Linear.Quaternion:$ccos from linear-1.19.1.3

Percentage Accurate: 100.0% → 100.0%

Time: 5.7s

Alternatives: 16

Speedup: 1.0×

• Could not converge on a ground truth

  1. x = 1.821131836098558e+256
  2. y = 4.929326243201338e+258

• 48.6% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Final simplification 100.0%

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

Alternative 2: 62.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y}{y} \cdot \sin x\\ t_1 := \left(x \cdot x\right) \cdot x\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(t\_1, \mathsf{fma}\left(-0.0001984126984126984 \cdot \left(x \cdot x\right), x \cdot x, -0.16666666666666666\right), x\right)\\ \mathbf{elif}\;t\_0 \leq 500:\\ \;\;\;\;1 \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, \mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right), x\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (/ (sinh y) y) (sin x))) (t_1 (* (* x x) x)))
   (if (<= t_0 (- INFINITY))
     (*
      1.0
      (fma
       t_1
       (fma (* -0.0001984126984126984 (* x x)) (* x x) -0.16666666666666666)
       x))
     (if (<= t_0 500.0)
       (* 1.0 (sin x))
       (*
        (fma t_1 (fma 0.008333333333333333 (* x x) -0.16666666666666666) x)
        1.0)))))

C

double code(double x, double y) {
	double t_0 = (sinh(y) / y) * sin(x);
	double t_1 = (x * x) * x;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = 1.0 * fma(t_1, fma((-0.0001984126984126984 * (x * x)), (x * x), -0.16666666666666666), x);
	} else if (t_0 <= 500.0) {
		tmp = 1.0 * sin(x);
	} else {
		tmp = fma(t_1, fma(0.008333333333333333, (x * x), -0.16666666666666666), x) * 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(sinh(y) / y) * sin(x))
	t_1 = Float64(Float64(x * x) * x)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(1.0 * fma(t_1, fma(Float64(-0.0001984126984126984 * Float64(x * x)), Float64(x * x), -0.16666666666666666), x));
	elseif (t_0 <= 500.0)
		tmp = Float64(1.0 * sin(x));
	else
		tmp = Float64(fma(t_1, fma(0.008333333333333333, Float64(x * x), -0.16666666666666666), x) * 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(1.0 * N[(t$95$1 * N[(N[(-0.0001984126984126984 * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 500.0], N[(1.0 * N[Sin[x], $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 * N[(0.008333333333333333 * N[(x * x), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] + x), $MachinePrecision] * 1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \sin x \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 2.8%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. pow-plus N/A

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

      8. lower-pow.f64 N/A

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

      9. metadata-eval N/A

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

      10. sub-neg N/A

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

      11. *-commutative N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 19.2

          \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, x \cdot x, 0.008333333333333333\right), \color{blue}{x \cdot x}, -0.16666666666666666\right), x\right) \cdot 1 \]

   8. Applied rewrites 19.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, x \cdot x, 0.008333333333333333\right), x \cdot x, -0.16666666666666666\right), x\right)} \cdot 1 \]
   9. Step-by-step derivation

   10. Applied rewrites 19.2%

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

   11. Taylor expanded in x around inf

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

   13. Applied rewrites 19.2%

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

   if -inf.0 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < 500

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

      \[\leadsto \sin x \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 96.0%

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

   if 500 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \sin x \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 2.6%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. pow-plus N/A

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

      8. lower-pow.f64 N/A

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

      9. metadata-eval N/A

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

      10. sub-neg N/A

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

      11. *-commutative N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 19.6

          \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, x \cdot x, 0.008333333333333333\right), \color{blue}{x \cdot x}, -0.16666666666666666\right), x\right) \cdot 1 \]

   8. Applied rewrites 19.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, x \cdot x, 0.008333333333333333\right), x \cdot x, -0.16666666666666666\right), x\right)} \cdot 1 \]
   9. Step-by-step derivation

   10. Applied rewrites 19.6%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 18.3%

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

4. Final simplification 58.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y}{y} \cdot \sin x \leq -\infty:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(\left(x \cdot x\right) \cdot x, \mathsf{fma}\left(-0.0001984126984126984 \cdot \left(x \cdot x\right), x \cdot x, -0.16666666666666666\right), x\right)\\ \mathbf{elif}\;\frac{\sinh y}{y} \cdot \sin x \leq 500:\\ \;\;\;\;1 \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(x \cdot x\right) \cdot x, \mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right), x\right) \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right)\\ \mathbf{if}\;\frac{\sinh y}{y} \cdot \sin x \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left({x}^{3}, -0.16666666666666666, x\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \sin x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma (* y y) 0.16666666666666666 1.0)))
   (if (<= (* (/ (sinh y) y) (sin x)) (- INFINITY))
     (* (fma (pow x 3.0) -0.16666666666666666 x) t_0)
     (* t_0 (sin x)))))

C

double code(double x, double y) {
	double t_0 = fma((y * y), 0.16666666666666666, 1.0);
	double tmp;
	if (((sinh(y) / y) * sin(x)) <= -((double) INFINITY)) {
		tmp = fma(pow(x, 3.0), -0.16666666666666666, x) * t_0;
	} else {
		tmp = t_0 * sin(x);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(Float64(y * y), 0.16666666666666666, 1.0)
	tmp = 0.0
	if (Float64(Float64(sinh(y) / y) * sin(x)) <= Float64(-Inf))
		tmp = Float64(fma((x ^ 3.0), -0.16666666666666666, x) * t_0);
	else
		tmp = Float64(t_0 * sin(x));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * y), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision], (-Infinity)], N[(N[(N[Power[x, 3.0], $MachinePrecision] * -0.16666666666666666 + x), $MachinePrecision] * t$95$0), $MachinePrecision], N[(t$95$0 * N[Sin[x], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 60.6

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

   5. Applied rewrites 60.6%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. unpow2 N/A

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

      7. cube-mult N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-pow.f64 56.5

         \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{3}}, -0.16666666666666666, x\right) \cdot \mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right) \]

   8. Applied rewrites 56.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, -0.16666666666666666, x\right)} \cdot \mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right) \]

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 81.3

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

   5. Applied rewrites 81.3%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y}{y} \cdot \sin x \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left({x}^{3}, -0.16666666666666666, x\right) \cdot \mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right) \cdot \sin x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 69.8% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \sin x \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 2.8%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. pow-plus N/A

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

      8. lower-pow.f64 N/A

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

      9. metadata-eval N/A

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

      10. sub-neg N/A

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

      11. *-commutative N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 19.2

          \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, x \cdot x, 0.008333333333333333\right), \color{blue}{x \cdot x}, -0.16666666666666666\right), x\right) \cdot 1 \]

   8. Applied rewrites 19.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, x \cdot x, 0.008333333333333333\right), x \cdot x, -0.16666666666666666\right), x\right)} \cdot 1 \]
   9. Step-by-step derivation

   10. Applied rewrites 19.2%

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

   11. Taylor expanded in x around inf

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

   13. Applied rewrites 19.2%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 81.3

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

   5. Applied rewrites 81.3%

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

4. Final simplification 66.0%

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

Alternative 5: 36.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x \cdot x\right) \cdot x\\ \mathbf{if}\;\frac{\sinh y}{y} \cdot \sin x \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(t\_0, \mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, x \cdot x, 0.008333333333333333\right), x \cdot x, -0.16666666666666666\right), x\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, \mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right), x\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* x x) x)))
   (if (<= (* (/ (sinh y) y) (sin x)) -0.01)
     (*
      (fma
       t_0
       (fma
        (fma -0.0001984126984126984 (* x x) 0.008333333333333333)
        (* x x)
        -0.16666666666666666)
       x)
      1.0)
     (*
      (fma t_0 (fma 0.008333333333333333 (* x x) -0.16666666666666666) x)
      1.0))))

C

double code(double x, double y) {
	double t_0 = (x * x) * x;
	double tmp;
	if (((sinh(y) / y) * sin(x)) <= -0.01) {
		tmp = fma(t_0, fma(fma(-0.0001984126984126984, (x * x), 0.008333333333333333), (x * x), -0.16666666666666666), x) * 1.0;
	} else {
		tmp = fma(t_0, fma(0.008333333333333333, (x * x), -0.16666666666666666), x) * 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(x * x) * x)
	tmp = 0.0
	if (Float64(Float64(sinh(y) / y) * sin(x)) <= -0.01)
		tmp = Float64(fma(t_0, fma(fma(-0.0001984126984126984, Float64(x * x), 0.008333333333333333), Float64(x * x), -0.16666666666666666), x) * 1.0);
	else
		tmp = Float64(fma(t_0, fma(0.008333333333333333, Float64(x * x), -0.16666666666666666), x) * 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision], -0.01], N[(N[(t$95$0 * N[(N[(-0.0001984126984126984 * N[(x * x), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] + x), $MachinePrecision] * 1.0), $MachinePrecision], N[(N[(t$95$0 * N[(0.008333333333333333 * N[(x * x), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] + x), $MachinePrecision] * 1.0), $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

      \[\leadsto \sin x \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 35.5%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. pow-plus N/A

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

      8. lower-pow.f64 N/A

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

      9. metadata-eval N/A

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

      10. sub-neg N/A

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

      11. *-commutative N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 13.1

          \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, x \cdot x, 0.008333333333333333\right), \color{blue}{x \cdot x}, -0.16666666666666666\right), x\right) \cdot 1 \]

   8. Applied rewrites 13.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, x \cdot x, 0.008333333333333333\right), x \cdot x, -0.16666666666666666\right), x\right)} \cdot 1 \]
   9. Step-by-step derivation

   10. Applied rewrites 13.1%

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

   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

      \[\leadsto \sin x \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 60.7%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. pow-plus N/A

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

      8. lower-pow.f64 N/A

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

      9. metadata-eval N/A

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

      10. sub-neg N/A

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

      11. *-commutative N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 43.4

          \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, x \cdot x, 0.008333333333333333\right), \color{blue}{x \cdot x}, -0.16666666666666666\right), x\right) \cdot 1 \]

   8. Applied rewrites 43.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, x \cdot x, 0.008333333333333333\right), x \cdot x, -0.16666666666666666\right), x\right)} \cdot 1 \]
   9. Step-by-step derivation

   10. Applied rewrites 43.4%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 42.8%

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

4. Final simplification 31.6%

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

Alternative 6: 35.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

      \[\leadsto \sin x \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 35.5%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. pow-plus N/A

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

      9. lower-pow.f64 N/A

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

      10. metadata-eval 10.3

          \[\leadsto \mathsf{fma}\left({x}^{\color{blue}{3}}, -0.16666666666666666, x\right) \cdot 1 \]

   8. Applied rewrites 10.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, -0.16666666666666666, x\right)} \cdot 1 \]
   9. Step-by-step derivation

   10. Applied rewrites 10.3%

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

   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

      \[\leadsto \sin x \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 60.7%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. pow-plus N/A

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

      8. lower-pow.f64 N/A

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

      9. metadata-eval N/A

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

      10. sub-neg N/A

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

      11. *-commutative N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 43.4

          \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, x \cdot x, 0.008333333333333333\right), \color{blue}{x \cdot x}, -0.16666666666666666\right), x\right) \cdot 1 \]

   8. Applied rewrites 43.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, x \cdot x, 0.008333333333333333\right), x \cdot x, -0.16666666666666666\right), x\right)} \cdot 1 \]
   9. Step-by-step derivation

   10. Applied rewrites 43.4%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 42.8%

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

4. Final simplification 30.5%

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

Alternative 7: 95.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.48:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984, y \cdot y, 0.008333333333333333\right), y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot \sin x\\ \mathbf{elif}\;y \leq 9.6 \cdot 10^{+51}:\\ \;\;\;\;\left(0.5 \cdot \sinh y\right) \cdot \left(\mathsf{fma}\left(-0.3333333333333333 \cdot x, x, 2\right) \cdot \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right) \cdot y\right) \cdot y, y \cdot y, 1\right) \cdot \sin x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.48)
   (*
    (fma
     (fma
      (fma 0.0001984126984126984 (* y y) 0.008333333333333333)
      (* y y)
      0.16666666666666666)
     (* y y)
     1.0)
    (sin x))
   (if (<= y 9.6e+51)
     (* (* 0.5 (sinh y)) (* (fma (* -0.3333333333333333 x) x 2.0) (/ x y)))
     (*
      (fma
       (* (* (fma (* y y) 0.0001984126984126984 0.008333333333333333) y) y)
       (* y y)
       1.0)
      (sin x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.48) {
		tmp = fma(fma(fma(0.0001984126984126984, (y * y), 0.008333333333333333), (y * y), 0.16666666666666666), (y * y), 1.0) * sin(x);
	} else if (y <= 9.6e+51) {
		tmp = (0.5 * sinh(y)) * (fma((-0.3333333333333333 * x), x, 2.0) * (x / y));
	} else {
		tmp = fma(((fma((y * y), 0.0001984126984126984, 0.008333333333333333) * y) * y), (y * y), 1.0) * sin(x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.48)
		tmp = Float64(fma(fma(fma(0.0001984126984126984, Float64(y * y), 0.008333333333333333), Float64(y * y), 0.16666666666666666), Float64(y * y), 1.0) * sin(x));
	elseif (y <= 9.6e+51)
		tmp = Float64(Float64(0.5 * sinh(y)) * Float64(fma(Float64(-0.3333333333333333 * x), x, 2.0) * Float64(x / y)));
	else
		tmp = Float64(fma(Float64(Float64(fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333) * y) * y), Float64(y * y), 1.0) * sin(x));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.48], N[(N[(N[(N[(0.0001984126984126984 * N[(y * y), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] * N[(y * y), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 9.6e+51], N[(N[(0.5 * N[Sinh[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(-0.3333333333333333 * x), $MachinePrecision] * x + 2.0), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] * y), $MachinePrecision] * y), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.48

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 93.9

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

   5. Applied rewrites 93.9%

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

   if 1.48 < y < 9.5999999999999994e51

   1. Initial program 99.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sinh.f64 N/A

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

      4. sinh-def N/A

         \[\leadsto \sin x \cdot \frac{\color{blue}{\frac{e^{y} - e^{\mathsf{neg}\left(y\right)}}{2}}}{y} \]

      5. associate-/l/ N/A

         \[\leadsto \sin x \cdot \color{blue}{\frac{e^{y} - e^{\mathsf{neg}\left(y\right)}}{y \cdot 2}} \]

      6. sinh-undef N/A

         \[\leadsto \sin x \cdot \frac{\color{blue}{2 \cdot \sinh y}}{y \cdot 2} \]

      7. lift-sinh.f64 N/A

         \[\leadsto \sin x \cdot \frac{2 \cdot \color{blue}{\sinh y}}{y \cdot 2} \]

      8. times-frac N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-sinh.f64 N/A

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

      14. sinh-def N/A

          \[\leadsto \left(\sin x \cdot \frac{2}{y}\right) \cdot \frac{\color{blue}{\frac{e^{y} - e^{\mathsf{neg}\left(y\right)}}{2}}}{2} \]

      15. clear-num N/A

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

      16. associate-/r/ N/A

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

      17. associate-/l* N/A

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

      18. sinh-def N/A

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

      19. lift-sinh.f64 N/A

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

      20. lower-*.f64 N/A

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

      21. metadata-eval 100.0

          \[\leadsto \left(\sin x \cdot \frac{2}{y}\right) \cdot \left(\color{blue}{0.5} \cdot \sinh y\right) \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\left(\sin x \cdot \frac{2}{y}\right) \cdot \left(0.5 \cdot \sinh y\right)} \]

   5. Taylor expanded in x around 0

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

      1. distribute-rgt-in N/A

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

      2. associate-*r/ N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. associate-*l* N/A

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

      6. associate-*l/ N/A

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

      7. *-lft-identity N/A

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

      8. distribute-rgt-out N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 77.0

          \[\leadsto \left(\frac{x}{y} \cdot \mathsf{fma}\left(\color{blue}{-0.3333333333333333 \cdot x}, x, 2\right)\right) \cdot \left(0.5 \cdot \sinh y\right) \]

   7. Applied rewrites 77.0%

      \[\leadsto \color{blue}{\left(\frac{x}{y} \cdot \mathsf{fma}\left(-0.3333333333333333 \cdot x, x, 2\right)\right)} \cdot \left(0.5 \cdot \sinh y\right) \]

   if 9.5999999999999994e51 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 100.0%

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

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.48:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984, y \cdot y, 0.008333333333333333\right), y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot \sin x\\ \mathbf{elif}\;y \leq 9.6 \cdot 10^{+51}:\\ \;\;\;\;\left(0.5 \cdot \sinh y\right) \cdot \left(\mathsf{fma}\left(-0.3333333333333333 \cdot x, x, 2\right) \cdot \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right) \cdot y\right) \cdot y, y \cdot y, 1\right) \cdot \sin x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 93.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.38:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right) \cdot y, y, 1\right) \cdot \sin x\\ \mathbf{elif}\;y \leq 9.6 \cdot 10^{+51}:\\ \;\;\;\;\left(0.5 \cdot \sinh y\right) \cdot \left(\mathsf{fma}\left(-0.3333333333333333 \cdot x, x, 2\right) \cdot \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right) \cdot y\right) \cdot y, y \cdot y, 1\right) \cdot \sin x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.38)
   (*
    (fma (* (fma 0.008333333333333333 (* y y) 0.16666666666666666) y) y 1.0)
    (sin x))
   (if (<= y 9.6e+51)
     (* (* 0.5 (sinh y)) (* (fma (* -0.3333333333333333 x) x 2.0) (/ x y)))
     (*
      (fma
       (* (* (fma (* y y) 0.0001984126984126984 0.008333333333333333) y) y)
       (* y y)
       1.0)
      (sin x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.38) {
		tmp = fma((fma(0.008333333333333333, (y * y), 0.16666666666666666) * y), y, 1.0) * sin(x);
	} else if (y <= 9.6e+51) {
		tmp = (0.5 * sinh(y)) * (fma((-0.3333333333333333 * x), x, 2.0) * (x / y));
	} else {
		tmp = fma(((fma((y * y), 0.0001984126984126984, 0.008333333333333333) * y) * y), (y * y), 1.0) * sin(x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.38)
		tmp = Float64(fma(Float64(fma(0.008333333333333333, Float64(y * y), 0.16666666666666666) * y), y, 1.0) * sin(x));
	elseif (y <= 9.6e+51)
		tmp = Float64(Float64(0.5 * sinh(y)) * Float64(fma(Float64(-0.3333333333333333 * x), x, 2.0) * Float64(x / y)));
	else
		tmp = Float64(fma(Float64(Float64(fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333) * y) * y), Float64(y * y), 1.0) * sin(x));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.38], N[(N[(N[(N[(0.008333333333333333 * N[(y * y), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * y), $MachinePrecision] * y + 1.0), $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 9.6e+51], N[(N[(0.5 * N[Sinh[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(-0.3333333333333333 * x), $MachinePrecision] * x + 2.0), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] * y), $MachinePrecision] * y), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.3799999999999999

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 91.7

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

   5. Applied rewrites 91.7%

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

   7. Applied rewrites 91.7%

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

   if 1.3799999999999999 < y < 9.5999999999999994e51

   1. Initial program 99.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sinh.f64 N/A

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

      4. sinh-def N/A

         \[\leadsto \sin x \cdot \frac{\color{blue}{\frac{e^{y} - e^{\mathsf{neg}\left(y\right)}}{2}}}{y} \]

      5. associate-/l/ N/A

         \[\leadsto \sin x \cdot \color{blue}{\frac{e^{y} - e^{\mathsf{neg}\left(y\right)}}{y \cdot 2}} \]

      6. sinh-undef N/A

         \[\leadsto \sin x \cdot \frac{\color{blue}{2 \cdot \sinh y}}{y \cdot 2} \]

      7. lift-sinh.f64 N/A

         \[\leadsto \sin x \cdot \frac{2 \cdot \color{blue}{\sinh y}}{y \cdot 2} \]

      8. times-frac N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-sinh.f64 N/A

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

      14. sinh-def N/A

          \[\leadsto \left(\sin x \cdot \frac{2}{y}\right) \cdot \frac{\color{blue}{\frac{e^{y} - e^{\mathsf{neg}\left(y\right)}}{2}}}{2} \]

      15. clear-num N/A

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

      16. associate-/r/ N/A

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

      17. associate-/l* N/A

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

      18. sinh-def N/A

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

      19. lift-sinh.f64 N/A

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

      20. lower-*.f64 N/A

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

      21. metadata-eval 100.0

          \[\leadsto \left(\sin x \cdot \frac{2}{y}\right) \cdot \left(\color{blue}{0.5} \cdot \sinh y\right) \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\left(\sin x \cdot \frac{2}{y}\right) \cdot \left(0.5 \cdot \sinh y\right)} \]

   5. Taylor expanded in x around 0

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

      1. distribute-rgt-in N/A

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

      2. associate-*r/ N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. associate-*l* N/A

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

      6. associate-*l/ N/A

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

      7. *-lft-identity N/A

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

      8. distribute-rgt-out N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 77.0

          \[\leadsto \left(\frac{x}{y} \cdot \mathsf{fma}\left(\color{blue}{-0.3333333333333333 \cdot x}, x, 2\right)\right) \cdot \left(0.5 \cdot \sinh y\right) \]

   7. Applied rewrites 77.0%

      \[\leadsto \color{blue}{\left(\frac{x}{y} \cdot \mathsf{fma}\left(-0.3333333333333333 \cdot x, x, 2\right)\right)} \cdot \left(0.5 \cdot \sinh y\right) \]

   if 9.5999999999999994e51 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 100.0%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.38:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right) \cdot y, y, 1\right) \cdot \sin x\\ \mathbf{elif}\;y \leq 9.6 \cdot 10^{+51}:\\ \;\;\;\;\left(0.5 \cdot \sinh y\right) \cdot \left(\mathsf{fma}\left(-0.3333333333333333 \cdot x, x, 2\right) \cdot \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right) \cdot y\right) \cdot y, y \cdot y, 1\right) \cdot \sin x\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 92.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.38:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right) \cdot y, y, 1\right) \cdot \sin x\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{+77}:\\ \;\;\;\;\left(0.5 \cdot \sinh y\right) \cdot \left(\mathsf{fma}\left(-0.3333333333333333 \cdot x, x, 2\right) \cdot \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right) \cdot y\right) \cdot y\right) \cdot \sin x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.38)
   (*
    (fma (* (fma 0.008333333333333333 (* y y) 0.16666666666666666) y) y 1.0)
    (sin x))
   (if (<= y 3.8e+77)
     (* (* 0.5 (sinh y)) (* (fma (* -0.3333333333333333 x) x 2.0) (/ x y)))
     (*
      (* (* (fma (* y y) 0.008333333333333333 0.16666666666666666) y) y)
      (sin x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.38) {
		tmp = fma((fma(0.008333333333333333, (y * y), 0.16666666666666666) * y), y, 1.0) * sin(x);
	} else if (y <= 3.8e+77) {
		tmp = (0.5 * sinh(y)) * (fma((-0.3333333333333333 * x), x, 2.0) * (x / y));
	} else {
		tmp = ((fma((y * y), 0.008333333333333333, 0.16666666666666666) * y) * y) * sin(x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.38)
		tmp = Float64(fma(Float64(fma(0.008333333333333333, Float64(y * y), 0.16666666666666666) * y), y, 1.0) * sin(x));
	elseif (y <= 3.8e+77)
		tmp = Float64(Float64(0.5 * sinh(y)) * Float64(fma(Float64(-0.3333333333333333 * x), x, 2.0) * Float64(x / y)));
	else
		tmp = Float64(Float64(Float64(fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666) * y) * y) * sin(x));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.38], N[(N[(N[(N[(0.008333333333333333 * N[(y * y), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * y), $MachinePrecision] * y + 1.0), $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.8e+77], N[(N[(0.5 * N[Sinh[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(-0.3333333333333333 * x), $MachinePrecision] * x + 2.0), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision] * y), $MachinePrecision] * y), $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.3799999999999999

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 91.7

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

   5. Applied rewrites 91.7%

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

   7. Applied rewrites 91.7%

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

   if 1.3799999999999999 < y < 3.8000000000000001e77

   1. Initial program 99.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sinh.f64 N/A

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

      4. sinh-def N/A

         \[\leadsto \sin x \cdot \frac{\color{blue}{\frac{e^{y} - e^{\mathsf{neg}\left(y\right)}}{2}}}{y} \]

      5. associate-/l/ N/A

         \[\leadsto \sin x \cdot \color{blue}{\frac{e^{y} - e^{\mathsf{neg}\left(y\right)}}{y \cdot 2}} \]

      6. sinh-undef N/A

         \[\leadsto \sin x \cdot \frac{\color{blue}{2 \cdot \sinh y}}{y \cdot 2} \]

      7. lift-sinh.f64 N/A

         \[\leadsto \sin x \cdot \frac{2 \cdot \color{blue}{\sinh y}}{y \cdot 2} \]

      8. times-frac N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-sinh.f64 N/A

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

      14. sinh-def N/A

          \[\leadsto \left(\sin x \cdot \frac{2}{y}\right) \cdot \frac{\color{blue}{\frac{e^{y} - e^{\mathsf{neg}\left(y\right)}}{2}}}{2} \]

      15. clear-num N/A

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

      16. associate-/r/ N/A

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

      17. associate-/l* N/A

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

      18. sinh-def N/A

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

      19. lift-sinh.f64 N/A

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

      20. lower-*.f64 N/A

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

      21. metadata-eval 94.4

          \[\leadsto \left(\sin x \cdot \frac{2}{y}\right) \cdot \left(\color{blue}{0.5} \cdot \sinh y\right) \]

   4. Applied rewrites 94.4%

      \[\leadsto \color{blue}{\left(\sin x \cdot \frac{2}{y}\right) \cdot \left(0.5 \cdot \sinh y\right)} \]

   5. Taylor expanded in x around 0

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

      1. distribute-rgt-in N/A

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

      2. associate-*r/ N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. associate-*l* N/A

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

      6. associate-*l/ N/A

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

      7. *-lft-identity N/A

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

      8. distribute-rgt-out N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 77.8

          \[\leadsto \left(\frac{x}{y} \cdot \mathsf{fma}\left(\color{blue}{-0.3333333333333333 \cdot x}, x, 2\right)\right) \cdot \left(0.5 \cdot \sinh y\right) \]

   7. Applied rewrites 77.8%

      \[\leadsto \color{blue}{\left(\frac{x}{y} \cdot \mathsf{fma}\left(-0.3333333333333333 \cdot x, x, 2\right)\right)} \cdot \left(0.5 \cdot \sinh y\right) \]

   if 3.8000000000000001e77 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 100.0%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.38:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right) \cdot y, y, 1\right) \cdot \sin x\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{+77}:\\ \;\;\;\;\left(0.5 \cdot \sinh y\right) \cdot \left(\mathsf{fma}\left(-0.3333333333333333 \cdot x, x, 2\right) \cdot \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right) \cdot y\right) \cdot y\right) \cdot \sin x\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 36.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x \cdot x\right) \cdot x\\ \mathbf{if}\;\sin x \leq 2 \cdot 10^{-295}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(t\_0, \mathsf{fma}\left(-0.0001984126984126984 \cdot \left(x \cdot x\right), x \cdot x, -0.16666666666666666\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, \mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right), x\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* x x) x)))
   (if (<= (sin x) 2e-295)
     (*
      1.0
      (fma
       t_0
       (fma (* -0.0001984126984126984 (* x x)) (* x x) -0.16666666666666666)
       x))
     (*
      (fma t_0 (fma 0.008333333333333333 (* x x) -0.16666666666666666) x)
      1.0))))

C

double code(double x, double y) {
	double t_0 = (x * x) * x;
	double tmp;
	if (sin(x) <= 2e-295) {
		tmp = 1.0 * fma(t_0, fma((-0.0001984126984126984 * (x * x)), (x * x), -0.16666666666666666), x);
	} else {
		tmp = fma(t_0, fma(0.008333333333333333, (x * x), -0.16666666666666666), x) * 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(x * x) * x)
	tmp = 0.0
	if (sin(x) <= 2e-295)
		tmp = Float64(1.0 * fma(t_0, fma(Float64(-0.0001984126984126984 * Float64(x * x)), Float64(x * x), -0.16666666666666666), x));
	else
		tmp = Float64(fma(t_0, fma(0.008333333333333333, Float64(x * x), -0.16666666666666666), x) * 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[N[Sin[x], $MachinePrecision], 2e-295], N[(1.0 * N[(t$95$0 * N[(N[(-0.0001984126984126984 * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * N[(0.008333333333333333 * N[(x * x), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] + x), $MachinePrecision] * 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 x) < 2.00000000000000012e-295

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \sin x \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 47.9%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. pow-plus N/A

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

      8. lower-pow.f64 N/A

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

      9. metadata-eval N/A

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

      10. sub-neg N/A

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

      11. *-commutative N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 30.8

          \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, x \cdot x, 0.008333333333333333\right), \color{blue}{x \cdot x}, -0.16666666666666666\right), x\right) \cdot 1 \]

   8. Applied rewrites 30.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, x \cdot x, 0.008333333333333333\right), x \cdot x, -0.16666666666666666\right), x\right)} \cdot 1 \]
   9. Step-by-step derivation

   10. Applied rewrites 30.8%

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

   11. Taylor expanded in x around inf

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

   13. Applied rewrites 30.8%

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

   if 2.00000000000000012e-295 < (sin.f64 x)

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \sin x \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 54.4%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. pow-plus N/A

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

      8. lower-pow.f64 N/A

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

      9. metadata-eval N/A

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

      10. sub-neg N/A

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

      11. *-commutative N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 33.1

          \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, x \cdot x, 0.008333333333333333\right), \color{blue}{x \cdot x}, -0.16666666666666666\right), x\right) \cdot 1 \]

   8. Applied rewrites 33.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, x \cdot x, 0.008333333333333333\right), x \cdot x, -0.16666666666666666\right), x\right)} \cdot 1 \]
   9. Step-by-step derivation

   10. Applied rewrites 33.1%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 32.4%

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

4. Final simplification 31.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin x \leq 2 \cdot 10^{-295}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(\left(x \cdot x\right) \cdot x, \mathsf{fma}\left(-0.0001984126984126984 \cdot \left(x \cdot x\right), x \cdot x, -0.16666666666666666\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(x \cdot x\right) \cdot x, \mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right), x\right) \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 92.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.97:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right) \cdot y, y, 1\right) \cdot \sin x\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{+77}:\\ \;\;\;\;\frac{2 \cdot x}{y} \cdot \left(0.5 \cdot \sinh y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right) \cdot y\right) \cdot y\right) \cdot \sin x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 0.97)
   (*
    (fma (* (fma 0.008333333333333333 (* y y) 0.16666666666666666) y) y 1.0)
    (sin x))
   (if (<= y 3.8e+77)
     (* (/ (* 2.0 x) y) (* 0.5 (sinh y)))
     (*
      (* (* (fma (* y y) 0.008333333333333333 0.16666666666666666) y) y)
      (sin x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 0.97) {
		tmp = fma((fma(0.008333333333333333, (y * y), 0.16666666666666666) * y), y, 1.0) * sin(x);
	} else if (y <= 3.8e+77) {
		tmp = ((2.0 * x) / y) * (0.5 * sinh(y));
	} else {
		tmp = ((fma((y * y), 0.008333333333333333, 0.16666666666666666) * y) * y) * sin(x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 0.97)
		tmp = Float64(fma(Float64(fma(0.008333333333333333, Float64(y * y), 0.16666666666666666) * y), y, 1.0) * sin(x));
	elseif (y <= 3.8e+77)
		tmp = Float64(Float64(Float64(2.0 * x) / y) * Float64(0.5 * sinh(y)));
	else
		tmp = Float64(Float64(Float64(fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666) * y) * y) * sin(x));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 0.97], N[(N[(N[(N[(0.008333333333333333 * N[(y * y), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * y), $MachinePrecision] * y + 1.0), $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.8e+77], N[(N[(N[(2.0 * x), $MachinePrecision] / y), $MachinePrecision] * N[(0.5 * N[Sinh[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision] * y), $MachinePrecision] * y), $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 0.96999999999999997

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 91.7

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

   5. Applied rewrites 91.7%

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

   7. Applied rewrites 91.7%

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

   if 0.96999999999999997 < y < 3.8000000000000001e77

   1. Initial program 99.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sinh.f64 N/A

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

      4. sinh-def N/A

         \[\leadsto \sin x \cdot \frac{\color{blue}{\frac{e^{y} - e^{\mathsf{neg}\left(y\right)}}{2}}}{y} \]

      5. associate-/l/ N/A

         \[\leadsto \sin x \cdot \color{blue}{\frac{e^{y} - e^{\mathsf{neg}\left(y\right)}}{y \cdot 2}} \]

      6. sinh-undef N/A

         \[\leadsto \sin x \cdot \frac{\color{blue}{2 \cdot \sinh y}}{y \cdot 2} \]

      7. lift-sinh.f64 N/A

         \[\leadsto \sin x \cdot \frac{2 \cdot \color{blue}{\sinh y}}{y \cdot 2} \]

      8. times-frac N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-sinh.f64 N/A

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

      14. sinh-def N/A

          \[\leadsto \left(\sin x \cdot \frac{2}{y}\right) \cdot \frac{\color{blue}{\frac{e^{y} - e^{\mathsf{neg}\left(y\right)}}{2}}}{2} \]

      15. clear-num N/A

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

      16. associate-/r/ N/A

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

      17. associate-/l* N/A

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

      18. sinh-def N/A

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

      19. lift-sinh.f64 N/A

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

      20. lower-*.f64 N/A

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

      21. metadata-eval 94.4

          \[\leadsto \left(\sin x \cdot \frac{2}{y}\right) \cdot \left(\color{blue}{0.5} \cdot \sinh y\right) \]

   4. Applied rewrites 94.4%

      \[\leadsto \color{blue}{\left(\sin x \cdot \frac{2}{y}\right) \cdot \left(0.5 \cdot \sinh y\right)} \]

   5. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 66.9

         \[\leadsto \left(\color{blue}{\frac{2}{y}} \cdot x\right) \cdot \left(0.5 \cdot \sinh y\right) \]

   7. Applied rewrites 66.9%

      \[\leadsto \color{blue}{\left(\frac{2}{y} \cdot x\right)} \cdot \left(0.5 \cdot \sinh y\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 66.9%

      \[\leadsto \frac{2 \cdot x}{\color{blue}{y}} \cdot \left(0.5 \cdot \sinh y\right) \]

   if 3.8000000000000001e77 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 100.0%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.97:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right) \cdot y, y, 1\right) \cdot \sin x\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{+77}:\\ \;\;\;\;\frac{2 \cdot x}{y} \cdot \left(0.5 \cdot \sinh y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right) \cdot y\right) \cdot y\right) \cdot \sin x\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 86.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.94:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right) \cdot \sin x\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{+77}:\\ \;\;\;\;\frac{2 \cdot x}{y} \cdot \left(0.5 \cdot \sinh y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right) \cdot y\right) \cdot y\right) \cdot \sin x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 0.94)
   (* (fma (* y y) 0.16666666666666666 1.0) (sin x))
   (if (<= y 3.8e+77)
     (* (/ (* 2.0 x) y) (* 0.5 (sinh y)))
     (*
      (* (* (fma (* y y) 0.008333333333333333 0.16666666666666666) y) y)
      (sin x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 0.94) {
		tmp = fma((y * y), 0.16666666666666666, 1.0) * sin(x);
	} else if (y <= 3.8e+77) {
		tmp = ((2.0 * x) / y) * (0.5 * sinh(y));
	} else {
		tmp = ((fma((y * y), 0.008333333333333333, 0.16666666666666666) * y) * y) * sin(x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 0.94)
		tmp = Float64(fma(Float64(y * y), 0.16666666666666666, 1.0) * sin(x));
	elseif (y <= 3.8e+77)
		tmp = Float64(Float64(Float64(2.0 * x) / y) * Float64(0.5 * sinh(y)));
	else
		tmp = Float64(Float64(Float64(fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666) * y) * y) * sin(x));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 0.94], N[(N[(N[(y * y), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.8e+77], N[(N[(N[(2.0 * x), $MachinePrecision] / y), $MachinePrecision] * N[(0.5 * N[Sinh[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision] * y), $MachinePrecision] * y), $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 0.93999999999999995

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 84.9

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

   5. Applied rewrites 84.9%

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

   if 0.93999999999999995 < y < 3.8000000000000001e77

   1. Initial program 99.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sinh.f64 N/A

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

      4. sinh-def N/A

         \[\leadsto \sin x \cdot \frac{\color{blue}{\frac{e^{y} - e^{\mathsf{neg}\left(y\right)}}{2}}}{y} \]

      5. associate-/l/ N/A

         \[\leadsto \sin x \cdot \color{blue}{\frac{e^{y} - e^{\mathsf{neg}\left(y\right)}}{y \cdot 2}} \]

      6. sinh-undef N/A

         \[\leadsto \sin x \cdot \frac{\color{blue}{2 \cdot \sinh y}}{y \cdot 2} \]

      7. lift-sinh.f64 N/A

         \[\leadsto \sin x \cdot \frac{2 \cdot \color{blue}{\sinh y}}{y \cdot 2} \]

      8. times-frac N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-sinh.f64 N/A

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

      14. sinh-def N/A

          \[\leadsto \left(\sin x \cdot \frac{2}{y}\right) \cdot \frac{\color{blue}{\frac{e^{y} - e^{\mathsf{neg}\left(y\right)}}{2}}}{2} \]

      15. clear-num N/A

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

      16. associate-/r/ N/A

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

      17. associate-/l* N/A

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

      18. sinh-def N/A

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

      19. lift-sinh.f64 N/A

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

      20. lower-*.f64 N/A

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

      21. metadata-eval 94.4

          \[\leadsto \left(\sin x \cdot \frac{2}{y}\right) \cdot \left(\color{blue}{0.5} \cdot \sinh y\right) \]

   4. Applied rewrites 94.4%

      \[\leadsto \color{blue}{\left(\sin x \cdot \frac{2}{y}\right) \cdot \left(0.5 \cdot \sinh y\right)} \]

   5. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 66.9

         \[\leadsto \left(\color{blue}{\frac{2}{y}} \cdot x\right) \cdot \left(0.5 \cdot \sinh y\right) \]

   7. Applied rewrites 66.9%

      \[\leadsto \color{blue}{\left(\frac{2}{y} \cdot x\right)} \cdot \left(0.5 \cdot \sinh y\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 66.9%

      \[\leadsto \frac{2 \cdot x}{\color{blue}{y}} \cdot \left(0.5 \cdot \sinh y\right) \]

   if 3.8000000000000001e77 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 100.0%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.94:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right) \cdot \sin x\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{+77}:\\ \;\;\;\;\frac{2 \cdot x}{y} \cdot \left(0.5 \cdot \sinh y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right) \cdot y\right) \cdot y\right) \cdot \sin x\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 83.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.94:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right) \cdot \sin x\\ \mathbf{elif}\;y \leq 3.4 \cdot 10^{+154}:\\ \;\;\;\;\frac{2 \cdot x}{y} \cdot \left(0.5 \cdot \sinh y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(0.16666666666666666 \cdot y\right) \cdot y\right) \cdot \sin x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 0.94)
   (* (fma (* y y) 0.16666666666666666 1.0) (sin x))
   (if (<= y 3.4e+154)
     (* (/ (* 2.0 x) y) (* 0.5 (sinh y)))
     (* (* (* 0.16666666666666666 y) y) (sin x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 0.94) {
		tmp = fma((y * y), 0.16666666666666666, 1.0) * sin(x);
	} else if (y <= 3.4e+154) {
		tmp = ((2.0 * x) / y) * (0.5 * sinh(y));
	} else {
		tmp = ((0.16666666666666666 * y) * y) * sin(x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 0.94)
		tmp = Float64(fma(Float64(y * y), 0.16666666666666666, 1.0) * sin(x));
	elseif (y <= 3.4e+154)
		tmp = Float64(Float64(Float64(2.0 * x) / y) * Float64(0.5 * sinh(y)));
	else
		tmp = Float64(Float64(Float64(0.16666666666666666 * y) * y) * sin(x));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 0.94], N[(N[(N[(y * y), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.4e+154], N[(N[(N[(2.0 * x), $MachinePrecision] / y), $MachinePrecision] * N[(0.5 * N[Sinh[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.16666666666666666 * y), $MachinePrecision] * y), $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 0.93999999999999995

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 84.9

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

   5. Applied rewrites 84.9%

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

   if 0.93999999999999995 < y < 3.39999999999999974e154

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sinh.f64 N/A

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

      4. sinh-def N/A

         \[\leadsto \sin x \cdot \frac{\color{blue}{\frac{e^{y} - e^{\mathsf{neg}\left(y\right)}}{2}}}{y} \]

      5. associate-/l/ N/A

         \[\leadsto \sin x \cdot \color{blue}{\frac{e^{y} - e^{\mathsf{neg}\left(y\right)}}{y \cdot 2}} \]

      6. sinh-undef N/A

         \[\leadsto \sin x \cdot \frac{\color{blue}{2 \cdot \sinh y}}{y \cdot 2} \]

      7. lift-sinh.f64 N/A

         \[\leadsto \sin x \cdot \frac{2 \cdot \color{blue}{\sinh y}}{y \cdot 2} \]

      8. times-frac N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-sinh.f64 N/A

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

      14. sinh-def N/A

          \[\leadsto \left(\sin x \cdot \frac{2}{y}\right) \cdot \frac{\color{blue}{\frac{e^{y} - e^{\mathsf{neg}\left(y\right)}}{2}}}{2} \]

      15. clear-num N/A

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

      16. associate-/r/ N/A

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

      17. associate-/l* N/A

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

      18. sinh-def N/A

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

      19. lift-sinh.f64 N/A

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

      20. lower-*.f64 N/A

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

      21. metadata-eval 94.1

          \[\leadsto \left(\sin x \cdot \frac{2}{y}\right) \cdot \left(\color{blue}{0.5} \cdot \sinh y\right) \]

   4. Applied rewrites 94.1%

      \[\leadsto \color{blue}{\left(\sin x \cdot \frac{2}{y}\right) \cdot \left(0.5 \cdot \sinh y\right)} \]

   5. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 73.7

         \[\leadsto \left(\color{blue}{\frac{2}{y}} \cdot x\right) \cdot \left(0.5 \cdot \sinh y\right) \]

   7. Applied rewrites 73.7%

      \[\leadsto \color{blue}{\left(\frac{2}{y} \cdot x\right)} \cdot \left(0.5 \cdot \sinh y\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 73.7%

      \[\leadsto \frac{2 \cdot x}{\color{blue}{y}} \cdot \left(0.5 \cdot \sinh y\right) \]

   if 3.39999999999999974e154 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in y around 0

      \[\leadsto \sin x \cdot \left(\left(\frac{1}{6} \cdot y\right) \cdot y\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 100.0%

       \[\leadsto \sin x \cdot \left(\left(0.16666666666666666 \cdot y\right) \cdot y\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.94:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right) \cdot \sin x\\ \mathbf{elif}\;y \leq 3.4 \cdot 10^{+154}:\\ \;\;\;\;\frac{2 \cdot x}{y} \cdot \left(0.5 \cdot \sinh y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(0.16666666666666666 \cdot y\right) \cdot y\right) \cdot \sin x\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 83.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.94:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right) \cdot \sin x\\ \mathbf{elif}\;y \leq 3.4 \cdot 10^{+154}:\\ \;\;\;\;\left(\frac{2}{y} \cdot x\right) \cdot \left(0.5 \cdot \sinh y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(0.16666666666666666 \cdot y\right) \cdot y\right) \cdot \sin x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 0.94)
   (* (fma (* y y) 0.16666666666666666 1.0) (sin x))
   (if (<= y 3.4e+154)
     (* (* (/ 2.0 y) x) (* 0.5 (sinh y)))
     (* (* (* 0.16666666666666666 y) y) (sin x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 0.94) {
		tmp = fma((y * y), 0.16666666666666666, 1.0) * sin(x);
	} else if (y <= 3.4e+154) {
		tmp = ((2.0 / y) * x) * (0.5 * sinh(y));
	} else {
		tmp = ((0.16666666666666666 * y) * y) * sin(x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 0.94)
		tmp = Float64(fma(Float64(y * y), 0.16666666666666666, 1.0) * sin(x));
	elseif (y <= 3.4e+154)
		tmp = Float64(Float64(Float64(2.0 / y) * x) * Float64(0.5 * sinh(y)));
	else
		tmp = Float64(Float64(Float64(0.16666666666666666 * y) * y) * sin(x));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 0.94], N[(N[(N[(y * y), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.4e+154], N[(N[(N[(2.0 / y), $MachinePrecision] * x), $MachinePrecision] * N[(0.5 * N[Sinh[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.16666666666666666 * y), $MachinePrecision] * y), $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 0.93999999999999995

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 84.9

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

   5. Applied rewrites 84.9%

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

   if 0.93999999999999995 < y < 3.39999999999999974e154

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sinh.f64 N/A

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

      4. sinh-def N/A

         \[\leadsto \sin x \cdot \frac{\color{blue}{\frac{e^{y} - e^{\mathsf{neg}\left(y\right)}}{2}}}{y} \]

      5. associate-/l/ N/A

         \[\leadsto \sin x \cdot \color{blue}{\frac{e^{y} - e^{\mathsf{neg}\left(y\right)}}{y \cdot 2}} \]

      6. sinh-undef N/A

         \[\leadsto \sin x \cdot \frac{\color{blue}{2 \cdot \sinh y}}{y \cdot 2} \]

      7. lift-sinh.f64 N/A

         \[\leadsto \sin x \cdot \frac{2 \cdot \color{blue}{\sinh y}}{y \cdot 2} \]

      8. times-frac N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-sinh.f64 N/A

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

      14. sinh-def N/A

          \[\leadsto \left(\sin x \cdot \frac{2}{y}\right) \cdot \frac{\color{blue}{\frac{e^{y} - e^{\mathsf{neg}\left(y\right)}}{2}}}{2} \]

      15. clear-num N/A

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

      16. associate-/r/ N/A

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

      17. associate-/l* N/A

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

      18. sinh-def N/A

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

      19. lift-sinh.f64 N/A

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

      20. lower-*.f64 N/A

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

      21. metadata-eval 94.1

          \[\leadsto \left(\sin x \cdot \frac{2}{y}\right) \cdot \left(\color{blue}{0.5} \cdot \sinh y\right) \]

   4. Applied rewrites 94.1%

      \[\leadsto \color{blue}{\left(\sin x \cdot \frac{2}{y}\right) \cdot \left(0.5 \cdot \sinh y\right)} \]

   5. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 73.7

         \[\leadsto \left(\color{blue}{\frac{2}{y}} \cdot x\right) \cdot \left(0.5 \cdot \sinh y\right) \]

   7. Applied rewrites 73.7%

      \[\leadsto \color{blue}{\left(\frac{2}{y} \cdot x\right)} \cdot \left(0.5 \cdot \sinh y\right) \]

   if 3.39999999999999974e154 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in y around 0

      \[\leadsto \sin x \cdot \left(\left(\frac{1}{6} \cdot y\right) \cdot y\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 100.0%

       \[\leadsto \sin x \cdot \left(\left(0.16666666666666666 \cdot y\right) \cdot y\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.94:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right) \cdot \sin x\\ \mathbf{elif}\;y \leq 3.4 \cdot 10^{+154}:\\ \;\;\;\;\left(\frac{2}{y} \cdot x\right) \cdot \left(0.5 \cdot \sinh y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(0.16666666666666666 \cdot y\right) \cdot y\right) \cdot \sin x\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 66.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2000000:\\ \;\;\;\;1 \cdot \sin x\\ \mathbf{elif}\;y \leq 7 \cdot 10^{+138}:\\ \;\;\;\;\mathsf{fma}\left(\left(x \cdot x\right) \cdot x, \mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, x \cdot x, 0.008333333333333333\right), x \cdot x, -0.16666666666666666\right), x\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(0.16666666666666666 \cdot y\right) \cdot y\right) \cdot \sin x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2000000.0)
   (* 1.0 (sin x))
   (if (<= y 7e+138)
     (*
      (fma
       (* (* x x) x)
       (fma
        (fma -0.0001984126984126984 (* x x) 0.008333333333333333)
        (* x x)
        -0.16666666666666666)
       x)
      1.0)
     (* (* (* 0.16666666666666666 y) y) (sin x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2000000.0) {
		tmp = 1.0 * sin(x);
	} else if (y <= 7e+138) {
		tmp = fma(((x * x) * x), fma(fma(-0.0001984126984126984, (x * x), 0.008333333333333333), (x * x), -0.16666666666666666), x) * 1.0;
	} else {
		tmp = ((0.16666666666666666 * y) * y) * sin(x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2000000.0)
		tmp = Float64(1.0 * sin(x));
	elseif (y <= 7e+138)
		tmp = Float64(fma(Float64(Float64(x * x) * x), fma(fma(-0.0001984126984126984, Float64(x * x), 0.008333333333333333), Float64(x * x), -0.16666666666666666), x) * 1.0);
	else
		tmp = Float64(Float64(Float64(0.16666666666666666 * y) * y) * sin(x));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2000000.0], N[(1.0 * N[Sin[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 7e+138], N[(N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * N[(N[(-0.0001984126984126984 * N[(x * x), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] + x), $MachinePrecision] * 1.0), $MachinePrecision], N[(N[(N[(0.16666666666666666 * y), $MachinePrecision] * y), $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 2e6

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \sin x \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 67.0%

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

   if 2e6 < y < 6.9999999999999996e138

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \sin x \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 2.8%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. pow-plus N/A

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

      8. lower-pow.f64 N/A

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

      9. metadata-eval N/A

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

      10. sub-neg N/A

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

      11. *-commutative N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 19.5

          \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, x \cdot x, 0.008333333333333333\right), \color{blue}{x \cdot x}, -0.16666666666666666\right), x\right) \cdot 1 \]

   8. Applied rewrites 19.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, x \cdot x, 0.008333333333333333\right), x \cdot x, -0.16666666666666666\right), x\right)} \cdot 1 \]
   9. Step-by-step derivation

   10. Applied rewrites 19.5%

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

   if 6.9999999999999996e138 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in y around 0

      \[\leadsto \sin x \cdot \left(\left(\frac{1}{6} \cdot y\right) \cdot y\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 92.0%

       \[\leadsto \sin x \cdot \left(\left(0.16666666666666666 \cdot y\right) \cdot y\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2000000:\\ \;\;\;\;1 \cdot \sin x\\ \mathbf{elif}\;y \leq 7 \cdot 10^{+138}:\\ \;\;\;\;\mathsf{fma}\left(\left(x \cdot x\right) \cdot x, \mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, x \cdot x, 0.008333333333333333\right), x \cdot x, -0.16666666666666666\right), x\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(0.16666666666666666 \cdot y\right) \cdot y\right) \cdot \sin x\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 34.1% accurate, 9.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\left(x \cdot x\right) \cdot -0.16666666666666666, x, x\right) \cdot 1 \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (* (fma (* (* x x) -0.16666666666666666) x x) 1.0))

C

double code(double x, double y) {
	return fma(((x * x) * -0.16666666666666666), x, x) * 1.0;
}

Julia

function code(x, y)
	return Float64(fma(Float64(Float64(x * x) * -0.16666666666666666), x, x) * 1.0)
end

Wolfram

code[x_, y_] := N[(N[(N[(N[(x * x), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * x + x), $MachinePrecision] * 1.0), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in y around 0

   \[\leadsto \sin x \cdot \color{blue}{1} \]
4. Step-by-step derivation

5. Applied rewrites 51.2%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. *-rgt-identity N/A

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

   6. lower-fma.f64 N/A

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

   7. *-commutative N/A

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

   8. pow-plus N/A

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

   9. lower-pow.f64 N/A

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

   10. metadata-eval 29.4

       \[\leadsto \mathsf{fma}\left({x}^{\color{blue}{3}}, -0.16666666666666666, x\right) \cdot 1 \]

8. Applied rewrites 29.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, -0.16666666666666666, x\right)} \cdot 1 \]
9. Step-by-step derivation

10. Applied rewrites 29.4%

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

11. Final simplification 29.4%

    \[\leadsto \mathsf{fma}\left(\left(x \cdot x\right) \cdot -0.16666666666666666, x, x\right) \cdot 1 \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024308 
(FPCore (x y)
  :name "Linear.Quaternion:$ccos from linear-1.19.1.3"
  :precision binary64
  (* (sin x) (/ (sinh y) y)))