}{y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)}}{y} \]

      2. +-commutative N/A

         \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)\right) \cdot \frac{y \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) + 1\right)}}{y} \]

      3. unpow2 N/A

         \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)\right) \cdot \frac{y \cdot \left(\color{blue}{\left(y \cdot y\right)} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) + 1\right)}{y} \]

      4. associate-*l* N/A

         \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)\right) \cdot \frac{y \cdot \left(\color{blue}{y \cdot \left(y \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)} + 1\right)}{y} \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)\right) \cdot \frac{y \cdot \color{blue}{\mathsf{fma}\left(y, y \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right), 1\right)}}{y} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)\right) \cdot \frac{y \cdot \mathsf{fma}\left(y, \color{blue}{y \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)}, 1\right)}{y} \]

      7. +-commutative N/A

         \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)\right) \cdot \frac{y \cdot \mathsf{fma}\left(y, y \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right) + \frac{1}{6}\right)}, 1\right)}{y} \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)\right) \cdot \frac{y \cdot \mathsf{fma}\left(y, y \cdot \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}, \frac{1}{6}\right)}, 1\right)}{y} \]

      9. unpow2 N/A

         \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)\right) \cdot \frac{y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}, \frac{1}{6}\right), 1\right)}{y} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)\right) \cdot \frac{y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}, \frac{1}{6}\right), 1\right)}{y} \]

      11. +-commutative N/A

          \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)\right) \cdot \frac{y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{\frac{1}{5040} \cdot {y}^{2} + \frac{1}{120}}, \frac{1}{6}\right), 1\right)}{y} \]

      12. *-commutative N/A

          \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)\right) \cdot \frac{y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{{y}^{2} \cdot \frac{1}{5040}} + \frac{1}{120}, \frac{1}{6}\right), 1\right)}{y} \]

      13. accelerator-lowering-fma.f64 N/A

          \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)\right) \cdot \frac{y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{1}{5040}, \frac{1}{120}\right)}, \frac{1}{6}\right), 1\right)}{y} \]

      14. unpow2 N/A

          \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)\right) \cdot \frac{y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right)}{y} \]

      15. *-lowering-*.f64 67.0

          \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right)\right) \cdot \frac{y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)}{y} \]

   8. Simplified 67.0%

      \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right)\right) \cdot \frac{\color{blue}{y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)}}{y} \]
   9. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)\right) \cdot \frac{y \cdot \mathsf{fma}\left(y, \color{blue}{\left(\left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{5040} + \frac{1}{120}\right)\right) \cdot y + \frac{1}{6} \cdot y}, 1\right)}{y} \]

      2. *-commutative N/A

         \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)\right) \cdot \frac{y \cdot \mathsf{fma}\left(y, \color{blue}{\left(\left(\left(y \cdot y\right) \cdot \frac{1}{5040} + \frac{1}{120}\right) \cdot \left(y \cdot y\right)\right)} \cdot y + \frac{1}{6} \cdot y, 1\right)}{y} \]

      3. associate-*l* N/A

         \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)\right) \cdot \frac{y \cdot \mathsf{fma}\left(y, \color{blue}{\left(\left(y \cdot y\right) \cdot \frac{1}{5040} + \frac{1}{120}\right) \cdot \left(\left(y \cdot y\right) \cdot y\right)} + \frac{1}{6} \cdot y, 1\right)}{y} \]

      4. pow3 N/A

         \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)\right) \cdot \frac{y \cdot \mathsf{fma}\left(y, \left(\left(y \cdot y\right) \cdot \frac{1}{5040} + \frac{1}{120}\right) \cdot \color{blue}{{y}^{3}} + \frac{1}{6} \cdot y, 1\right)}{y} \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)\right) \cdot \frac{y \cdot \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(\left(y \cdot y\right) \cdot \frac{1}{5040} + \frac{1}{120}, {y}^{3}, \frac{1}{6} \cdot y\right)}, 1\right)}{y} \]

      6. associate-*l* N/A

         \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)\right) \cdot \frac{y \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(\color{blue}{y \cdot \left(y \cdot \frac{1}{5040}\right)} + \frac{1}{120}, {y}^{3}, \frac{1}{6} \cdot y\right), 1\right)}{y} \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)\right) \cdot \frac{y \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(y, y \cdot \frac{1}{5040}, \frac{1}{120}\right)}, {y}^{3}, \frac{1}{6} \cdot y\right), 1\right)}{y} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)\right) \cdot \frac{y \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(\mathsf{fma}\left(y, \color{blue}{y \cdot \frac{1}{5040}}, \frac{1}{120}\right), {y}^{3}, \frac{1}{6} \cdot y\right), 1\right)}{y} \]

      9. cube-unmult N/A

         \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)\right) \cdot \frac{y \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \frac{1}{5040}, \frac{1}{120}\right), \color{blue}{y \cdot \left(y \cdot y\right)}, \frac{1}{6} \cdot y\right), 1\right)}{y} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)\right) \cdot \frac{y \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \frac{1}{5040}, \frac{1}{120}\right), \color{blue}{y \cdot \left(y \cdot y\right)}, \frac{1}{6} \cdot y\right), 1\right)}{y} \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)\right) \cdot \frac{y \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \frac{1}{5040}, \frac{1}{120}\right), y \cdot \color{blue}{\left(y \cdot y\right)}, \frac{1}{6} \cdot y\right), 1\right)}{y} \]

      12. *-commutative N/A

          \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)\right) \cdot \frac{y \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \frac{1}{5040}, \frac{1}{120}\right), y \cdot \left(y \cdot y\right), \color{blue}{y \cdot \frac{1}{6}}\right), 1\right)}{y} \]

      13. *-lowering-*.f64 67.0

          \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right)\right) \cdot \frac{y \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right), y \cdot \left(y \cdot y\right), \color{blue}{y \cdot 0.16666666666666666}\right), 1\right)}{y} \]

   10. Applied egg-rr 67.0%

       \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right)\right) \cdot \frac{y \cdot \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right), y \cdot \left(y \cdot y\right), y \cdot 0.16666666666666666\right)}, 1\right)}{y} \]

   if 8.0000000000000002e-54 < (sin.f64 x)

   1. Initial program 100.0%

      \[\sin x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \sin x \cdot \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \sin x \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) + 1\right)} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \sin x \cdot \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right), 1\right)} \]

      3. unpow2 N/A

         \[\leadsto \sin x \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right), 1\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \sin x \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right), 1\right) \]

      5. +-commutative N/A

         \[\leadsto \sin x \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{{y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right) + \frac{1}{6}}, 1\right) \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \sin x \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}, \frac{1}{6}\right)}, 1\right) \]

      7. unpow2 N/A

         \[\leadsto \sin x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}, \frac{1}{6}\right), 1\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \sin x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}, \frac{1}{6}\right), 1\right) \]

      9. +-commutative N/A

         \[\leadsto \sin x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \color{blue}{\frac{1}{5040} \cdot {y}^{2} + \frac{1}{120}}, \frac{1}{6}\right), 1\right) \]

      10. unpow2 N/A

          \[\leadsto \sin x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \frac{1}{5040} \cdot \color{blue}{\left(y \cdot y\right)} + \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      11. associate-*r* N/A

          \[\leadsto \sin x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \color{blue}{\left(\frac{1}{5040} \cdot y\right) \cdot y} + \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      12. *-commutative N/A

          \[\leadsto \sin x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \color{blue}{y \cdot \left(\frac{1}{5040} \cdot y\right)} + \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      13. accelerator-lowering-fma.f64 N/A

          \[\leadsto \sin x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \color{blue}{\mathsf{fma}\left(y, \frac{1}{5040} \cdot y, \frac{1}{120}\right)}, \frac{1}{6}\right), 1\right) \]

      14. *-commutative N/A

          \[\leadsto \sin x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, \color{blue}{y \cdot \frac{1}{5040}}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      15. *-lowering-*.f64 95.4

          \[\leadsto \sin x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, \color{blue}{y \cdot 0.0001984126984126984}, 0.008333333333333333\right), 0.16666666666666666\right), 1\right) \]

   5. Simplified 95.4%

      \[\leadsto \sin x \cdot \color{blue}{\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right)\right)} \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right)\right)} \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      2. +-commutative N/A

         \[\leadsto \left(x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) + 1\right)}\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      3. unpow2 N/A

         \[\leadsto \left(x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) + 1\right)\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      4. associate-*l* N/A

         \[\leadsto \left(x \cdot \left(\color{blue}{x \cdot \left(x \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right)} + 1\right)\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(x \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right), 1\right)}\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \left(x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)}, 1\right)\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      7. sub-neg N/A

         \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot \color{blue}{\left(\frac{1}{120} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)}, 1\right)\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      8. *-commutative N/A

         \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{1}{120}} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right), 1\right)\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      9. unpow2 N/A

         \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{120} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right), 1\right)\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      10. associate-*l* N/A

          \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot \left(\color{blue}{x \cdot \left(x \cdot \frac{1}{120}\right)} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right), 1\right)\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      11. *-commutative N/A

          \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot \left(x \cdot \color{blue}{\left(\frac{1}{120} \cdot x\right)} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right), 1\right)\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      12. metadata-eval N/A

          \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot \left(x \cdot \left(\frac{1}{120} \cdot x\right) + \color{blue}{\frac{-1}{6}}\right), 1\right)\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      13. accelerator-lowering-fma.f64 N/A

          \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot \color{blue}{\mathsf{fma}\left(x, \frac{1}{120} \cdot x, \frac{-1}{6}\right)}, 1\right)\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      14. *-commutative N/A

          \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{120}}, \frac{-1}{6}\right), 1\right)\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      15. *-lowering-*.f64 47.3

          \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot 0.008333333333333333}, -0.16666666666666666\right), 1\right)\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right) \]

   8. Simplified 47.3%

      \[\leadsto \color{blue}{\left(x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.008333333333333333, -0.16666666666666666\right), 1\right)\right)} \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin x \leq 8 \cdot 10^{-54}:\\ \;\;\;\;\left(x \cdot \mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right)\right) \cdot \frac{y \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right), y \cdot \left(y \cdot y\right), y \cdot 0.16666666666666666\right), 1\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right) \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.008333333333333333, -0.16666666666666666\right), 1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 58.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin x \leq 8 \cdot 10^{-54}:\\ \;\;\;\;\left(x \cdot \mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right)\right) \cdot \frac{y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right) \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.008333333333333333, -0.16666666666666666\right), 1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sin x) 8e-54)
   (*
    (* x (fma x (* x -0.16666666666666666) 1.0))
    (/
     (*
      y
      (fma
       y
       (*
        y
        (fma
         y
         (* y (fma (* y y) 0.0001984126984126984 0.008333333333333333))
         0.16666666666666666))
       1.0))
     y))
   (*
    (fma
     (* y y)
     (fma
      (* y y)
      (fma y (* y 0.0001984126984126984) 0.008333333333333333)
      0.16666666666666666)
     1.0)
    (*
     x
     (fma
      x
      (* x (fma x (* x 0.008333333333333333) -0.16666666666666666))
      1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (sin(x) <= 8e-54) {
		tmp = (x * fma(x, (x * -0.16666666666666666), 1.0)) * ((y * fma(y, (y * fma(y, (y * fma((y * y), 0.0001984126984126984, 0.008333333333333333)), 0.16666666666666666)), 1.0)) / y);
	} else {
		tmp = fma((y * y), fma((y * y), fma(y, (y * 0.0001984126984126984), 0.008333333333333333), 0.16666666666666666), 1.0) * (x * fma(x, (x * fma(x, (x * 0.008333333333333333), -0.16666666666666666)), 1.0));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (sin(x) <= 8e-54)
		tmp = Float64(Float64(x * fma(x, Float64(x * -0.16666666666666666), 1.0)) * Float64(Float64(y * fma(y, Float64(y * fma(y, Float64(y * fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333)), 0.16666666666666666)), 1.0)) / y));
	else
		tmp = Float64(fma(Float64(y * y), fma(Float64(y * y), fma(y, Float64(y * 0.0001984126984126984), 0.008333333333333333), 0.16666666666666666), 1.0) * Float64(x * fma(x, Float64(x * fma(x, Float64(x * 0.008333333333333333), -0.16666666666666666)), 1.0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[Sin[x], $MachinePrecision], 8e-54], N[(N[(x * N[(x * N[(x * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(y * N[(y * N[(y * N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * 0.0001984126984126984), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x * N[(x * N[(x * N[(x * N[(x * 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 x) < 8.0000000000000002e-54

   1. Initial program 100.0%

      \[\sin x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)} \cdot \frac{\sinh y}{y} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)} \cdot \frac{\sinh y}{y} \]

      2. +-commutative N/A

         \[\leadsto \left(x \cdot \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2} + 1\right)}\right) \cdot \frac{\sinh y}{y} \]

      3. *-commutative N/A

         \[\leadsto \left(x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{-1}{6}} + 1\right)\right) \cdot \frac{\sinh y}{y} \]

      4. unpow2 N/A

         \[\leadsto \left(x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{-1}{6} + 1\right)\right) \cdot \frac{\sinh y}{y} \]

      5. associate-*l* N/A

         \[\leadsto \left(x \cdot \left(\color{blue}{x \cdot \left(x \cdot \frac{-1}{6}\right)} + 1\right)\right) \cdot \frac{\sinh y}{y} \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(x \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)}\right) \cdot \frac{\sinh y}{y} \]

      7. *-lowering-*.f64 70.8

         \[\leadsto \left(x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot -0.16666666666666666}, 1\right)\right) \cdot \frac{\sinh y}{y} \]

   5. Simplified 70.8%

      \[\leadsto \color{blue}{\left(x \cdot \mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right)\right)} \cdot \frac{\sinh y}{y} \]

   6. Taylor expanded in y around 0

      \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)\right) \cdot \frac{\color{blue}{y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)}}{y} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)\right) \cdot \frac{\color{blue}{y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)}}{y} \]

      2. +-commutative N/A

         \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)\right) \cdot \frac{y \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) + 1\right)}}{y} \]

      3. unpow2 N/A

         \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)\right) \cdot \frac{y \cdot \left(\color{blue}{\left(y \cdot y\right)} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) + 1\right)}{y} \]

      4. associate-*l* N/A

         \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)\right) \cdot \frac{y \cdot \left(\color{blue}{y \cdot \left(y \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)} + 1\right)}{y} \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)\right) \cdot \frac{y \cdot \color{blue}{\mathsf{fma}\left(y, y \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right), 1\right)}}{y} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)\right) \cdot \frac{y \cdot \mathsf{fma}\left(y, \color{blue}{y \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)}, 1\right)}{y} \]

      7. +-commutative N/A

         \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)\right) \cdot \frac{y \cdot \mathsf{fma}\left(y, y \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right) + \frac{1}{6}\right)}, 1\right)}{y} \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)\right) \cdot \frac{y \cdot \mathsf{fma}\left(y, y \cdot \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}, \frac{1}{6}\right)}, 1\right)}{y} \]

      9. unpow2 N/A

         \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)\right) \cdot \frac{y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}, \frac{1}{6}\right), 1\right)}{y} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)\right) \cdot \frac{y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}, \frac{1}{6}\right), 1\right)}{y} \]

      11. +-commutative N/A

          \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)\right) \cdot \frac{y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{\frac{1}{5040} \cdot {y}^{2} + \frac{1}{120}}, \frac{1}{6}\right), 1\right)}{y} \]

      12. *-commutative N/A

          \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)\right) \cdot \frac{y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{{y}^{2} \cdot \frac{1}{5040}} + \frac{1}{120}, \frac{1}{6}\right), 1\right)}{y} \]

      13. accelerator-lowering-fma.f64 N/A

          \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)\right) \cdot \frac{y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{1}{5040}, \frac{1}{120}\right)}, \frac{1}{6}\right), 1\right)}{y} \]

      14. unpow2 N/A

          \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)\right) \cdot \frac{y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right)}{y} \]

      15. *-lowering-*.f64 67.0

          \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right)\right) \cdot \frac{y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)}{y} \]

   8. Simplified 67.0%

      \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right)\right) \cdot \frac{\color{blue}{y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)}}{y} \]

   9. Taylor expanded in y around 0

      \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)\right) \cdot \frac{y \cdot \mathsf{fma}\left(y, \color{blue}{y \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)}, 1\right)}{y} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)\right) \cdot \frac{y \cdot \mathsf{fma}\left(y, \color{blue}{y \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)}, 1\right)}{y} \]

       2. +-commutative N/A

          \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)\right) \cdot \frac{y \cdot \mathsf{fma}\left(y, y \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right) + \frac{1}{6}\right)}, 1\right)}{y} \]

       3. unpow2 N/A

          \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)\right) \cdot \frac{y \cdot \mathsf{fma}\left(y, y \cdot \left(\color{blue}{\left(y \cdot y\right)} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right) + \frac{1}{6}\right), 1\right)}{y} \]

       4. associate-*l* N/A

          \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)\right) \cdot \frac{y \cdot \mathsf{fma}\left(y, y \cdot \left(\color{blue}{y \cdot \left(y \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)} + \frac{1}{6}\right), 1\right)}{y} \]

       5. accelerator-lowering-fma.f64 N/A

          \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)\right) \cdot \frac{y \cdot \mathsf{fma}\left(y, y \cdot \color{blue}{\mathsf{fma}\left(y, y \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right), \frac{1}{6}\right)}, 1\right)}{y} \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)\right) \cdot \frac{y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, \color{blue}{y \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)}, \frac{1}{6}\right), 1\right)}{y} \]

       7. +-commutative N/A

          \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)\right) \cdot \frac{y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \color{blue}{\left(\frac{1}{5040} \cdot {y}^{2} + \frac{1}{120}\right)}, \frac{1}{6}\right), 1\right)}{y} \]

       8. *-commutative N/A

          \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)\right) \cdot \frac{y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \left(\color{blue}{{y}^{2} \cdot \frac{1}{5040}} + \frac{1}{120}\right), \frac{1}{6}\right), 1\right)}{y} \]

       9. accelerator-lowering-fma.f64 N/A

          \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)\right) \cdot \frac{y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{1}{5040}, \frac{1}{120}\right)}, \frac{1}{6}\right), 1\right)}{y} \]

       10. unpow2 N/A

           \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)\right) \cdot \frac{y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right)}{y} \]

       11. *-lowering-*.f64 67.0

           \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right)\right) \cdot \frac{y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)}{y} \]

   11. Simplified 67.0%

       \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right)\right) \cdot \frac{y \cdot \mathsf{fma}\left(y, \color{blue}{y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right)}, 1\right)}{y} \]

   if 8.0000000000000002e-54 < (sin.f64 x)

   1. Initial program 100.0%

      \[\sin x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \sin x \cdot \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \sin x \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) + 1\right)} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \sin x \cdot \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right), 1\right)} \]

      3. unpow2 N/A

         \[\leadsto \sin x \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right), 1\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \sin x \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right), 1\right) \]

      5. +-commutative N/A

         \[\leadsto \sin x \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{{y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right) + \frac{1}{6}}, 1\right) \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \sin x \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}, \frac{1}{6}\right)}, 1\right) \]

      7. unpow2 N/A

         \[\leadsto \sin x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}, \frac{1}{6}\right), 1\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \sin x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}, \frac{1}{6}\right), 1\right) \]

      9. +-commutative N/A

         \[\leadsto \sin x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \color{blue}{\frac{1}{5040} \cdot {y}^{2} + \frac{1}{120}}, \frac{1}{6}\right), 1\right) \]

      10. unpow2 N/A

          \[\leadsto \sin x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \frac{1}{5040} \cdot \color{blue}{\left(y \cdot y\right)} + \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      11. associate-*r* N/A

          \[\leadsto \sin x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \color{blue}{\left(\frac{1}{5040} \cdot y\right) \cdot y} + \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      12. *-commutative N/A

          \[\leadsto \sin x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \color{blue}{y \cdot \left(\frac{1}{5040} \cdot y\right)} + \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      13. accelerator-lowering-fma.f64 N/A

          \[\leadsto \sin x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \color{blue}{\mathsf{fma}\left(y, \frac{1}{5040} \cdot y, \frac{1}{120}\right)}, \frac{1}{6}\right), 1\right) \]

      14. *-commutative N/A

          \[\leadsto \sin x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, \color{blue}{y \cdot \frac{1}{5040}}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      15. *-lowering-*.f64 95.4

          \[\leadsto \sin x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, \color{blue}{y \cdot 0.0001984126984126984}, 0.008333333333333333\right), 0.16666666666666666\right), 1\right) \]

   5. Simplified 95.4%

      \[\leadsto \sin x \cdot \color{blue}{\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right)\right)} \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right)\right)} \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      2. +-commutative N/A

         \[\leadsto \left(x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) + 1\right)}\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      3. unpow2 N/A

         \[\leadsto \left(x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) + 1\right)\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      4. associate-*l* N/A

         \[\leadsto \left(x \cdot \left(\color{blue}{x \cdot \left(x \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right)} + 1\right)\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(x \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right), 1\right)}\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \left(x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)}, 1\right)\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      7. sub-neg N/A

         \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot \color{blue}{\left(\frac{1}{120} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)}, 1\right)\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      8. *-commutative N/A

         \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{1}{120}} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right), 1\right)\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      9. unpow2 N/A

         \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{120} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right), 1\right)\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      10. associate-*l* N/A

          \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot \left(\color{blue}{x \cdot \left(x \cdot \frac{1}{120}\right)} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right), 1\right)\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      11. *-commutative N/A

          \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot \left(x \cdot \color{blue}{\left(\frac{1}{120} \cdot x\right)} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right), 1\right)\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      12. metadata-eval N/A

          \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot \left(x \cdot \left(\frac{1}{120} \cdot x\right) + \color{blue}{\frac{-1}{6}}\right), 1\right)\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      13. accelerator-lowering-fma.f64 N/A

          \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot \color{blue}{\mathsf{fma}\left(x, \frac{1}{120} \cdot x, \frac{-1}{6}\right)}, 1\right)\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      14. *-commutative N/A

          \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{120}}, \frac{-1}{6}\right), 1\right)\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      15. *-lowering-*.f64 47.3

          \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot 0.008333333333333333}, -0.16666666666666666\right), 1\right)\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right) \]

   8. Simplified 47.3%

      \[\leadsto \color{blue}{\left(x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.008333333333333333, -0.16666666666666666\right), 1\right)\right)} \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin x \leq 8 \cdot 10^{-54}:\\ \;\;\;\;\left(x \cdot \mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right)\right) \cdot \frac{y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right) \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.008333333333333333, -0.16666666666666666\right), 1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 55.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin x \leq -0.02:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, -0.001388888888888889, -0.027777777777777776\right), -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sin x) -0.02)
   (*
    (* x x)
    (*
     x
     (fma
      (* y y)
      (fma (* y y) -0.001388888888888889 -0.027777777777777776)
      -0.16666666666666666)))
   (*
    x
    (fma (* y y) (fma (* y y) 0.008333333333333333 0.16666666666666666) 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (sin(x) <= -0.02) {
		tmp = (x * x) * (x * fma((y * y), fma((y * y), -0.001388888888888889, -0.027777777777777776), -0.16666666666666666));
	} else {
		tmp = x * fma((y * y), fma((y * y), 0.008333333333333333, 0.16666666666666666), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (sin(x) <= -0.02)
		tmp = Float64(Float64(x * x) * Float64(x * fma(Float64(y * y), fma(Float64(y * y), -0.001388888888888889, -0.027777777777777776), -0.16666666666666666)));
	else
		tmp = Float64(x * fma(Float64(y * y), fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[Sin[x], $MachinePrecision], -0.02], N[(N[(x * x), $MachinePrecision] * N[(x * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * -0.001388888888888889 + -0.027777777777777776), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 x) < -0.0200000000000000004

   1. Initial program 100.0%

      \[\sin x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)} \cdot \frac{\sinh y}{y} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)} \cdot \frac{\sinh y}{y} \]

      2. +-commutative N/A

         \[\leadsto \left(x \cdot \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2} + 1\right)}\right) \cdot \frac{\sinh y}{y} \]

      3. *-commutative N/A

         \[\leadsto \left(x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{-1}{6}} + 1\right)\right) \cdot \frac{\sinh y}{y} \]

      4. unpow2 N/A

         \[\leadsto \left(x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{-1}{6} + 1\right)\right) \cdot \frac{\sinh y}{y} \]

      5. associate-*l* N/A

         \[\leadsto \left(x \cdot \left(\color{blue}{x \cdot \left(x \cdot \frac{-1}{6}\right)} + 1\right)\right) \cdot \frac{\sinh y}{y} \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(x \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)}\right) \cdot \frac{\sinh y}{y} \]

      7. *-lowering-*.f64 22.5

         \[\leadsto \left(x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot -0.16666666666666666}, 1\right)\right) \cdot \frac{\sinh y}{y} \]

   5. Simplified 22.5%

      \[\leadsto \color{blue}{\left(x \cdot \mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right)\right)} \cdot \frac{\sinh y}{y} \]

   6. Taylor expanded in y around 0

      \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)\right) \cdot \frac{\color{blue}{y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)}}{y} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)\right) \cdot \frac{\color{blue}{y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)}}{y} \]

      2. +-commutative N/A

         \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)\right) \cdot \frac{y \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) + 1\right)}}{y} \]

      3. unpow2 N/A

         \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)\right) \cdot \frac{y \cdot \left(\color{blue}{\left(y \cdot y\right)} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) + 1\right)}{y} \]

      4. associate-*l* N/A

         \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)\right) \cdot \frac{y \cdot \left(\color{blue}{y \cdot \left(y \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)} + 1\right)}{y} \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)\right) \cdot \frac{y \cdot \color{blue}{\mathsf{fma}\left(y, y \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right), 1\right)}}{y} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)\right) \cdot \frac{y \cdot \mathsf{fma}\left(y, \color{blue}{y \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)}, 1\right)}{y} \]

      7. +-commutative N/A

         \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)\right) \cdot \frac{y \cdot \mathsf{fma}\left(y, y \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right) + \frac{1}{6}\right)}, 1\right)}{y} \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)\right) \cdot \frac{y \cdot \mathsf{fma}\left(y, y \cdot \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}, \frac{1}{6}\right)}, 1\right)}{y} \]

      9. unpow2 N/A

         \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)\right) \cdot \frac{y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}, \frac{1}{6}\right), 1\right)}{y} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)\right) \cdot \frac{y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}, \frac{1}{6}\right), 1\right)}{y} \]

      11. +-commutative N/A

          \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)\right) \cdot \frac{y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{\frac{1}{5040} \cdot {y}^{2} + \frac{1}{120}}, \frac{1}{6}\right), 1\right)}{y} \]

      12. *-commutative N/A

          \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)\right) \cdot \frac{y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{{y}^{2} \cdot \frac{1}{5040}} + \frac{1}{120}, \frac{1}{6}\right), 1\right)}{y} \]

      13. accelerator-lowering-fma.f64 N/A

          \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)\right) \cdot \frac{y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{1}{5040}, \frac{1}{120}\right)}, \frac{1}{6}\right), 1\right)}{y} \]

      14. unpow2 N/A

          \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)\right) \cdot \frac{y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right)}{y} \]

      15. *-lowering-*.f64 22.5

          \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right)\right) \cdot \frac{y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)}{y} \]

   8. Simplified 22.5%

      \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right)\right) \cdot \frac{\color{blue}{y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)}}{y} \]

   9. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right) + {y}^{2} \cdot \left(\frac{1}{120} \cdot \left(x \cdot \left({y}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right) + \frac{1}{6} \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right)} \]
   10. Step-by-step derivation

       1. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(x, 1 + \frac{-1}{6} \cdot {x}^{2}, {y}^{2} \cdot \left(\frac{1}{120} \cdot \left(x \cdot \left({y}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right) + \frac{1}{6} \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right)\right)} \]

       2. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{-1}{6} \cdot {x}^{2} + 1}, {y}^{2} \cdot \left(\frac{1}{120} \cdot \left(x \cdot \left({y}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right) + \frac{1}{6} \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right)\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x, \color{blue}{{x}^{2} \cdot \frac{-1}{6}} + 1, {y}^{2} \cdot \left(\frac{1}{120} \cdot \left(x \cdot \left({y}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right) + \frac{1}{6} \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right)\right) \]

       4. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{-1}{6}, 1\right)}, {y}^{2} \cdot \left(\frac{1}{120} \cdot \left(x \cdot \left({y}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right) + \frac{1}{6} \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{6}, 1\right), {y}^{2} \cdot \left(\frac{1}{120} \cdot \left(x \cdot \left({y}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right) + \frac{1}{6} \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{6}, 1\right), {y}^{2} \cdot \left(\frac{1}{120} \cdot \left(x \cdot \left({y}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right) + \frac{1}{6} \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot x, \frac{-1}{6}, 1\right), \color{blue}{{y}^{2} \cdot \left(\frac{1}{120} \cdot \left(x \cdot \left({y}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right) + \frac{1}{6} \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right)}\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot x, \frac{-1}{6}, 1\right), \color{blue}{\left(y \cdot y\right)} \cdot \left(\frac{1}{120} \cdot \left(x \cdot \left({y}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right) + \frac{1}{6} \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot x, \frac{-1}{6}, 1\right), \color{blue}{\left(y \cdot y\right)} \cdot \left(\frac{1}{120} \cdot \left(x \cdot \left({y}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right) + \frac{1}{6} \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right)\right) \]

       10. +-commutative N/A

           \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot x, \frac{-1}{6}, 1\right), \left(y \cdot y\right) \cdot \color{blue}{\left(\frac{1}{6} \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right) + \frac{1}{120} \cdot \left(x \cdot \left({y}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right)\right)}\right) \]

       11. *-commutative N/A

           \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot x, \frac{-1}{6}, 1\right), \left(y \cdot y\right) \cdot \left(\color{blue}{\left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right) \cdot \frac{1}{6}} + \frac{1}{120} \cdot \left(x \cdot \left({y}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right)\right)\right) \]

       12. *-commutative N/A

           \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot x, \frac{-1}{6}, 1\right), \left(y \cdot y\right) \cdot \left(\left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right) \cdot \frac{1}{6} + \frac{1}{120} \cdot \left(x \cdot \color{blue}{\left(\left(1 + \frac{-1}{6} \cdot {x}^{2}\right) \cdot {y}^{2}\right)}\right)\right)\right) \]

       13. associate-*r* N/A

           \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot x, \frac{-1}{6}, 1\right), \left(y \cdot y\right) \cdot \left(\left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right) \cdot \frac{1}{6} + \frac{1}{120} \cdot \color{blue}{\left(\left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right) \cdot {y}^{2}\right)}\right)\right) \]

   11. Simplified 21.0%

       \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right), \left(y \cdot y\right) \cdot \left(\left(x \cdot \mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right)\right) \cdot \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right)\right)\right)} \]

   12. Taylor expanded in x around inf

       \[\leadsto \color{blue}{{x}^{3} \cdot \left(\frac{-1}{6} \cdot \left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) - \frac{1}{6}\right)} \]
   13. Step-by-step derivation

       1. unpow3 N/A

          \[\leadsto \color{blue}{\left(\left(x \cdot x\right) \cdot x\right)} \cdot \left(\frac{-1}{6} \cdot \left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) - \frac{1}{6}\right) \]

       2. unpow2 N/A

          \[\leadsto \left(\color{blue}{{x}^{2}} \cdot x\right) \cdot \left(\frac{-1}{6} \cdot \left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) - \frac{1}{6}\right) \]

       3. associate-*l* N/A

          \[\leadsto \color{blue}{{x}^{2} \cdot \left(x \cdot \left(\frac{-1}{6} \cdot \left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) - \frac{1}{6}\right)\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{{x}^{2} \cdot \left(x \cdot \left(\frac{-1}{6} \cdot \left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) - \frac{1}{6}\right)\right)} \]

       5. unpow2 N/A

          \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot \left(x \cdot \left(\frac{-1}{6} \cdot \left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) - \frac{1}{6}\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot \left(x \cdot \left(\frac{-1}{6} \cdot \left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) - \frac{1}{6}\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \left(x \cdot x\right) \cdot \color{blue}{\left(x \cdot \left(\frac{-1}{6} \cdot \left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) - \frac{1}{6}\right)\right)} \]

       8. sub-neg N/A

          \[\leadsto \left(x \cdot x\right) \cdot \left(x \cdot \color{blue}{\left(\frac{-1}{6} \cdot \left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)}\right) \]

       9. *-commutative N/A

          \[\leadsto \left(x \cdot x\right) \cdot \left(x \cdot \left(\color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) \cdot \frac{-1}{6}} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)\right) \]

       10. associate-*l* N/A

           \[\leadsto \left(x \cdot x\right) \cdot \left(x \cdot \left(\color{blue}{{y}^{2} \cdot \left(\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) \cdot \frac{-1}{6}\right)} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)\right) \]

       11. *-commutative N/A

           \[\leadsto \left(x \cdot x\right) \cdot \left(x \cdot \left({y}^{2} \cdot \color{blue}{\left(\frac{-1}{6} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)\right) \]

       12. metadata-eval N/A

           \[\leadsto \left(x \cdot x\right) \cdot \left(x \cdot \left({y}^{2} \cdot \left(\frac{-1}{6} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) + \color{blue}{\frac{-1}{6}}\right)\right) \]

       13. accelerator-lowering-fma.f64 N/A

           \[\leadsto \left(x \cdot x\right) \cdot \left(x \cdot \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{-1}{6} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right), \frac{-1}{6}\right)}\right) \]

   14. Simplified 21.2%

       \[\leadsto \color{blue}{\left(x \cdot x\right) \cdot \left(x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, -0.001388888888888889, -0.027777777777777776\right), -0.16666666666666666\right)\right)} \]

   if -0.0200000000000000004 < (sin.f64 x)

   1. Initial program 100.0%

      \[\sin x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)} \cdot \frac{\sinh y}{y} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)} \cdot \frac{\sinh y}{y} \]

      2. +-commutative N/A

         \[\leadsto \left(x \cdot \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2} + 1\right)}\right) \cdot \frac{\sinh y}{y} \]

      3. *-commutative N/A

         \[\leadsto \left(x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{-1}{6}} + 1\right)\right) \cdot \frac{\sinh y}{y} \]

      4. unpow2 N/A

         \[\leadsto \left(x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{-1}{6} + 1\right)\right) \cdot \frac{\sinh y}{y} \]

      5. associate-*l* N/A

         \[\leadsto \left(x \cdot \left(\color{blue}{x \cdot \left(x \cdot \frac{-1}{6}\right)} + 1\right)\right) \cdot \frac{\sinh y}{y} \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(x \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)}\right) \cdot \frac{\sinh y}{y} \]

      7. *-lowering-*.f64 76.2

         \[\leadsto \left(x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot -0.16666666666666666}, 1\right)\right) \cdot \frac{\sinh y}{y} \]

   5. Simplified 76.2%

      \[\leadsto \color{blue}{\left(x \cdot \mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right)\right)} \cdot \frac{\sinh y}{y} \]

   6. Taylor expanded in y around 0

      \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)\right) \cdot \frac{\color{blue}{y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)}}{y} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)\right) \cdot \frac{\color{blue}{y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)}}{y} \]

      2. +-commutative N/A

         \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)\right) \cdot \frac{y \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) + 1\right)}}{y} \]

      3. unpow2 N/A

         \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)\right) \cdot \frac{y \cdot \left(\color{blue}{\left(y \cdot y\right)} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) + 1\right)}{y} \]

      4. associate-*l* N/A

         \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)\right) \cdot \frac{y \cdot \left(\color{blue}{y \cdot \left(y \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)} + 1\right)}{y} \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)\right) \cdot \frac{y \cdot \color{blue}{\mathsf{fma}\left(y, y \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right), 1\right)}}{y} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)\right) \cdot \frac{y \cdot \mathsf{fma}\left(y, \color{blue}{y \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)}, 1\right)}{y} \]

      7. +-commutative N/A

         \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)\right) \cdot \frac{y \cdot \mathsf{fma}\left(y, y \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right) + \frac{1}{6}\right)}, 1\right)}{y} \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)\right) \cdot \frac{y \cdot \mathsf{fma}\left(y, y \cdot \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}, \frac{1}{6}\right)}, 1\right)}{y} \]

      9. unpow2 N/A

         \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)\right) \cdot \frac{y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}, \frac{1}{6}\right), 1\right)}{y} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)\right) \cdot \frac{y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}, \frac{1}{6}\right), 1\right)}{y} \]

      11. +-commutative N/A

          \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)\right) \cdot \frac{y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{\frac{1}{5040} \cdot {y}^{2} + \frac{1}{120}}, \frac{1}{6}\right), 1\right)}{y} \]

      12. *-commutative N/A

          \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)\right) \cdot \frac{y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{{y}^{2} \cdot \frac{1}{5040}} + \frac{1}{120}, \frac{1}{6}\right), 1\right)}{y} \]

      13. accelerator-lowering-fma.f64 N/A

          \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)\right) \cdot \frac{y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{1}{5040}, \frac{1}{120}\right)}, \frac{1}{6}\right), 1\right)}{y} \]

      14. unpow2 N/A

          \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)\right) \cdot \frac{y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right)}{y} \]

      15. *-lowering-*.f64 71.8

          \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right)\right) \cdot \frac{y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)}{y} \]

   8. Simplified 71.8%

      \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right)\right) \cdot \frac{\color{blue}{y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)}}{y} \]

   9. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right) + {y}^{2} \cdot \left(\frac{1}{120} \cdot \left(x \cdot \left({y}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right) + \frac{1}{6} \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right)} \]
   10. Step-by-step derivation

       1. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(x, 1 + \frac{-1}{6} \cdot {x}^{2}, {y}^{2} \cdot \left(\frac{1}{120} \cdot \left(x \cdot \left({y}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right) + \frac{1}{6} \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right)\right)} \]

       2. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{-1}{6} \cdot {x}^{2} + 1}, {y}^{2} \cdot \left(\frac{1}{120} \cdot \left(x \cdot \left({y}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right) + \frac{1}{6} \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right)\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x, \color{blue}{{x}^{2} \cdot \frac{-1}{6}} + 1, {y}^{2} \cdot \left(\frac{1}{120} \cdot \left(x \cdot \left({y}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right) + \frac{1}{6} \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right)\right) \]

       4. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{-1}{6}, 1\right)}, {y}^{2} \cdot \left(\frac{1}{120} \cdot \left(x \cdot \left({y}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right) + \frac{1}{6} \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{6}, 1\right), {y}^{2} \cdot \left(\frac{1}{120} \cdot \left(x \cdot \left({y}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right) + \frac{1}{6} \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{6}, 1\right), {y}^{2} \cdot \left(\frac{1}{120} \cdot \left(x \cdot \left({y}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right) + \frac{1}{6} \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot x, \frac{-1}{6}, 1\right), \color{blue}{{y}^{2} \cdot \left(\frac{1}{120} \cdot \left(x \cdot \left({y}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right) + \frac{1}{6} \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right)}\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot x, \frac{-1}{6}, 1\right), \color{blue}{\left(y \cdot y\right)} \cdot \left(\frac{1}{120} \cdot \left(x \cdot \left({y}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right) + \frac{1}{6} \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot x, \frac{-1}{6}, 1\right), \color{blue}{\left(y \cdot y\right)} \cdot \left(\frac{1}{120} \cdot \left(x \cdot \left({y}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right) + \frac{1}{6} \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right)\right) \]

       10. +-commutative N/A

           \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot x, \frac{-1}{6}, 1\right), \left(y \cdot y\right) \cdot \color{blue}{\left(\frac{1}{6} \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right) + \frac{1}{120} \cdot \left(x \cdot \left({y}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right)\right)}\right) \]

       11. *-commutative N/A

           \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot x, \frac{-1}{6}, 1\right), \left(y \cdot y\right) \cdot \left(\color{blue}{\left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right) \cdot \frac{1}{6}} + \frac{1}{120} \cdot \left(x \cdot \left({y}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right)\right)\right) \]

       12. *-commutative N/A

           \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot x, \frac{-1}{6}, 1\right), \left(y \cdot y\right) \cdot \left(\left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right) \cdot \frac{1}{6} + \frac{1}{120} \cdot \left(x \cdot \color{blue}{\left(\left(1 + \frac{-1}{6} \cdot {x}^{2}\right) \cdot {y}^{2}\right)}\right)\right)\right) \]

       13. associate-*r* N/A

           \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot x, \frac{-1}{6}, 1\right), \left(y \cdot y\right) \cdot \left(\left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right) \cdot \frac{1}{6} + \frac{1}{120} \cdot \color{blue}{\left(\left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right) \cdot {y}^{2}\right)}\right)\right) \]

   11. Simplified 59.7%

       \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right), \left(y \cdot y\right) \cdot \left(\left(x \cdot \mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right)\right) \cdot \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right)\right)\right)} \]

   12. Taylor expanded in x around 0

       \[\leadsto \color{blue}{x \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} \]
   13. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{x \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} \]

       2. +-commutative N/A

          \[\leadsto x \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) + 1\right)} \]

       3. accelerator-lowering-fma.f64 N/A

          \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{1}{6} + \frac{1}{120} \cdot {y}^{2}, 1\right)} \]

       4. unpow2 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{6} + \frac{1}{120} \cdot {y}^{2}, 1\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{6} + \frac{1}{120} \cdot {y}^{2}, 1\right) \]

       6. +-commutative N/A

          \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{\frac{1}{120} \cdot {y}^{2} + \frac{1}{6}}, 1\right) \]

       7. *-commutative N/A

          \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{{y}^{2} \cdot \frac{1}{120}} + \frac{1}{6}, 1\right) \]

       8. accelerator-lowering-fma.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{1}{120}, \frac{1}{6}\right)}, 1\right) \]

       9. unpow2 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

       10. *-lowering-*.f64 66.6

           \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, 0.008333333333333333, 0.16666666666666666\right), 1\right) \]

   14. Simplified 66.6%

       \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 54.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin x \leq -0.02:\\ \;\;\;\;\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(y \cdot y, -0.027777777777777776, -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sin x) -0.02)
   (* (* x (* x x)) (fma (* y y) -0.027777777777777776 -0.16666666666666666))
   (*
    x
    (fma (* y y) (fma (* y y) 0.008333333333333333 0.16666666666666666) 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (sin(x) <= -0.02) {
		tmp = (x * (x * x)) * fma((y * y), -0.027777777777777776, -0.16666666666666666);
	} else {
		tmp = x * fma((y * y), fma((y * y), 0.008333333333333333, 0.16666666666666666), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (sin(x) <= -0.02)
		tmp = Float64(Float64(x * Float64(x * x)) * fma(Float64(y * y), -0.027777777777777776, -0.16666666666666666));
	else
		tmp = Float64(x * fma(Float64(y * y), fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[Sin[x], $MachinePrecision], -0.02], N[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * -0.027777777777777776 + -0.16666666666666666), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 x) < -0.0200000000000000004

   1. Initial program 100.0%

      \[\sin x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)} \cdot \frac{\sinh y}{y} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)} \cdot \frac{\sinh y}{y} \]

      2. +-commutative N/A

         \[\leadsto \left(x \cdot \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2} + 1\right)}\right) \cdot \frac{\sinh y}{y} \]

      3. *-commutative N/A

         \[\leadsto \left(x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{-1}{6}} + 1\right)\right) \cdot \frac{\sinh y}{y} \]

      4. unpow2 N/A

         \[\leadsto \left(x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{-1}{6} + 1\right)\right) \cdot \frac{\sinh y}{y} \]

      5. associate-*l* N/A

         \[\leadsto \left(x \cdot \left(\color{blue}{x \cdot \left(x \cdot \frac{-1}{6}\right)} + 1\right)\right) \cdot \frac{\sinh y}{y} \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(x \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)}\right) \cdot \frac{\sinh y}{y} \]

      7. *-lowering-*.f64 22.5

         \[\leadsto \left(x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot -0.16666666666666666}, 1\right)\right) \cdot \frac{\sinh y}{y} \]

   5. Simplified 22.5%

      \[\leadsto \color{blue}{\left(x \cdot \mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right)\right)} \cdot \frac{\sinh y}{y} \]

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\left(\frac{-1}{6} \cdot {x}^{3}\right)} \cdot \frac{\sinh y}{y} \]
   7. Step-by-step derivation

      1. unpow3 N/A

         \[\leadsto \left(\frac{-1}{6} \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot x\right)}\right) \cdot \frac{\sinh y}{y} \]

      2. unpow2 N/A

         \[\leadsto \left(\frac{-1}{6} \cdot \left(\color{blue}{{x}^{2}} \cdot x\right)\right) \cdot \frac{\sinh y}{y} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(\frac{-1}{6} \cdot {x}^{2}\right) \cdot x\right)} \cdot \frac{\sinh y}{y} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot \left(\frac{-1}{6} \cdot {x}^{2}\right)\right)} \cdot \frac{\sinh y}{y} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot \left(\frac{-1}{6} \cdot {x}^{2}\right)\right)} \cdot \frac{\sinh y}{y} \]

      6. *-commutative N/A

         \[\leadsto \left(x \cdot \color{blue}{\left({x}^{2} \cdot \frac{-1}{6}\right)}\right) \cdot \frac{\sinh y}{y} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \left(x \cdot \color{blue}{\left({x}^{2} \cdot \frac{-1}{6}\right)}\right) \cdot \frac{\sinh y}{y} \]

      8. unpow2 N/A

         \[\leadsto \left(x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{-1}{6}\right)\right) \cdot \frac{\sinh y}{y} \]

      9. *-lowering-*.f64 22.5

         \[\leadsto \left(x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot -0.16666666666666666\right)\right) \cdot \frac{\sinh y}{y} \]

   8. Simplified 22.5%

      \[\leadsto \color{blue}{\left(x \cdot \left(\left(x \cdot x\right) \cdot -0.16666666666666666\right)\right)} \cdot \frac{\sinh y}{y} \]

   9. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{-1}{6} \cdot {x}^{3} + \frac{-1}{36} \cdot \left({x}^{3} \cdot {y}^{2}\right)} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{-1}{6} \cdot {x}^{3} + \frac{-1}{36} \cdot \color{blue}{\left({y}^{2} \cdot {x}^{3}\right)} \]

       2. associate-*r* N/A

          \[\leadsto \frac{-1}{6} \cdot {x}^{3} + \color{blue}{\left(\frac{-1}{36} \cdot {y}^{2}\right) \cdot {x}^{3}} \]

       3. metadata-eval N/A

          \[\leadsto \frac{-1}{6} \cdot {x}^{3} + \left(\color{blue}{\left(\frac{-1}{6} \cdot \frac{1}{6}\right)} \cdot {y}^{2}\right) \cdot {x}^{3} \]

       4. associate-*r* N/A

          \[\leadsto \frac{-1}{6} \cdot {x}^{3} + \color{blue}{\left(\frac{-1}{6} \cdot \left(\frac{1}{6} \cdot {y}^{2}\right)\right)} \cdot {x}^{3} \]

       5. distribute-rgt-out N/A

          \[\leadsto \color{blue}{{x}^{3} \cdot \left(\frac{-1}{6} + \frac{-1}{6} \cdot \left(\frac{1}{6} \cdot {y}^{2}\right)\right)} \]

       6. metadata-eval N/A

          \[\leadsto {x}^{3} \cdot \left(\color{blue}{\frac{-1}{6} \cdot 1} + \frac{-1}{6} \cdot \left(\frac{1}{6} \cdot {y}^{2}\right)\right) \]

       7. distribute-lft-in N/A

          \[\leadsto {x}^{3} \cdot \color{blue}{\left(\frac{-1}{6} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)} \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{{x}^{3} \cdot \left(\frac{-1}{6} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)} \]

       9. cube-mult N/A

          \[\leadsto \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot \left(\frac{-1}{6} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \left(x \cdot \color{blue}{{x}^{2}}\right) \cdot \left(\frac{-1}{6} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \color{blue}{\left(x \cdot {x}^{2}\right)} \cdot \left(\frac{-1}{6} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right) \]

       12. unpow2 N/A

           \[\leadsto \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(\frac{-1}{6} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(\frac{-1}{6} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right) \]

       14. distribute-rgt-in N/A

           \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(1 \cdot \frac{-1}{6} + \left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \frac{-1}{6}\right)} \]

       15. metadata-eval N/A

           \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\color{blue}{\frac{-1}{6}} + \left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \frac{-1}{6}\right) \]

       16. +-commutative N/A

           \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(\left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \frac{-1}{6} + \frac{-1}{6}\right)} \]

       17. *-commutative N/A

           \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\color{blue}{\left({y}^{2} \cdot \frac{1}{6}\right)} \cdot \frac{-1}{6} + \frac{-1}{6}\right) \]

       18. associate-*l* N/A

           \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\color{blue}{{y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{-1}{6}\right)} + \frac{-1}{6}\right) \]

       19. metadata-eval N/A

           \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left({y}^{2} \cdot \color{blue}{\frac{-1}{36}} + \frac{-1}{6}\right) \]

       20. accelerator-lowering-fma.f64 N/A

           \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{-1}{36}, \frac{-1}{6}\right)} \]

       21. unpow2 N/A

           \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{-1}{36}, \frac{-1}{6}\right) \]

       22. *-lowering-*.f64 21.1

           \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, -0.027777777777777776, -0.16666666666666666\right) \]

   11. Simplified 21.1%

       \[\leadsto \color{blue}{\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(y \cdot y, -0.027777777777777776, -0.16666666666666666\right)} \]

   if -0.0200000000000000004 < (sin.f64 x)

   1. Initial program 100.0%

      \[\sin x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)} \cdot \frac{\sinh y}{y} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)} \cdot \frac{\sinh y}{y} \]

      2. +-commutative N/A

         \[\leadsto \left(x \cdot \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2} + 1\right)}\right) \cdot \frac{\sinh y}{y} \]

      3. *-commutative N/A

         \[\leadsto \left(x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{-1}{6}} + 1\right)\right) \cdot \frac{\sinh y}{y} \]

      4. unpow2 N/A

         \[\leadsto \left(x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{-1}{6} + 1\right)\right) \cdot \frac{\sinh y}{y} \]

      5. associate-*l* N/A

         \[\leadsto \left(x \cdot \left(\color{blue}{x \cdot \left(x \cdot \frac{-1}{6}\right)} + 1\right)\right) \cdot \frac{\sinh y}{y} \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(x \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)}\right) \cdot \frac{\sinh y}{y} \]

      7. *-lowering-*.f64 76.2

         \[\leadsto \left(x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot -0.16666666666666666}, 1\right)\right) \cdot \frac{\sinh y}{y} \]

   5. Simplified 76.2%

      \[\leadsto \color{blue}{\left(x \cdot \mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right)\right)} \cdot \frac{\sinh y}{y} \]

   6. Taylor expanded in y around 0

      \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)\right) \cdot \frac{\color{blue}{y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)}}{y} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)\right) \cdot \frac{\color{blue}{y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)}}{y} \]

      2. +-commutative N/A

         \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)\right) \cdot \frac{y \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) + 1\right)}}{y} \]

      3. unpow2 N/A

         \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)\right) \cdot \frac{y \cdot \left(\color{blue}{\left(y \cdot y\right)} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) + 1\right)}{y} \]

      4. associate-*l* N/A

         \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)\right) \cdot \frac{y \cdot \left(\color{blue}{y \cdot \left(y \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)} + 1\right)}{y} \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)\right) \cdot \frac{y \cdot \color{blue}{\mathsf{fma}\left(y, y \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right), 1\right)}}{y} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)\right) \cdot \frac{y \cdot \mathsf{fma}\left(y, \color{blue}{y \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)}, 1\right)}{y} \]

      7. +-commutative N/A

         \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)\right) \cdot \frac{y \cdot \mathsf{fma}\left(y, y \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right) + \frac{1}{6}\right)}, 1\right)}{y} \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)\right) \cdot \frac{y \cdot \mathsf{fma}\left(y, y \cdot \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}, \frac{1}{6}\right)}, 1\right)}{y} \]

      9. unpow2 N/A

         \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)\right) \cdot \frac{y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}, \frac{1}{6}\right), 1\right)}{y} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)\right) \cdot \frac{y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}, \frac{1}{6}\right), 1\right)}{y} \]

      11. +-commutative N/A

          \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)\right) \cdot \frac{y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{\frac{1}{5040} \cdot {y}^{2} + \frac{1}{120}}, \frac{1}{6}\right), 1\right)}{y} \]

      12. *-commutative N/A

          \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)\right) \cdot \frac{y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{{y}^{2} \cdot \frac{1}{5040}} + \frac{1}{120}, \frac{1}{6}\right), 1\right)}{y} \]

      13. accelerator-lowering-fma.f64 N/A

          \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)\right) \cdot \frac{y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{1}{5040}, \frac{1}{120}\right)}, \frac{1}{6}\right), 1\right)}{y} \]

      14. unpow2 N/A

          \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)\right) \cdot \frac{y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right)}{y} \]

      15. *-lowering-*.f64 71.8

          \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right)\right) \cdot \frac{y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)}{y} \]

   8. Simplified 71.8%

      \[\leadsto \left(x \cdot \mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right)\right) \cdot \frac{\color{blue}{y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)}}{y} \]

   9. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right) + {y}^{2} \cdot \left(\frac{1}{120} \cdot \left(x \cdot \left({y}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right) + \frac{1}{6} \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right)} \]
   10. Step-by-step derivation

       1. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(x, 1 + \frac{-1}{6} \cdot {x}^{2}, {y}^{2} \cdot \left(\frac{1}{120} \cdot \left(x \cdot \left({y}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right) + \frac{1}{6} \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right)\right)} \]

       2. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{-1}{6} \cdot {x}^{2} + 1}, {y}^{2} \cdot \left(\frac{1}{120} \cdot \left(x \cdot \left({y}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right) + \frac{1}{6} \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right)\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x, \color{blue}{{x}^{2} \cdot \frac{-1}{6}} + 1, {y}^{2} \cdot \left(\frac{1}{120} \cdot \left(x \cdot \left({y}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right) + \frac{1}{6} \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right)\right) \]

       4. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{-1}{6}, 1\right)}, {y}^{2} \cdot \left(\frac{1}{120} \cdot \left(x \cdot \left({y}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right) + \frac{1}{6} \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{6}, 1\right), {y}^{2} \cdot \left(\frac{1}{120} \cdot \left(x \cdot \left({y}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right) + \frac{1}{6} \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{6}, 1\right), {y}^{2} \cdot \left(\frac{1}{120} \cdot \left(x \cdot \left({y}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right) + \frac{1}{6} \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot x, \frac{-1}{6}, 1\right), \color{blue}{{y}^{2} \cdot \left(\frac{1}{120} \cdot \left(x \cdot \left({y}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right) + \frac{1}{6} \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right)}\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot x, \frac{-1}{6}, 1\right), \color{blue}{\left(y \cdot y\right)} \cdot \left(\frac{1}{120} \cdot \left(x \cdot \left({y}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right) + \frac{1}{6} \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot x, \frac{-1}{6}, 1\right), \color{blue}{\left(y \cdot y\right)} \cdot \left(\frac{1}{120} \cdot \left(x \cdot \left({y}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right) + \frac{1}{6} \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right)\right) \]

       10. +-commutative N/A

           \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot x, \frac{-1}{6}, 1\right), \left(y \cdot y\right) \cdot \color{blue}{\left(\frac{1}{6} \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right) + \frac{1}{120} \cdot \left(x \cdot \left({y}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right)\right)}\right) \]

       11. *-commutative N/A

           \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot x, \frac{-1}{6}, 1\right), \left(y \cdot y\right) \cdot \left(\color{blue}{\left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right) \cdot \frac{1}{6}} + \frac{1}{120} \cdot \left(x \cdot \left({y}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right)\right)\right) \]

       12. *-commutative N/A

           \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot x, \frac{-1}{6}, 1\right), \left(y \cdot y\right) \cdot \left(\left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right) \cdot \frac{1}{6} + \frac{1}{120} \cdot \left(x \cdot \color{blue}{\left(\left(1 + \frac{-1}{6} \cdot {x}^{2}\right) \cdot {y}^{2}\right)}\right)\right)\right) \]

       13. associate-*r* N/A

           \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot x, \frac{-1}{6}, 1\right), \left(y \cdot y\right) \cdot \left(\left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right) \cdot \frac{1}{6} + \frac{1}{120} \cdot \color{blue}{\left(\left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right) \cdot {y}^{2}\right)}\right)\right) \]

   11. Simplified 59.7%

       \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right), \left(y \cdot y\right) \cdot \left(\left(x \cdot \mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right)\right) \cdot \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right)\right)\right)} \]

   12. Taylor expanded in x around 0

       \[\leadsto \color{blue}{x \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} \]
   13. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{x \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} \]

       2. +-commutative N/A

          \[\leadsto x \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) + 1\right)} \]

       3. accelerator-lowering-fma.f64 N/A

          \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{1}{6} + \frac{1}{120} \cdot {y}^{2}, 1\right)} \]

       4. unpow2 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{6} + \frac{1}{120} \cdot {y}^{2}, 1\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{6} + \frac{1}{120} \cdot {y}^{2}, 1\right) \]

       6. +-commutative N/A

          \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{\frac{1}{120} \cdot {y}^{2} + \frac{1}{6}}, 1\right) \]

       7. *-commutative N/A

          \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{{y}^{2} \cdot \frac{1}{120}} + \frac{1}{6}, 1\right) \]

       8. accelerator-lowering-fma.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{1}{120}, \frac{1}{6}\right)}, 1\right) \]

       9. unpow2 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

       10. *-lowering-*.f64 66.6

           \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, 0.008333333333333333, 0.16666666666666666\right), 1\right) \]

   14. Simplified 66.6%

       \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 34.2% accurate, 12.8× speedup

Math

\[\begin{array}{l} \\ x \cdot \mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* x (fma x (* x -0.16666666666666666) 1.0)))

C

double code(double x, double y) {
	return x * fma(x, (x * -0.16666666666666666), 1.0);
}

Julia

function code(x, y)
	return Float64(x * fma(x, Float64(x * -0.16666666666666666), 1.0))
end

Wolfram

code[x_, y_] := N[(x * N[(x * N[(x * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\sin x \cdot \frac{\sinh y}{y} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\sin x} \]
4. Step-by-step derivation

   1. sin-lowering-sin.f64 53.9

      \[\leadsto \color{blue}{\sin x} \]

5. Simplified 53.9%

   \[\leadsto \color{blue}{\sin x} \]

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)} \]
7. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)} \]

   2. +-commutative N/A

      \[\leadsto x \cdot \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2} + 1\right)} \]

   3. *-commutative N/A

      \[\leadsto x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{-1}{6}} + 1\right) \]

   4. unpow2 N/A

      \[\leadsto x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{-1}{6} + 1\right) \]

   5. associate-*l* N/A

      \[\leadsto x \cdot \left(\color{blue}{x \cdot \left(x \cdot \frac{-1}{6}\right)} + 1\right) \]

   6. *-commutative N/A

      \[\leadsto x \cdot \left(x \cdot \color{blue}{\left(\frac{-1}{6} \cdot x\right)} + 1\right) \]

   7. accelerator-lowering-fma.f64 N/A

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(x, \frac{-1}{6} \cdot x, 1\right)} \]

   8. *-commutative N/A

      \[\leadsto x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{-1}{6}}, 1\right) \]

   9. *-lowering-*.f64 36.3

      \[\leadsto x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot -0.16666666666666666}, 1\right) \]

8. Simplified 36.3%

   \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right)} \]
9. Add Preprocessing

Alternative 21: 26.3% accurate, 217.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

(FPCore (x y) :precision binary64 x)

C

double code(double x, double y) {
	return x;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x
end function

Java

public static double code(double x, double y) {
	return x;
}

Python

def code(x, y):
	return x

Julia

function code(x, y)
	return x
end

MATLAB

function tmp = code(x, y)
	tmp = x;
end

Wolfram

code[x_, y_] := x

Derivation

1. Initial program 100.0%

   \[\sin x \cdot \frac{\sinh y}{y} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\sin x} \]
4. Step-by-step derivation

   1. sin-lowering-sin.f64 53.9

      \[\leadsto \color{blue}{\sin x} \]

5. Simplified 53.9%

   \[\leadsto \color{blue}{\sin x} \]

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x} \]
7. Step-by-step derivation

8. Simplified 29.0%

   \[\leadsto \color{blue}{x} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024196 
(FPCore (x y)
  :name "Linear.Quaternion:$ccos from linear-1.19.1.3"
  :precision binary64
  (* (sin x) (/ (sinh y) y)))