Linear.Quaternion:$ccos from linear-1.19.1.3

Percentage Accurate: 100.0% → 100.0%

Time: 15.2s

Alternatives: 21

Speedup: 1.0×

• 49.9% 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: 75.0% accurate, 0.4× speedup

Math

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

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 (* -0.16666666666666666 (* x x))))
     (if (<= t_1 1.0)
       (*
        (sin x)
        (fma
         y
         (*
          y
          (fma
           y
           (* y (fma y (* y 0.0001984126984126984) 0.008333333333333333))
           0.16666666666666666))
         1.0))
       (*
        t_0
        (fma
         (fma 0.008333333333333333 (* x x) -0.16666666666666666)
         (* x (* x x))
         x))))))

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 * (-0.16666666666666666 * (x * x)));
	} else if (t_1 <= 1.0) {
		tmp = sin(x) * fma(y, (y * fma(y, (y * fma(y, (y * 0.0001984126984126984), 0.008333333333333333)), 0.16666666666666666)), 1.0);
	} else {
		tmp = t_0 * fma(fma(0.008333333333333333, (x * x), -0.16666666666666666), (x * (x * x)), x);
	}
	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(-0.16666666666666666 * Float64(x * x))));
	elseif (t_1 <= 1.0)
		tmp = Float64(sin(x) * fma(y, Float64(y * fma(y, Float64(y * fma(y, Float64(y * 0.0001984126984126984), 0.008333333333333333)), 0.16666666666666666)), 1.0));
	else
		tmp = Float64(t_0 * fma(fma(0.008333333333333333, Float64(x * x), -0.16666666666666666), Float64(x * Float64(x * x)), x));
	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[(-0.16666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1.0], N[(N[Sin[x], $MachinePrecision] * N[(y * N[(y * N[(y * N[(y * N[(y * N[(y * 0.0001984126984126984), $MachinePrecision] + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(N[(0.008333333333333333 * N[(x * x), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $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 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. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 74.0

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

   5. Simplified 74.0%

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

   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. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 20.5

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

   8. Simplified 20.5%

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

   if -inf.0 < (*.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

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

   5. Simplified 99.3%

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

   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

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 72.1

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

   5. Simplified 72.1%

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

4. Final simplification 70.4%

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

Alternative 3: 82.6% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, y)
	t_0 = Float64(sin(x) * Float64(sinh(y) / y))
	t_1 = fma(y, Float64(y * fma(y, Float64(y * fma(y, Float64(y * 0.0001984126984126984), 0.008333333333333333)), 0.16666666666666666)), 1.0)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(t_1 * fma(Float64(x * x), Float64(x * fma(Float64(x * x), fma(x, Float64(x * -0.0001984126984126984), 0.008333333333333333), -0.16666666666666666)), x));
	elseif (t_0 <= 1.0)
		tmp = Float64(sin(x) * fma(y, Float64(y * 0.16666666666666666), 1.0));
	else
		tmp = Float64(t_1 * fma(Float64(x * x), Float64(x * fma(x, Float64(x * 0.008333333333333333), -0.16666666666666666)), x));
	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]}, Block[{t$95$1 = N[(y * N[(y * N[(y * N[(y * N[(y * N[(y * 0.0001984126984126984), $MachinePrecision] + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(t$95$1 * N[(N[(x * x), $MachinePrecision] * N[(x * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * -0.0001984126984126984), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1.0], N[(N[Sin[x], $MachinePrecision] * N[(y * N[(y * 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(x * 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + x), $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

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

   5. Simplified 85.5%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-*l* N/A

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

      4. *-lft-identity N/A

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

      5. lower-fma.f64 N/A

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

   8. Simplified 62.1%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. +-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sin.f64 99.3

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

   5. Simplified 99.3%

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

   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 y around 0

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

   5. Simplified 74.9%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-*l* N/A

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

      4. *-lft-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. sub-neg N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. metadata-eval N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-*.f64 53.3

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

   8. Simplified 53.3%

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

4. Final simplification 77.7%

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

Alternative 4: 82.3% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, y)
	t_0 = Float64(sin(x) * Float64(sinh(y) / y))
	t_1 = fma(y, Float64(y * fma(y, Float64(y * fma(y, Float64(y * 0.0001984126984126984), 0.008333333333333333)), 0.16666666666666666)), 1.0)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(t_1 * fma(Float64(x * x), Float64(x * fma(Float64(x * x), fma(x, Float64(x * -0.0001984126984126984), 0.008333333333333333), -0.16666666666666666)), x));
	elseif (t_0 <= 1.0)
		tmp = sin(x);
	else
		tmp = Float64(t_1 * fma(Float64(x * x), Float64(x * fma(x, Float64(x * 0.008333333333333333), -0.16666666666666666)), x));
	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]}, Block[{t$95$1 = N[(y * N[(y * N[(y * N[(y * N[(y * N[(y * 0.0001984126984126984), $MachinePrecision] + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(t$95$1 * N[(N[(x * x), $MachinePrecision] * N[(x * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * -0.0001984126984126984), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1.0], N[Sin[x], $MachinePrecision], N[(t$95$1 * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(x * 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + x), $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

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

   5. Simplified 85.5%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-*l* N/A

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

      4. *-lft-identity N/A

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

      5. lower-fma.f64 N/A

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

   8. Simplified 62.1%

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

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

   1. Initial program 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. lower-sin.f64 99.2

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

   5. Simplified 99.2%

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

   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 y around 0

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

   5. Simplified 74.9%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-*l* N/A

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

      4. *-lft-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. sub-neg N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. metadata-eval N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-*.f64 53.3

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

   8. Simplified 53.3%

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

4. Final simplification 77.7%

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

Alternative 5: 41.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin x \cdot \frac{\sinh y}{y} \leq -0.01:\\ \;\;\;\;x \cdot \left(-0.16666666666666666 \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, y \cdot \left(y \cdot 0.16666666666666666\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (sin x) (/ (sinh y) y)) -0.01)
   (* x (* -0.16666666666666666 (* x x)))
   (fma x (* y (* y 0.16666666666666666)) x)))

C

double code(double x, double y) {
	double tmp;
	if ((sin(x) * (sinh(y) / y)) <= -0.01) {
		tmp = x * (-0.16666666666666666 * (x * x));
	} else {
		tmp = fma(x, (y * (y * 0.16666666666666666)), x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(sin(x) * Float64(sinh(y) / y)) <= -0.01)
		tmp = Float64(x * Float64(-0.16666666666666666 * Float64(x * x)));
	else
		tmp = fma(x, Float64(y * Float64(y * 0.16666666666666666)), x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Sin[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], -0.01], N[(x * N[(-0.16666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(y * N[(y * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < -0.0100000000000000002

   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. lower-sin.f64 33.4

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

   5. Simplified 33.4%

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

   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. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 11.3

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

   8. Simplified 11.3%

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

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{-1}{6} \cdot {x}^{3}} \]
   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. lower-*.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 11.0

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

   11. Simplified 11.0%

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

   if -0.0100000000000000002 < (*.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

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. +-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sin.f64 79.9

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

   5. Simplified 79.9%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-*l* N/A

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

      4. *-lft-identity N/A

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

      5. lower-fma.f64 N/A

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

   8. Simplified 56.9%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. distribute-lft-in N/A

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

       3. *-rgt-identity N/A

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

       4. lower-fma.f64 N/A

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

       5. *-commutative N/A

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

       6. unpow2 N/A

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

       7. associate-*l* N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 55.2

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

   11. Simplified 55.2%

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

Alternative 6: 58.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin x \leq 10^{-64}:\\ \;\;\;\;\mathsf{fma}\left(x, -0.16666666666666666 \cdot \left(x \cdot x\right), x\right) \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \left(y \cdot y\right) \cdot \mathsf{fma}\left(0.0001984126984126984, \left(y \cdot y\right) \cdot \left(y \cdot y\right), 0.16666666666666666\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sin x) 1e-64)
   (*
    (fma x (* -0.16666666666666666 (* x x)) x)
    (/
     (fma
      (fma
       y
       (* y (fma (* y y) 0.0001984126984126984 0.008333333333333333))
       0.16666666666666666)
      (* y (* y y))
      y)
     y))
   (fma
    x
    (*
     (* y y)
     (fma 0.0001984126984126984 (* (* y y) (* y y)) 0.16666666666666666))
    x)))

C

double code(double x, double y) {
	double tmp;
	if (sin(x) <= 1e-64) {
		tmp = fma(x, (-0.16666666666666666 * (x * x)), x) * (fma(fma(y, (y * fma((y * y), 0.0001984126984126984, 0.008333333333333333)), 0.16666666666666666), (y * (y * y)), y) / y);
	} else {
		tmp = fma(x, ((y * y) * fma(0.0001984126984126984, ((y * y) * (y * y)), 0.16666666666666666)), x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (sin(x) <= 1e-64)
		tmp = Float64(fma(x, Float64(-0.16666666666666666 * Float64(x * x)), x) * Float64(fma(fma(y, Float64(y * fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333)), 0.16666666666666666), Float64(y * Float64(y * y)), y) / y));
	else
		tmp = fma(x, Float64(Float64(y * y) * fma(0.0001984126984126984, Float64(Float64(y * y) * Float64(y * y)), 0.16666666666666666)), x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[Sin[x], $MachinePrecision], 1e-64], N[(N[(x * N[(-0.16666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] * N[(N[(N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(y * y), $MachinePrecision] * N[(0.0001984126984126984 * N[(N[(y * y), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 x) < 9.99999999999999965e-65

   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. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 70.7

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

   5. Simplified 70.7%

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

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. unpow2 N/A

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

      7. unpow3 N/A

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

      8. lower-fma.f64 N/A

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

   8. Simplified 62.1%

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

   if 9.99999999999999965e-65 < (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

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

   5. Simplified 91.2%

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

   6. Taylor expanded in y around inf

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

      1. unpow3 N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 91.2

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

   8. Simplified 91.2%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. distribute-lft-in N/A

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

       3. *-rgt-identity N/A

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

       4. lower-fma.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 N/A

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

       8. +-commutative N/A

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

       9. lower-fma.f64 N/A

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

       10. metadata-eval N/A

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

       11. pow-sqr N/A

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

       12. lower-*.f64 N/A

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

       13. unpow2 N/A

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

       14. lower-*.f64 N/A

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

       15. unpow2 N/A

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

       16. lower-*.f64 32.3

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

   11. Simplified 32.3%

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

Alternative 7: 58.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin x \leq 2 \cdot 10^{-276}:\\ \;\;\;\;x \cdot \left(\mathsf{fma}\left(-0.16666666666666666, x \cdot x, 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \left(y \cdot y\right) \cdot \mathsf{fma}\left(0.0001984126984126984, \left(y \cdot y\right) \cdot \left(y \cdot y\right), 0.16666666666666666\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sin x) 2e-276)
   (*
    x
    (*
     (fma -0.16666666666666666 (* x x) 1.0)
     (fma
      (* y y)
      (fma
       y
       (* y (fma (* y y) 0.0001984126984126984 0.008333333333333333))
       0.16666666666666666)
      1.0)))
   (fma
    x
    (*
     (* y y)
     (fma 0.0001984126984126984 (* (* y y) (* y y)) 0.16666666666666666))
    x)))

C

double code(double x, double y) {
	double tmp;
	if (sin(x) <= 2e-276) {
		tmp = x * (fma(-0.16666666666666666, (x * x), 1.0) * fma((y * y), fma(y, (y * fma((y * y), 0.0001984126984126984, 0.008333333333333333)), 0.16666666666666666), 1.0));
	} else {
		tmp = fma(x, ((y * y) * fma(0.0001984126984126984, ((y * y) * (y * y)), 0.16666666666666666)), x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (sin(x) <= 2e-276)
		tmp = Float64(x * Float64(fma(-0.16666666666666666, Float64(x * x), 1.0) * fma(Float64(y * y), fma(y, Float64(y * fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333)), 0.16666666666666666), 1.0)));
	else
		tmp = fma(x, Float64(Float64(y * y) * fma(0.0001984126984126984, Float64(Float64(y * y) * Float64(y * y)), 0.16666666666666666)), x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[Sin[x], $MachinePrecision], 2e-276], N[(x * N[(N[(-0.16666666666666666 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * 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]), $MachinePrecision], N[(x * N[(N[(y * y), $MachinePrecision] * N[(0.0001984126984126984 * N[(N[(y * y), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 x) < 2e-276

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Simplified 91.9%

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

   6. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt1-in N/A

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

   8. Simplified 56.5%

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

   if 2e-276 < (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

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

   5. Simplified 86.6%

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

   6. Taylor expanded in y around inf

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

      1. unpow3 N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 86.6

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

   8. Simplified 86.6%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. distribute-lft-in N/A

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

       3. *-rgt-identity N/A

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

       4. lower-fma.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 N/A

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

       8. +-commutative N/A

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

       9. lower-fma.f64 N/A

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

       10. metadata-eval N/A

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

       11. pow-sqr N/A

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

       12. lower-*.f64 N/A

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

       13. unpow2 N/A

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

       14. lower-*.f64 N/A

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

       15. unpow2 N/A

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

       16. lower-*.f64 47.9

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

   11. Simplified 47.9%

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

Alternative 8: 58.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(0.0001984126984126984, y \cdot y, 0.008333333333333333\right), 0.16666666666666666\right)\right)\\ \mathbf{if}\;\sin x \leq -0.01:\\ \;\;\;\;\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(-0.16666666666666666, t\_0, -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, t\_0, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (*
          y
          (*
           y
           (fma
            (* y y)
            (fma 0.0001984126984126984 (* y y) 0.008333333333333333)
            0.16666666666666666)))))
   (if (<= (sin x) -0.01)
     (* (* x (* x x)) (fma -0.16666666666666666 t_0 -0.16666666666666666))
     (fma x t_0 x))))

C

double code(double x, double y) {
	double t_0 = y * (y * fma((y * y), fma(0.0001984126984126984, (y * y), 0.008333333333333333), 0.16666666666666666));
	double tmp;
	if (sin(x) <= -0.01) {
		tmp = (x * (x * x)) * fma(-0.16666666666666666, t_0, -0.16666666666666666);
	} else {
		tmp = fma(x, t_0, x);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(y * Float64(y * fma(Float64(y * y), fma(0.0001984126984126984, Float64(y * y), 0.008333333333333333), 0.16666666666666666)))
	tmp = 0.0
	if (sin(x) <= -0.01)
		tmp = Float64(Float64(x * Float64(x * x)) * fma(-0.16666666666666666, t_0, -0.16666666666666666));
	else
		tmp = fma(x, t_0, x);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * N[(0.0001984126984126984 * N[(y * y), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Sin[x], $MachinePrecision], -0.01], N[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(-0.16666666666666666 * t$95$0 + -0.16666666666666666), $MachinePrecision]), $MachinePrecision], N[(x * t$95$0 + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 x) < -0.0100000000000000002

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Simplified 94.6%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 21.7

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

   8. Simplified 21.7%

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

   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. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. lower-*.f64 N/A

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

       5. cube-mult N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 N/A

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

       10. +-commutative N/A

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

       11. distribute-lft-in N/A

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

       12. metadata-eval N/A

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

       13. lower-fma.f64 N/A

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

   11. Simplified 21.7%

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

   if -0.0100000000000000002 < (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

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

   5. Simplified 87.7%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 65.4

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

   8. Simplified 65.4%

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

   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. +-commutative N/A

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

       2. distribute-lft-in N/A

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

       3. *-rgt-identity N/A

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

       4. lower-fma.f64 N/A

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

   11. Simplified 64.1%

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

Alternative 9: 58.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin x \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(x, -0.16666666666666666 \cdot \left(x \cdot x\right), x\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, y \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(0.0001984126984126984, y \cdot y, 0.008333333333333333\right), 0.16666666666666666\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sin x) -0.01)
   (*
    (fma x (* -0.16666666666666666 (* x x)) x)
    (fma (* y y) (fma y (* y 0.008333333333333333) 0.16666666666666666) 1.0))
   (fma
    x
    (*
     y
     (*
      y
      (fma
       (* y y)
       (fma 0.0001984126984126984 (* y y) 0.008333333333333333)
       0.16666666666666666)))
    x)))

C

double code(double x, double y) {
	double tmp;
	if (sin(x) <= -0.01) {
		tmp = fma(x, (-0.16666666666666666 * (x * x)), x) * fma((y * y), fma(y, (y * 0.008333333333333333), 0.16666666666666666), 1.0);
	} else {
		tmp = fma(x, (y * (y * fma((y * y), fma(0.0001984126984126984, (y * y), 0.008333333333333333), 0.16666666666666666))), x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (sin(x) <= -0.01)
		tmp = Float64(fma(x, Float64(-0.16666666666666666 * Float64(x * x)), x) * fma(Float64(y * y), fma(y, Float64(y * 0.008333333333333333), 0.16666666666666666), 1.0));
	else
		tmp = fma(x, Float64(y * Float64(y * fma(Float64(y * y), fma(0.0001984126984126984, Float64(y * y), 0.008333333333333333), 0.16666666666666666))), x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[Sin[x], $MachinePrecision], -0.01], N[(N[(x * N[(-0.16666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(x * N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * N[(0.0001984126984126984 * N[(y * y), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 x) < -0.0100000000000000002

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. distribute-lft-in N/A

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

      2. associate-+r+ N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt1-in N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt-in N/A

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

      9. +-commutative N/A

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

      10. associate-+l+ N/A

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

      11. *-commutative N/A

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

      12. distribute-lft-in N/A

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

   5. Simplified 89.3%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 21.7

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

   8. Simplified 21.7%

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

   if -0.0100000000000000002 < (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

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

   5. Simplified 87.7%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 65.4

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

   8. Simplified 65.4%

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

   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. +-commutative N/A

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

       2. distribute-lft-in N/A

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

       3. *-rgt-identity N/A

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

       4. lower-fma.f64 N/A

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

   11. Simplified 64.1%

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

Alternative 10: 94.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 12.2:\\ \;\;\;\;\sin x \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)\\ \mathbf{elif}\;y \leq 10^{+52}:\\ \;\;\;\;\frac{\sinh y}{y} \cdot \mathsf{fma}\left(x, -0.16666666666666666 \cdot \left(x \cdot x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \mathsf{fma}\left(y, y \cdot \left(0.0001984126984126984 \cdot \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right)\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 12.2)
   (*
    (sin x)
    (fma
     y
     (*
      y
      (fma
       y
       (* y (fma y (* y 0.0001984126984126984) 0.008333333333333333))
       0.16666666666666666))
     1.0))
   (if (<= y 1e+52)
     (* (/ (sinh y) y) (fma x (* -0.16666666666666666 (* x x)) x))
     (*
      (sin x)
      (fma y (* y (* 0.0001984126984126984 (* (* y y) (* y y)))) 1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 12.2) {
		tmp = sin(x) * fma(y, (y * fma(y, (y * fma(y, (y * 0.0001984126984126984), 0.008333333333333333)), 0.16666666666666666)), 1.0);
	} else if (y <= 1e+52) {
		tmp = (sinh(y) / y) * fma(x, (-0.16666666666666666 * (x * x)), x);
	} else {
		tmp = sin(x) * fma(y, (y * (0.0001984126984126984 * ((y * y) * (y * y)))), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 12.2)
		tmp = Float64(sin(x) * fma(y, Float64(y * fma(y, Float64(y * fma(y, Float64(y * 0.0001984126984126984), 0.008333333333333333)), 0.16666666666666666)), 1.0));
	elseif (y <= 1e+52)
		tmp = Float64(Float64(sinh(y) / y) * fma(x, Float64(-0.16666666666666666 * Float64(x * x)), x));
	else
		tmp = Float64(sin(x) * fma(y, Float64(y * Float64(0.0001984126984126984 * Float64(Float64(y * y) * Float64(y * y)))), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 12.2], N[(N[Sin[x], $MachinePrecision] * N[(y * N[(y * N[(y * N[(y * N[(y * N[(y * 0.0001984126984126984), $MachinePrecision] + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1e+52], N[(N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision] * N[(x * N[(-0.16666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], N[(N[Sin[x], $MachinePrecision] * N[(y * N[(y * N[(0.0001984126984126984 * N[(N[(y * y), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 12.199999999999999

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Simplified 94.9%

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

   if 12.199999999999999 < y < 9.9999999999999999e51

   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. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 88.9

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

   5. Simplified 88.9%

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

   if 9.9999999999999999e51 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around inf

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

      1. unpow3 N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 100.0

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

   8. Simplified 100.0%

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

   9. Taylor expanded in y around inf

      \[\leadsto \sin x \cdot \mathsf{fma}\left(y, \color{blue}{\frac{1}{5040} \cdot {y}^{5}}, 1\right) \]
   10. Step-by-step derivation

       1. metadata-eval N/A

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

       2. pow-plus N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. metadata-eval N/A

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

       8. pow-sqr N/A

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

       9. lower-*.f64 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f64 N/A

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

       12. unpow2 N/A

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

       13. lower-*.f64 100.0

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

   11. Simplified 100.0%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 12.2:\\ \;\;\;\;\sin x \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)\\ \mathbf{elif}\;y \leq 10^{+52}:\\ \;\;\;\;\frac{\sinh y}{y} \cdot \mathsf{fma}\left(x, -0.16666666666666666 \cdot \left(x \cdot x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \mathsf{fma}\left(y, y \cdot \left(0.0001984126984126984 \cdot \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right)\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 94.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 12.2:\\ \;\;\;\;\sin x \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \left(0.0001984126984126984 \cdot \left(y \cdot y\right)\right), 0.16666666666666666\right), 1\right)\\ \mathbf{elif}\;y \leq 10^{+52}:\\ \;\;\;\;\frac{\sinh y}{y} \cdot \mathsf{fma}\left(x, -0.16666666666666666 \cdot \left(x \cdot x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \mathsf{fma}\left(y, y \cdot \left(0.0001984126984126984 \cdot \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right)\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 12.2)
   (*
    (sin x)
    (fma
     y
     (* y (fma y (* y (* 0.0001984126984126984 (* y y))) 0.16666666666666666))
     1.0))
   (if (<= y 1e+52)
     (* (/ (sinh y) y) (fma x (* -0.16666666666666666 (* x x)) x))
     (*
      (sin x)
      (fma y (* y (* 0.0001984126984126984 (* (* y y) (* y y)))) 1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 12.2) {
		tmp = sin(x) * fma(y, (y * fma(y, (y * (0.0001984126984126984 * (y * y))), 0.16666666666666666)), 1.0);
	} else if (y <= 1e+52) {
		tmp = (sinh(y) / y) * fma(x, (-0.16666666666666666 * (x * x)), x);
	} else {
		tmp = sin(x) * fma(y, (y * (0.0001984126984126984 * ((y * y) * (y * y)))), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 12.2)
		tmp = Float64(sin(x) * fma(y, Float64(y * fma(y, Float64(y * Float64(0.0001984126984126984 * Float64(y * y))), 0.16666666666666666)), 1.0));
	elseif (y <= 1e+52)
		tmp = Float64(Float64(sinh(y) / y) * fma(x, Float64(-0.16666666666666666 * Float64(x * x)), x));
	else
		tmp = Float64(sin(x) * fma(y, Float64(y * Float64(0.0001984126984126984 * Float64(Float64(y * y) * Float64(y * y)))), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 12.2], N[(N[Sin[x], $MachinePrecision] * N[(y * N[(y * N[(y * N[(y * N[(0.0001984126984126984 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1e+52], N[(N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision] * N[(x * N[(-0.16666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], N[(N[Sin[x], $MachinePrecision] * N[(y * N[(y * N[(0.0001984126984126984 * N[(N[(y * y), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 12.199999999999999

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Simplified 94.9%

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

   6. Taylor expanded in y around inf

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

      1. unpow3 N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 94.9

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

   8. Simplified 94.9%

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

   if 12.199999999999999 < y < 9.9999999999999999e51

   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. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 88.9

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

   5. Simplified 88.9%

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

   if 9.9999999999999999e51 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around inf

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

      1. unpow3 N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 100.0

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

   8. Simplified 100.0%

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

   9. Taylor expanded in y around inf

      \[\leadsto \sin x \cdot \mathsf{fma}\left(y, \color{blue}{\frac{1}{5040} \cdot {y}^{5}}, 1\right) \]
   10. Step-by-step derivation

       1. metadata-eval N/A

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

       2. pow-plus N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. metadata-eval N/A

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

       8. pow-sqr N/A

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

       9. lower-*.f64 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f64 N/A

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

       12. unpow2 N/A

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

       13. lower-*.f64 100.0

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

   11. Simplified 100.0%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 12.2:\\ \;\;\;\;\sin x \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \left(0.0001984126984126984 \cdot \left(y \cdot y\right)\right), 0.16666666666666666\right), 1\right)\\ \mathbf{elif}\;y \leq 10^{+52}:\\ \;\;\;\;\frac{\sinh y}{y} \cdot \mathsf{fma}\left(x, -0.16666666666666666 \cdot \left(x \cdot x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \mathsf{fma}\left(y, y \cdot \left(0.0001984126984126984 \cdot \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right)\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 92.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 12.2:\\ \;\;\;\;\sin x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)\\ \mathbf{elif}\;y \leq 10^{+52}:\\ \;\;\;\;\frac{\sinh y}{y} \cdot \mathsf{fma}\left(x, -0.16666666666666666 \cdot \left(x \cdot x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \mathsf{fma}\left(y, y \cdot \left(0.0001984126984126984 \cdot \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right)\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 12.2)
   (*
    (sin x)
    (fma (* y y) (fma y (* y 0.008333333333333333) 0.16666666666666666) 1.0))
   (if (<= y 1e+52)
     (* (/ (sinh y) y) (fma x (* -0.16666666666666666 (* x x)) x))
     (*
      (sin x)
      (fma y (* y (* 0.0001984126984126984 (* (* y y) (* y y)))) 1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 12.2) {
		tmp = sin(x) * fma((y * y), fma(y, (y * 0.008333333333333333), 0.16666666666666666), 1.0);
	} else if (y <= 1e+52) {
		tmp = (sinh(y) / y) * fma(x, (-0.16666666666666666 * (x * x)), x);
	} else {
		tmp = sin(x) * fma(y, (y * (0.0001984126984126984 * ((y * y) * (y * y)))), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 12.2)
		tmp = Float64(sin(x) * fma(Float64(y * y), fma(y, Float64(y * 0.008333333333333333), 0.16666666666666666), 1.0));
	elseif (y <= 1e+52)
		tmp = Float64(Float64(sinh(y) / y) * fma(x, Float64(-0.16666666666666666 * Float64(x * x)), x));
	else
		tmp = Float64(sin(x) * fma(y, Float64(y * Float64(0.0001984126984126984 * Float64(Float64(y * y) * Float64(y * y)))), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 12.2], N[(N[Sin[x], $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1e+52], N[(N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision] * N[(x * N[(-0.16666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], N[(N[Sin[x], $MachinePrecision] * N[(y * N[(y * N[(0.0001984126984126984 * N[(N[(y * y), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 12.199999999999999

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. distribute-lft-in N/A

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

      2. associate-+r+ N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt1-in N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt-in N/A

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

      9. +-commutative N/A

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

      10. associate-+l+ N/A

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

      11. *-commutative N/A

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

      12. distribute-lft-in N/A

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

   5. Simplified 91.9%

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

   if 12.199999999999999 < y < 9.9999999999999999e51

   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. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 88.9

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

   5. Simplified 88.9%

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

   if 9.9999999999999999e51 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around inf

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

      1. unpow3 N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 100.0

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

   8. Simplified 100.0%

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

   9. Taylor expanded in y around inf

      \[\leadsto \sin x \cdot \mathsf{fma}\left(y, \color{blue}{\frac{1}{5040} \cdot {y}^{5}}, 1\right) \]
   10. Step-by-step derivation

       1. metadata-eval N/A

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

       2. pow-plus N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. metadata-eval N/A

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

       8. pow-sqr N/A

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

       9. lower-*.f64 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f64 N/A

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

       12. unpow2 N/A

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

       13. lower-*.f64 100.0

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

   11. Simplified 100.0%

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

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 12.2:\\ \;\;\;\;\sin x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)\\ \mathbf{elif}\;y \leq 10^{+52}:\\ \;\;\;\;\frac{\sinh y}{y} \cdot \mathsf{fma}\left(x, -0.16666666666666666 \cdot \left(x \cdot x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \mathsf{fma}\left(y, y \cdot \left(0.0001984126984126984 \cdot \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right)\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 92.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 12.2:\\ \;\;\;\;\sin x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)\\ \mathbf{elif}\;y \leq 10^{+76}:\\ \;\;\;\;\frac{\sinh y}{y} \cdot \mathsf{fma}\left(x, -0.16666666666666666 \cdot \left(x \cdot x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \left(y \cdot \left(y \cdot \left(0.008333333333333333 \cdot \left(y \cdot y\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 12.2)
   (*
    (sin x)
    (fma (* y y) (fma y (* y 0.008333333333333333) 0.16666666666666666) 1.0))
   (if (<= y 1e+76)
     (* (/ (sinh y) y) (fma x (* -0.16666666666666666 (* x x)) x))
     (* (sin x) (* y (* y (* 0.008333333333333333 (* y y))))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 12.2) {
		tmp = sin(x) * fma((y * y), fma(y, (y * 0.008333333333333333), 0.16666666666666666), 1.0);
	} else if (y <= 1e+76) {
		tmp = (sinh(y) / y) * fma(x, (-0.16666666666666666 * (x * x)), x);
	} else {
		tmp = sin(x) * (y * (y * (0.008333333333333333 * (y * y))));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 12.2)
		tmp = Float64(sin(x) * fma(Float64(y * y), fma(y, Float64(y * 0.008333333333333333), 0.16666666666666666), 1.0));
	elseif (y <= 1e+76)
		tmp = Float64(Float64(sinh(y) / y) * fma(x, Float64(-0.16666666666666666 * Float64(x * x)), x));
	else
		tmp = Float64(sin(x) * Float64(y * Float64(y * Float64(0.008333333333333333 * Float64(y * y)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 12.2], N[(N[Sin[x], $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1e+76], N[(N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision] * N[(x * N[(-0.16666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], N[(N[Sin[x], $MachinePrecision] * N[(y * N[(y * N[(0.008333333333333333 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 12.199999999999999

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. distribute-lft-in N/A

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

      2. associate-+r+ N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt1-in N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt-in N/A

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

      9. +-commutative N/A

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

      10. associate-+l+ N/A

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

      11. *-commutative N/A

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

      12. distribute-lft-in N/A

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

   5. Simplified 91.9%

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

   if 12.199999999999999 < y < 1e76

   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. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 91.7

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

   5. Simplified 91.7%

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

   if 1e76 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. distribute-lft-in N/A

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

      2. associate-+r+ N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt1-in N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt-in N/A

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

      9. +-commutative N/A

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

      10. associate-+l+ N/A

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

      11. *-commutative N/A

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

      12. distribute-lft-in N/A

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 100.0

          \[\leadsto \sin x \cdot \left(y \cdot \left(y \cdot \left(\color{blue}{\left(y \cdot y\right)} \cdot 0.008333333333333333\right)\right)\right) \]

   8. Simplified 100.0%

      \[\leadsto \color{blue}{\sin x \cdot \left(y \cdot \left(y \cdot \left(\left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 12.2:\\ \;\;\;\;\sin x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)\\ \mathbf{elif}\;y \leq 10^{+76}:\\ \;\;\;\;\frac{\sinh y}{y} \cdot \mathsf{fma}\left(x, -0.16666666666666666 \cdot \left(x \cdot x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \left(y \cdot \left(y \cdot \left(0.008333333333333333 \cdot \left(y \cdot y\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 57.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin x \leq -0.01:\\ \;\;\;\;x \cdot \left(\mathsf{fma}\left(y, y \cdot 0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(-0.16666666666666666, x \cdot x, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, y \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(0.0001984126984126984, y \cdot y, 0.008333333333333333\right), 0.16666666666666666\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sin x) -0.01)
   (*
    x
    (*
     (fma y (* y 0.16666666666666666) 1.0)
     (fma -0.16666666666666666 (* x x) 1.0)))
   (fma
    x
    (*
     y
     (*
      y
      (fma
       (* y y)
       (fma 0.0001984126984126984 (* y y) 0.008333333333333333)
       0.16666666666666666)))
    x)))

C

double code(double x, double y) {
	double tmp;
	if (sin(x) <= -0.01) {
		tmp = x * (fma(y, (y * 0.16666666666666666), 1.0) * fma(-0.16666666666666666, (x * x), 1.0));
	} else {
		tmp = fma(x, (y * (y * fma((y * y), fma(0.0001984126984126984, (y * y), 0.008333333333333333), 0.16666666666666666))), x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (sin(x) <= -0.01)
		tmp = Float64(x * Float64(fma(y, Float64(y * 0.16666666666666666), 1.0) * fma(-0.16666666666666666, Float64(x * x), 1.0)));
	else
		tmp = fma(x, Float64(y * Float64(y * fma(Float64(y * y), fma(0.0001984126984126984, Float64(y * y), 0.008333333333333333), 0.16666666666666666))), x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[Sin[x], $MachinePrecision], -0.01], N[(x * N[(N[(y * N[(y * 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * N[(-0.16666666666666666 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * N[(0.0001984126984126984 * N[(y * y), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 x) < -0.0100000000000000002

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. +-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sin.f64 74.0

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

   5. Simplified 74.0%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt1-in N/A

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

      10. +-commutative N/A

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

   8. Simplified 21.7%

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

   if -0.0100000000000000002 < (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

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

   5. Simplified 87.7%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 65.4

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

   8. Simplified 65.4%

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

   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. +-commutative N/A

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

       2. distribute-lft-in N/A

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

       3. *-rgt-identity N/A

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

       4. lower-fma.f64 N/A

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

   11. Simplified 64.1%

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

4. Final simplification 52.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin x \leq -0.01:\\ \;\;\;\;x \cdot \left(\mathsf{fma}\left(y, y \cdot 0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(-0.16666666666666666, x \cdot x, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, y \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(0.0001984126984126984, y \cdot y, 0.008333333333333333\right), 0.16666666666666666\right)\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 57.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin x \leq -0.01:\\ \;\;\;\;x \cdot \left(\mathsf{fma}\left(y, y \cdot 0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(-0.16666666666666666, x \cdot x, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \left(y \cdot y\right) \cdot \mathsf{fma}\left(0.0001984126984126984, \left(y \cdot y\right) \cdot \left(y \cdot y\right), 0.16666666666666666\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sin x) -0.01)
   (*
    x
    (*
     (fma y (* y 0.16666666666666666) 1.0)
     (fma -0.16666666666666666 (* x x) 1.0)))
   (fma
    x
    (*
     (* y y)
     (fma 0.0001984126984126984 (* (* y y) (* y y)) 0.16666666666666666))
    x)))

C

double code(double x, double y) {
	double tmp;
	if (sin(x) <= -0.01) {
		tmp = x * (fma(y, (y * 0.16666666666666666), 1.0) * fma(-0.16666666666666666, (x * x), 1.0));
	} else {
		tmp = fma(x, ((y * y) * fma(0.0001984126984126984, ((y * y) * (y * y)), 0.16666666666666666)), x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (sin(x) <= -0.01)
		tmp = Float64(x * Float64(fma(y, Float64(y * 0.16666666666666666), 1.0) * fma(-0.16666666666666666, Float64(x * x), 1.0)));
	else
		tmp = fma(x, Float64(Float64(y * y) * fma(0.0001984126984126984, Float64(Float64(y * y) * Float64(y * y)), 0.16666666666666666)), x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[Sin[x], $MachinePrecision], -0.01], N[(x * N[(N[(y * N[(y * 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * N[(-0.16666666666666666 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(y * y), $MachinePrecision] * N[(0.0001984126984126984 * N[(N[(y * y), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 x) < -0.0100000000000000002

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. +-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sin.f64 74.0

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

   5. Simplified 74.0%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt1-in N/A

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

      10. +-commutative N/A

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

   8. Simplified 21.7%

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

   if -0.0100000000000000002 < (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

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

   5. Simplified 87.7%

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

   6. Taylor expanded in y around inf

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

      1. unpow3 N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 87.7

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

   8. Simplified 87.7%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. distribute-lft-in N/A

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

       3. *-rgt-identity N/A

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

       4. lower-fma.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 N/A

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

       8. +-commutative N/A

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

       9. lower-fma.f64 N/A

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

       10. metadata-eval N/A

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

       11. pow-sqr N/A

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

       12. lower-*.f64 N/A

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

       13. unpow2 N/A

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

       14. lower-*.f64 N/A

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

       15. unpow2 N/A

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

       16. lower-*.f64 64.1

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

   11. Simplified 64.1%

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

4. Final simplification 52.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin x \leq -0.01:\\ \;\;\;\;x \cdot \left(\mathsf{fma}\left(y, y \cdot 0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(-0.16666666666666666, x \cdot x, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \left(y \cdot y\right) \cdot \mathsf{fma}\left(0.0001984126984126984, \left(y \cdot y\right) \cdot \left(y \cdot y\right), 0.16666666666666666\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 52.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin x \leq -0.01:\\ \;\;\;\;x \cdot \left(\mathsf{fma}\left(y, y \cdot 0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(-0.16666666666666666, x \cdot x, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), x \cdot \left(y \cdot y\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sin x) -0.01)
   (*
    x
    (*
     (fma y (* y 0.16666666666666666) 1.0)
     (fma -0.16666666666666666 (* x x) 1.0)))
   (fma
    (fma (* y y) 0.008333333333333333 0.16666666666666666)
    (* x (* y y))
    x)))

C

double code(double x, double y) {
	double tmp;
	if (sin(x) <= -0.01) {
		tmp = x * (fma(y, (y * 0.16666666666666666), 1.0) * fma(-0.16666666666666666, (x * x), 1.0));
	} else {
		tmp = fma(fma((y * y), 0.008333333333333333, 0.16666666666666666), (x * (y * y)), x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (sin(x) <= -0.01)
		tmp = Float64(x * Float64(fma(y, Float64(y * 0.16666666666666666), 1.0) * fma(-0.16666666666666666, Float64(x * x), 1.0)));
	else
		tmp = fma(fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666), Float64(x * Float64(y * y)), x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[Sin[x], $MachinePrecision], -0.01], N[(x * N[(N[(y * N[(y * 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * N[(-0.16666666666666666 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision] * N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 x) < -0.0100000000000000002

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. +-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sin.f64 74.0

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

   5. Simplified 74.0%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt1-in N/A

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

      10. +-commutative N/A

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

   8. Simplified 21.7%

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

   if -0.0100000000000000002 < (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

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

      1. distribute-lft-in N/A

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

      2. associate-+r+ N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt1-in N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt-in N/A

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

      9. +-commutative N/A

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

      10. associate-+l+ N/A

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

      11. *-commutative N/A

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

      12. distribute-lft-in N/A

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

   5. Simplified 83.5%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 57.9

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

   8. Simplified 57.9%

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

4. Final simplification 48.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin x \leq -0.01:\\ \;\;\;\;x \cdot \left(\mathsf{fma}\left(y, y \cdot 0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(-0.16666666666666666, x \cdot x, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), x \cdot \left(y \cdot y\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 89.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1750:\\ \;\;\;\;\sin x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)\\ \mathbf{elif}\;y \leq 10^{+76}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot 0.16666666666666666, 1\right), \mathsf{fma}\left(x, -0.16666666666666666 \cdot \left(x \cdot x\right), x\right), x \cdot \left(\left(x \cdot x\right) \cdot \left(\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \left(\mathsf{fma}\left(y \cdot y, -3.306878306878307 \cdot 10^{-5}, -0.001388888888888889\right) + \frac{\mathsf{fma}\left(0.0001984126984126984, y \cdot y, 0.008333333333333333\right)}{x \cdot x}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \left(y \cdot \left(y \cdot \left(0.008333333333333333 \cdot \left(y \cdot y\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1750.0)
   (*
    (sin x)
    (fma (* y y) (fma y (* y 0.008333333333333333) 0.16666666666666666) 1.0))
   (if (<= y 1e+76)
     (fma
      (fma y (* y 0.16666666666666666) 1.0)
      (fma x (* -0.16666666666666666 (* x x)) x)
      (*
       x
       (*
        (* x x)
        (*
         (* (* y y) (* y y))
         (+
          (fma (* y y) -3.306878306878307e-5 -0.001388888888888889)
          (/
           (fma 0.0001984126984126984 (* y y) 0.008333333333333333)
           (* x x)))))))
     (* (sin x) (* y (* y (* 0.008333333333333333 (* y y))))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1750.0) {
		tmp = sin(x) * fma((y * y), fma(y, (y * 0.008333333333333333), 0.16666666666666666), 1.0);
	} else if (y <= 1e+76) {
		tmp = fma(fma(y, (y * 0.16666666666666666), 1.0), fma(x, (-0.16666666666666666 * (x * x)), x), (x * ((x * x) * (((y * y) * (y * y)) * (fma((y * y), -3.306878306878307e-5, -0.001388888888888889) + (fma(0.0001984126984126984, (y * y), 0.008333333333333333) / (x * x)))))));
	} else {
		tmp = sin(x) * (y * (y * (0.008333333333333333 * (y * y))));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1750.0)
		tmp = Float64(sin(x) * fma(Float64(y * y), fma(y, Float64(y * 0.008333333333333333), 0.16666666666666666), 1.0));
	elseif (y <= 1e+76)
		tmp = fma(fma(y, Float64(y * 0.16666666666666666), 1.0), fma(x, Float64(-0.16666666666666666 * Float64(x * x)), x), Float64(x * Float64(Float64(x * x) * Float64(Float64(Float64(y * y) * Float64(y * y)) * Float64(fma(Float64(y * y), -3.306878306878307e-5, -0.001388888888888889) + Float64(fma(0.0001984126984126984, Float64(y * y), 0.008333333333333333) / Float64(x * x)))))));
	else
		tmp = Float64(sin(x) * Float64(y * Float64(y * Float64(0.008333333333333333 * Float64(y * y)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1750.0], N[(N[Sin[x], $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1e+76], N[(N[(y * N[(y * 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x * N[(-0.16666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] + N[(x * N[(N[(x * x), $MachinePrecision] * N[(N[(N[(y * y), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(y * y), $MachinePrecision] * -3.306878306878307e-5 + -0.001388888888888889), $MachinePrecision] + N[(N[(0.0001984126984126984 * N[(y * y), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[x], $MachinePrecision] * N[(y * N[(y * N[(0.008333333333333333 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 1750

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. distribute-lft-in N/A

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

      2. associate-+r+ N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt1-in N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt-in N/A

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

      9. +-commutative N/A

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

      10. associate-+l+ N/A

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

      11. *-commutative N/A

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

      12. distribute-lft-in N/A

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

   5. Simplified 91.4%

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

   if 1750 < y < 1e76

   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. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 91.3

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

   5. Simplified 91.3%

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

   6. Taylor expanded in y around 0

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

   8. Simplified 40.9%

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

   9. Taylor expanded in y around 0

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

   11. Simplified 40.9%

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

   12. Taylor expanded in x around inf

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \frac{1}{6}, 1\right), \mathsf{fma}\left(x, \frac{-1}{6} \cdot \left(x \cdot x\right), x\right), \color{blue}{{x}^{3} \cdot \left(\frac{-1}{6} \cdot \left({y}^{4} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) + \frac{{y}^{4} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)}{{x}^{2}}\right)}\right) \]
   13. Step-by-step derivation

       1. cube-mult N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 N/A

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

       8. *-commutative N/A

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

       9. associate-*l* N/A

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

       10. *-commutative N/A

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

       11. associate-/l* N/A

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

   14. Simplified 56.9%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot 0.16666666666666666, 1\right), \mathsf{fma}\left(x, -0.16666666666666666 \cdot \left(x \cdot x\right), x\right), \color{blue}{x \cdot \left(\left(x \cdot x\right) \cdot \left(\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \left(\mathsf{fma}\left(y \cdot y, -3.306878306878307 \cdot 10^{-5}, -0.001388888888888889\right) + \frac{\mathsf{fma}\left(0.0001984126984126984, y \cdot y, 0.008333333333333333\right)}{x \cdot x}\right)\right)\right)}\right) \]

   if 1e76 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. distribute-lft-in N/A

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

      2. associate-+r+ N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt1-in N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt-in N/A

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

      9. +-commutative N/A

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

      10. associate-+l+ N/A

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

      11. *-commutative N/A

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

      12. distribute-lft-in N/A

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 100.0

          \[\leadsto \sin x \cdot \left(y \cdot \left(y \cdot \left(\color{blue}{\left(y \cdot y\right)} \cdot 0.008333333333333333\right)\right)\right) \]

   8. Simplified 100.0%

      \[\leadsto \color{blue}{\sin x \cdot \left(y \cdot \left(y \cdot \left(\left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1750:\\ \;\;\;\;\sin x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)\\ \mathbf{elif}\;y \leq 10^{+76}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot 0.16666666666666666, 1\right), \mathsf{fma}\left(x, -0.16666666666666666 \cdot \left(x \cdot x\right), x\right), x \cdot \left(\left(x \cdot x\right) \cdot \left(\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \left(\mathsf{fma}\left(y \cdot y, -3.306878306878307 \cdot 10^{-5}, -0.001388888888888889\right) + \frac{\mathsf{fma}\left(0.0001984126984126984, y \cdot y, 0.008333333333333333\right)}{x \cdot x}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \left(y \cdot \left(y \cdot \left(0.008333333333333333 \cdot \left(y \cdot y\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 83.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y, y \cdot 0.16666666666666666, 1\right)\\ \mathbf{if}\;y \leq 1750:\\ \;\;\;\;\sin x \cdot t\_0\\ \mathbf{elif}\;y \leq 10^{+76}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, \mathsf{fma}\left(x, -0.16666666666666666 \cdot \left(x \cdot x\right), x\right), x \cdot \left(\left(x \cdot x\right) \cdot \left(\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \left(\mathsf{fma}\left(y \cdot y, -3.306878306878307 \cdot 10^{-5}, -0.001388888888888889\right) + \frac{\mathsf{fma}\left(0.0001984126984126984, y \cdot y, 0.008333333333333333\right)}{x \cdot x}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \left(y \cdot \left(y \cdot \left(0.008333333333333333 \cdot \left(y \cdot y\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma y (* y 0.16666666666666666) 1.0)))
   (if (<= y 1750.0)
     (* (sin x) t_0)
     (if (<= y 1e+76)
       (fma
        t_0
        (fma x (* -0.16666666666666666 (* x x)) x)
        (*
         x
         (*
          (* x x)
          (*
           (* (* y y) (* y y))
           (+
            (fma (* y y) -3.306878306878307e-5 -0.001388888888888889)
            (/
             (fma 0.0001984126984126984 (* y y) 0.008333333333333333)
             (* x x)))))))
       (* (sin x) (* y (* y (* 0.008333333333333333 (* y y)))))))))

C

double code(double x, double y) {
	double t_0 = fma(y, (y * 0.16666666666666666), 1.0);
	double tmp;
	if (y <= 1750.0) {
		tmp = sin(x) * t_0;
	} else if (y <= 1e+76) {
		tmp = fma(t_0, fma(x, (-0.16666666666666666 * (x * x)), x), (x * ((x * x) * (((y * y) * (y * y)) * (fma((y * y), -3.306878306878307e-5, -0.001388888888888889) + (fma(0.0001984126984126984, (y * y), 0.008333333333333333) / (x * x)))))));
	} else {
		tmp = sin(x) * (y * (y * (0.008333333333333333 * (y * y))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(y, Float64(y * 0.16666666666666666), 1.0)
	tmp = 0.0
	if (y <= 1750.0)
		tmp = Float64(sin(x) * t_0);
	elseif (y <= 1e+76)
		tmp = fma(t_0, fma(x, Float64(-0.16666666666666666 * Float64(x * x)), x), Float64(x * Float64(Float64(x * x) * Float64(Float64(Float64(y * y) * Float64(y * y)) * Float64(fma(Float64(y * y), -3.306878306878307e-5, -0.001388888888888889) + Float64(fma(0.0001984126984126984, Float64(y * y), 0.008333333333333333) / Float64(x * x)))))));
	else
		tmp = Float64(sin(x) * Float64(y * Float64(y * Float64(0.008333333333333333 * Float64(y * y)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(y * 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[y, 1750.0], N[(N[Sin[x], $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[y, 1e+76], N[(t$95$0 * N[(x * N[(-0.16666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] + N[(x * N[(N[(x * x), $MachinePrecision] * N[(N[(N[(y * y), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(y * y), $MachinePrecision] * -3.306878306878307e-5 + -0.001388888888888889), $MachinePrecision] + N[(N[(0.0001984126984126984 * N[(y * y), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[x], $MachinePrecision] * N[(y * N[(y * N[(0.008333333333333333 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 1750

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. +-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sin.f64 83.2

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

   5. Simplified 83.2%

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

   if 1750 < y < 1e76

   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. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 91.3

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

   5. Simplified 91.3%

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

   6. Taylor expanded in y around 0

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

   8. Simplified 40.9%

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

   9. Taylor expanded in y around 0

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

   11. Simplified 40.9%

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

   12. Taylor expanded in x around inf

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \frac{1}{6}, 1\right), \mathsf{fma}\left(x, \frac{-1}{6} \cdot \left(x \cdot x\right), x\right), \color{blue}{{x}^{3} \cdot \left(\frac{-1}{6} \cdot \left({y}^{4} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) + \frac{{y}^{4} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)}{{x}^{2}}\right)}\right) \]
   13. Step-by-step derivation

       1. cube-mult N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 N/A

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

       8. *-commutative N/A

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

       9. associate-*l* N/A

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

       10. *-commutative N/A

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

       11. associate-/l* N/A

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

   14. Simplified 56.9%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot 0.16666666666666666, 1\right), \mathsf{fma}\left(x, -0.16666666666666666 \cdot \left(x \cdot x\right), x\right), \color{blue}{x \cdot \left(\left(x \cdot x\right) \cdot \left(\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \left(\mathsf{fma}\left(y \cdot y, -3.306878306878307 \cdot 10^{-5}, -0.001388888888888889\right) + \frac{\mathsf{fma}\left(0.0001984126984126984, y \cdot y, 0.008333333333333333\right)}{x \cdot x}\right)\right)\right)}\right) \]

   if 1e76 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. distribute-lft-in N/A

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

      2. associate-+r+ N/A

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

      3. associate-*r* N/A

         \[\leadsto \left(\sin x + {y}^{2} \cdot \color{blue}{\left(\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot \sin x\right)}\right) + {y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x\right) \]

      4. associate-*r* N/A

         \[\leadsto \left(\sin x + \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2}\right)\right) \cdot \sin x}\right) + {y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x\right) \]

      5. distribute-rgt1-in N/A

         \[\leadsto \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2}\right) + 1\right) \cdot \sin x} + {y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x\right) \]

      6. associate-*r* N/A

         \[\leadsto \left({y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2}\right) + 1\right) \cdot \sin x + \color{blue}{\left({y}^{2} \cdot \frac{1}{6}\right) \cdot \sin x} \]

      7. *-commutative N/A

         \[\leadsto \left({y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2}\right) + 1\right) \cdot \sin x + \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right)} \cdot \sin x \]

      8. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\sin x \cdot \left(\left({y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2}\right) + 1\right) + \frac{1}{6} \cdot {y}^{2}\right)} \]

      9. +-commutative N/A

         \[\leadsto \sin x \cdot \color{blue}{\left(\frac{1}{6} \cdot {y}^{2} + \left({y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2}\right) + 1\right)\right)} \]

      10. associate-+l+ N/A

          \[\leadsto \sin x \cdot \color{blue}{\left(\left(\frac{1}{6} \cdot {y}^{2} + {y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2}\right)\right) + 1\right)} \]

      11. *-commutative N/A

          \[\leadsto \sin x \cdot \left(\left(\color{blue}{{y}^{2} \cdot \frac{1}{6}} + {y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2}\right)\right) + 1\right) \]

      12. distribute-lft-in N/A

          \[\leadsto \sin x \cdot \left(\color{blue}{{y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)} + 1\right) \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\sin x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)} \]

   6. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{1}{120} \cdot \left({y}^{4} \cdot \sin x\right)} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{120} \cdot {y}^{4}\right) \cdot \sin x} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\sin x \cdot \left(\frac{1}{120} \cdot {y}^{4}\right)} \]

      3. metadata-eval N/A

         \[\leadsto \sin x \cdot \left(\frac{1}{120} \cdot {y}^{\color{blue}{\left(2 \cdot 2\right)}}\right) \]

      4. pow-sqr N/A

         \[\leadsto \sin x \cdot \left(\frac{1}{120} \cdot \color{blue}{\left({y}^{2} \cdot {y}^{2}\right)}\right) \]

      5. associate-*l* N/A

         \[\leadsto \sin x \cdot \color{blue}{\left(\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot {y}^{2}\right)} \]

      6. *-commutative N/A

         \[\leadsto \sin x \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2}\right)\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sin x \cdot \left({y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2}\right)\right)} \]

      8. lower-sin.f64 N/A

         \[\leadsto \color{blue}{\sin x} \cdot \left({y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \sin x \cdot \left(\color{blue}{\left(y \cdot y\right)} \cdot \left(\frac{1}{120} \cdot {y}^{2}\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \sin x \cdot \color{blue}{\left(y \cdot \left(y \cdot \left(\frac{1}{120} \cdot {y}^{2}\right)\right)\right)} \]

      11. *-commutative N/A

          \[\leadsto \sin x \cdot \left(y \cdot \color{blue}{\left(\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot y\right)}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \sin x \cdot \color{blue}{\left(y \cdot \left(\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot y\right)\right)} \]

      13. *-commutative N/A

          \[\leadsto \sin x \cdot \left(y \cdot \color{blue}{\left(y \cdot \left(\frac{1}{120} \cdot {y}^{2}\right)\right)}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \sin x \cdot \left(y \cdot \color{blue}{\left(y \cdot \left(\frac{1}{120} \cdot {y}^{2}\right)\right)}\right) \]

      15. *-commutative N/A

          \[\leadsto \sin x \cdot \left(y \cdot \left(y \cdot \color{blue}{\left({y}^{2} \cdot \frac{1}{120}\right)}\right)\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \sin x \cdot \left(y \cdot \left(y \cdot \color{blue}{\left({y}^{2} \cdot \frac{1}{120}\right)}\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \sin x \cdot \left(y \cdot \left(y \cdot \left(\color{blue}{\left(y \cdot y\right)} \cdot \frac{1}{120}\right)\right)\right) \]

      18. lower-*.f64 100.0

          \[\leadsto \sin x \cdot \left(y \cdot \left(y \cdot \left(\color{blue}{\left(y \cdot y\right)} \cdot 0.008333333333333333\right)\right)\right) \]

   8. Simplified 100.0%

      \[\leadsto \color{blue}{\sin x \cdot \left(y \cdot \left(y \cdot \left(\left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1750:\\ \;\;\;\;\sin x \cdot \mathsf{fma}\left(y, y \cdot 0.16666666666666666, 1\right)\\ \mathbf{elif}\;y \leq 10^{+76}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot 0.16666666666666666, 1\right), \mathsf{fma}\left(x, -0.16666666666666666 \cdot \left(x \cdot x\right), x\right), x \cdot \left(\left(x \cdot x\right) \cdot \left(\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \left(\mathsf{fma}\left(y \cdot y, -3.306878306878307 \cdot 10^{-5}, -0.001388888888888889\right) + \frac{\mathsf{fma}\left(0.0001984126984126984, y \cdot y, 0.008333333333333333\right)}{x \cdot x}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \left(y \cdot \left(y \cdot \left(0.008333333333333333 \cdot \left(y \cdot y\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 51.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin x \leq -0.01:\\ \;\;\;\;x \cdot \left(-0.16666666666666666 \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), x \cdot \left(y \cdot y\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sin x) -0.01)
   (* x (* -0.16666666666666666 (* x x)))
   (fma
    (fma (* y y) 0.008333333333333333 0.16666666666666666)
    (* x (* y y))
    x)))

C

double code(double x, double y) {
	double tmp;
	if (sin(x) <= -0.01) {
		tmp = x * (-0.16666666666666666 * (x * x));
	} else {
		tmp = fma(fma((y * y), 0.008333333333333333, 0.16666666666666666), (x * (y * y)), x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (sin(x) <= -0.01)
		tmp = Float64(x * Float64(-0.16666666666666666 * Float64(x * x)));
	else
		tmp = fma(fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666), Float64(x * Float64(y * y)), x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[Sin[x], $MachinePrecision], -0.01], N[(x * N[(-0.16666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision] * N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 x) < -0.0100000000000000002

   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. lower-sin.f64 51.0

         \[\leadsto \color{blue}{\sin x} \]

   5. Simplified 51.0%

      \[\leadsto \color{blue}{\sin x} \]

   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. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2} + 1\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{x \cdot \left(\frac{-1}{6} \cdot {x}^{2}\right) + x \cdot 1} \]

      3. *-rgt-identity N/A

         \[\leadsto x \cdot \left(\frac{-1}{6} \cdot {x}^{2}\right) + \color{blue}{x} \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{-1}{6} \cdot {x}^{2}, x\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{-1}{6} \cdot {x}^{2}}, x\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(x, \frac{-1}{6} \cdot \color{blue}{\left(x \cdot x\right)}, x\right) \]

      7. lower-*.f64 16.4

         \[\leadsto \mathsf{fma}\left(x, -0.16666666666666666 \cdot \color{blue}{\left(x \cdot x\right)}, x\right) \]

   8. Simplified 16.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, -0.16666666666666666 \cdot \left(x \cdot x\right), x\right)} \]

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{-1}{6} \cdot {x}^{3}} \]
   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. lower-*.f64 N/A

          \[\leadsto \color{blue}{x \cdot \left(\frac{-1}{6} \cdot {x}^{2}\right)} \]

       6. lower-*.f64 N/A

          \[\leadsto x \cdot \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2}\right)} \]

       7. unpow2 N/A

          \[\leadsto x \cdot \left(\frac{-1}{6} \cdot \color{blue}{\left(x \cdot x\right)}\right) \]

       8. lower-*.f64 16.4

          \[\leadsto x \cdot \left(-0.16666666666666666 \cdot \color{blue}{\left(x \cdot x\right)}\right) \]

   11. Simplified 16.4%

       \[\leadsto \color{blue}{x \cdot \left(-0.16666666666666666 \cdot \left(x \cdot x\right)\right)} \]

   if -0.0100000000000000002 < (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

      \[\leadsto \color{blue}{\sin x + {y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{6} \cdot \sin x\right)} \]
   4. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \sin x + \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right)\right) + {y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x\right)\right)} \]

      2. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\sin x + {y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right)\right)\right) + {y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x\right)} \]

      3. associate-*r* N/A

         \[\leadsto \left(\sin x + {y}^{2} \cdot \color{blue}{\left(\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot \sin x\right)}\right) + {y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x\right) \]

      4. associate-*r* N/A

         \[\leadsto \left(\sin x + \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2}\right)\right) \cdot \sin x}\right) + {y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x\right) \]

      5. distribute-rgt1-in N/A

         \[\leadsto \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2}\right) + 1\right) \cdot \sin x} + {y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x\right) \]

      6. associate-*r* N/A

         \[\leadsto \left({y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2}\right) + 1\right) \cdot \sin x + \color{blue}{\left({y}^{2} \cdot \frac{1}{6}\right) \cdot \sin x} \]

      7. *-commutative N/A

         \[\leadsto \left({y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2}\right) + 1\right) \cdot \sin x + \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right)} \cdot \sin x \]

      8. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\sin x \cdot \left(\left({y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2}\right) + 1\right) + \frac{1}{6} \cdot {y}^{2}\right)} \]

      9. +-commutative N/A

         \[\leadsto \sin x \cdot \color{blue}{\left(\frac{1}{6} \cdot {y}^{2} + \left({y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2}\right) + 1\right)\right)} \]

      10. associate-+l+ N/A

          \[\leadsto \sin x \cdot \color{blue}{\left(\left(\frac{1}{6} \cdot {y}^{2} + {y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2}\right)\right) + 1\right)} \]

      11. *-commutative N/A

          \[\leadsto \sin x \cdot \left(\left(\color{blue}{{y}^{2} \cdot \frac{1}{6}} + {y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2}\right)\right) + 1\right) \]

      12. distribute-lft-in N/A

          \[\leadsto \sin x \cdot \left(\color{blue}{{y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)} + 1\right) \]

   5. Simplified 83.5%

      \[\leadsto \color{blue}{\sin x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) \cdot x} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) + 1\right)} \cdot x \]

      3. distribute-lft1-in N/A

         \[\leadsto \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) \cdot x + x} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) \cdot {y}^{2}\right)} \cdot x + x \]

      5. associate-*l* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) \cdot \left({y}^{2} \cdot x\right)} + x \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}, {y}^{2} \cdot x, x\right)} \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{120} \cdot {y}^{2} + \frac{1}{6}}, {y}^{2} \cdot x, x\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{y}^{2} \cdot \frac{1}{120}} + \frac{1}{6}, {y}^{2} \cdot x, x\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left({y}^{2}, \frac{1}{120}, \frac{1}{6}\right)}, {y}^{2} \cdot x, x\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{120}, \frac{1}{6}\right), {y}^{2} \cdot x, x\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{120}, \frac{1}{6}\right), {y}^{2} \cdot x, x\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{1}{120}, \frac{1}{6}\right), \color{blue}{x \cdot {y}^{2}}, x\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{1}{120}, \frac{1}{6}\right), \color{blue}{x \cdot {y}^{2}}, x\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{1}{120}, \frac{1}{6}\right), x \cdot \color{blue}{\left(y \cdot y\right)}, x\right) \]

      15. lower-*.f64 57.9

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), x \cdot \color{blue}{\left(y \cdot y\right)}, x\right) \]

   8. Simplified 57.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), x \cdot \left(y \cdot y\right), x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 34.2% accurate, 12.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x, -0.16666666666666666 \cdot \left(x \cdot x\right), x\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (fma x (* -0.16666666666666666 (* x x)) x))

C

double code(double x, double y) {
	return fma(x, (-0.16666666666666666 * (x * x)), x);
}

Julia

function code(x, y)
	return fma(x, Float64(-0.16666666666666666 * Float64(x * x)), x)
end

Wolfram

code[x_, y_] := N[(x * N[(-0.16666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $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. lower-sin.f64 48.7

      \[\leadsto \color{blue}{\sin x} \]

5. Simplified 48.7%

   \[\leadsto \color{blue}{\sin x} \]

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. +-commutative N/A

      \[\leadsto x \cdot \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2} + 1\right)} \]

   2. distribute-lft-in N/A

      \[\leadsto \color{blue}{x \cdot \left(\frac{-1}{6} \cdot {x}^{2}\right) + x \cdot 1} \]

   3. *-rgt-identity N/A

      \[\leadsto x \cdot \left(\frac{-1}{6} \cdot {x}^{2}\right) + \color{blue}{x} \]

   4. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{-1}{6} \cdot {x}^{2}, x\right)} \]

   5. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{-1}{6} \cdot {x}^{2}}, x\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(x, \frac{-1}{6} \cdot \color{blue}{\left(x \cdot x\right)}, x\right) \]

   7. lower-*.f64 30.7

      \[\leadsto \mathsf{fma}\left(x, -0.16666666666666666 \cdot \color{blue}{\left(x \cdot x\right)}, x\right) \]

8. Simplified 30.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, -0.16666666666666666 \cdot \left(x \cdot x\right), x\right)} \]
9. Add Preprocessing

Alternative 21: 10.5% accurate, 13.6× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(-0.16666666666666666 \cdot \left(x \cdot x\right)\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* x (* -0.16666666666666666 (* x x))))

C

double code(double x, double y) {
	return x * (-0.16666666666666666 * (x * x));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x * ((-0.16666666666666666d0) * (x * x))
end function

Java

public static double code(double x, double y) {
	return x * (-0.16666666666666666 * (x * x));
}

Python

def code(x, y):
	return x * (-0.16666666666666666 * (x * x))

Julia

function code(x, y)
	return Float64(x * Float64(-0.16666666666666666 * Float64(x * x)))
end

MATLAB

function tmp = code(x, y)
	tmp = x * (-0.16666666666666666 * (x * x));
end

Wolfram

code[x_, y_] := N[(x * N[(-0.16666666666666666 * N[(x * x), $MachinePrecision]), $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. lower-sin.f64 48.7

      \[\leadsto \color{blue}{\sin x} \]

5. Simplified 48.7%

   \[\leadsto \color{blue}{\sin x} \]

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. +-commutative N/A

      \[\leadsto x \cdot \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2} + 1\right)} \]

   2. distribute-lft-in N/A

      \[\leadsto \color{blue}{x \cdot \left(\frac{-1}{6} \cdot {x}^{2}\right) + x \cdot 1} \]

   3. *-rgt-identity N/A

      \[\leadsto x \cdot \left(\frac{-1}{6} \cdot {x}^{2}\right) + \color{blue}{x} \]

   4. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{-1}{6} \cdot {x}^{2}, x\right)} \]

   5. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{-1}{6} \cdot {x}^{2}}, x\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(x, \frac{-1}{6} \cdot \color{blue}{\left(x \cdot x\right)}, x\right) \]

   7. lower-*.f64 30.7

      \[\leadsto \mathsf{fma}\left(x, -0.16666666666666666 \cdot \color{blue}{\left(x \cdot x\right)}, x\right) \]

8. Simplified 30.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, -0.16666666666666666 \cdot \left(x \cdot x\right), x\right)} \]

9. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{-1}{6} \cdot {x}^{3}} \]
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. lower-*.f64 N/A

       \[\leadsto \color{blue}{x \cdot \left(\frac{-1}{6} \cdot {x}^{2}\right)} \]

    6. lower-*.f64 N/A

       \[\leadsto x \cdot \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2}\right)} \]

    7. unpow2 N/A

       \[\leadsto x \cdot \left(\frac{-1}{6} \cdot \color{blue}{\left(x \cdot x\right)}\right) \]

    8. lower-*.f64 10.2

       \[\leadsto x \cdot \left(-0.16666666666666666 \cdot \color{blue}{\left(x \cdot x\right)}\right) \]

11. Simplified 10.2%

    \[\leadsto \color{blue}{x \cdot \left(-0.16666666666666666 \cdot \left(x \cdot x\right)\right)} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024215 
(FPCore (x y)
  :name "Linear.Quaternion:$ccos from linear-1.19.1.3"
  :precision binary64
  (* (sin x) (/ (sinh y) y)))