Linear.Quaternion:$ccos from linear-1.19.1.3

Percentage Accurate: 100.0% → 100.0%

Time: 14.9s

Alternatives: 24

Speedup: 1.0×

• Could not converge on a ground truth

  1. x = 5.502930441122139e+191
  2. y = 1.0633592768671405e+20

• 50.0% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. lift-sin.f64 N/A

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

   2. lift-sinh.f64 N/A

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

   3. clear-num N/A

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

   4. un-div-inv N/A

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

   5. lower-/.f64 N/A

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

   6. lower-/.f64 100.0

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

4. Applied egg-rr 100.0%

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

Alternative 2: 81.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin x \cdot \frac{\sinh y}{y}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;x \cdot \left(\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.008333333333333333\right)\right)\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\sin x \cdot \mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right) \cdot \mathsf{fma}\left(x \cdot x, x \cdot \mathsf{fma}\left(x, x \cdot 0.008333333333333333, -0.16666666666666666\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sin x) (/ (sinh y) y))))
   (if (<= t_0 (- INFINITY))
     (*
      x
      (*
       (* (* y y) (* y y))
       (fma (* x x) -0.001388888888888889 0.008333333333333333)))
     (if (<= t_0 1.0)
       (* (sin x) (fma 0.16666666666666666 (* y y) 1.0))
       (*
        (fma
         y
         (*
          y
          (fma
           (* y y)
           (fma (* y y) 0.0001984126984126984 0.008333333333333333)
           0.16666666666666666))
         1.0)
        (fma
         (* x x)
         (* x (fma x (* x 0.008333333333333333) -0.16666666666666666))
         x))))))

C

double code(double x, double y) {
	double t_0 = sin(x) * (sinh(y) / y);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = x * (((y * y) * (y * y)) * fma((x * x), -0.001388888888888889, 0.008333333333333333));
	} else if (t_0 <= 1.0) {
		tmp = sin(x) * fma(0.16666666666666666, (y * y), 1.0);
	} else {
		tmp = fma(y, (y * fma((y * y), fma((y * y), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666)), 1.0) * fma((x * x), (x * fma(x, (x * 0.008333333333333333), -0.16666666666666666)), x);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(x) * Float64(sinh(y) / y))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(x * Float64(Float64(Float64(y * y) * Float64(y * y)) * fma(Float64(x * x), -0.001388888888888889, 0.008333333333333333)));
	elseif (t_0 <= 1.0)
		tmp = Float64(sin(x) * fma(0.16666666666666666, Float64(y * y), 1.0));
	else
		tmp = Float64(fma(y, Float64(y * fma(Float64(y * y), fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666)), 1.0) * fma(Float64(x * x), Float64(x * fma(x, Float64(x * 0.008333333333333333), -0.16666666666666666)), x));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(x * N[(N[(N[(y * y), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.001388888888888889 + 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1.0], N[(N[Sin[x], $MachinePrecision] * N[(0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(x * 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. distribute-rgt-in N/A

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

      2. *-rgt-identity N/A

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

      3. distribute-rgt-in N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out N/A

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

      7. +-commutative N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. distribute-lft-in N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sin.f64 N/A

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

      13. +-commutative N/A

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

   5. Simplified 81.4%

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

   6. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt1-in N/A

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

      6. +-commutative N/A

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

      7. lower-*.f64 N/A

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

   8. Simplified 60.2%

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

   9. Taylor expanded in y around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. lower-*.f64 N/A

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

       5. metadata-eval N/A

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

       6. pow-sqr N/A

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

       7. lower-*.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f64 N/A

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

       12. +-commutative N/A

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

       13. distribute-rgt-in N/A

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

       14. *-commutative N/A

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

       15. associate-*l* N/A

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

       16. metadata-eval N/A

           \[\leadsto x \cdot \left(\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \left({x}^{2} \cdot \color{blue}{\frac{-1}{720}} + 1 \cdot \frac{1}{120}\right)\right) \]

       17. metadata-eval N/A

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

       18. metadata-eval N/A

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

       19. lower-fma.f64 N/A

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

       20. unpow2 N/A

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

       21. lower-*.f64 N/A

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

       22. metadata-eval 60.2

           \[\leadsto x \cdot \left(\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{-0.001388888888888889}, 0.008333333333333333\right)\right) \]

   11. Simplified 60.2%

       \[\leadsto x \cdot \color{blue}{\left(\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.008333333333333333\right)\right)} \]

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-lft-identity N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-in N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 98.9

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

   5. Simplified 98.9%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. distribute-lft-in N/A

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

      2. associate-+r+ N/A

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

      3. *-lft-identity N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. distribute-rgt-in N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. distribute-rgt-out N/A

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

      11. +-commutative N/A

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

   5. Simplified 85.0%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. sub-neg N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. metadata-eval N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-*.f64 63.1

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

   8. Simplified 63.1%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin x \cdot \frac{\sinh y}{y} \leq -\infty:\\ \;\;\;\;x \cdot \left(\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.008333333333333333\right)\right)\\ \mathbf{elif}\;\sin x \cdot \frac{\sinh y}{y} \leq 1:\\ \;\;\;\;\sin x \cdot \mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right) \cdot \mathsf{fma}\left(x \cdot x, x \cdot \mathsf{fma}\left(x, x \cdot 0.008333333333333333, -0.16666666666666666\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin x \cdot \frac{\sinh y}{y}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;x \cdot \left(\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.008333333333333333\right)\right)\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\sin x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right) \cdot \mathsf{fma}\left(x \cdot x, x \cdot \mathsf{fma}\left(x, x \cdot 0.008333333333333333, -0.16666666666666666\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sin x) (/ (sinh y) y))))
   (if (<= t_0 (- INFINITY))
     (*
      x
      (*
       (* (* y y) (* y y))
       (fma (* x x) -0.001388888888888889 0.008333333333333333)))
     (if (<= t_0 1.0)
       (sin x)
       (*
        (fma
         y
         (*
          y
          (fma
           (* y y)
           (fma (* y y) 0.0001984126984126984 0.008333333333333333)
           0.16666666666666666))
         1.0)
        (fma
         (* x x)
         (* x (fma x (* x 0.008333333333333333) -0.16666666666666666))
         x))))))

C

double code(double x, double y) {
	double t_0 = sin(x) * (sinh(y) / y);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = x * (((y * y) * (y * y)) * fma((x * x), -0.001388888888888889, 0.008333333333333333));
	} else if (t_0 <= 1.0) {
		tmp = sin(x);
	} else {
		tmp = fma(y, (y * fma((y * y), fma((y * y), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666)), 1.0) * fma((x * x), (x * fma(x, (x * 0.008333333333333333), -0.16666666666666666)), x);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(x) * Float64(sinh(y) / y))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(x * Float64(Float64(Float64(y * y) * Float64(y * y)) * fma(Float64(x * x), -0.001388888888888889, 0.008333333333333333)));
	elseif (t_0 <= 1.0)
		tmp = sin(x);
	else
		tmp = Float64(fma(y, Float64(y * fma(Float64(y * y), fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666)), 1.0) * fma(Float64(x * x), Float64(x * fma(x, Float64(x * 0.008333333333333333), -0.16666666666666666)), x));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(x * N[(N[(N[(y * y), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.001388888888888889 + 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1.0], N[Sin[x], $MachinePrecision], N[(N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(x * 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. distribute-rgt-in N/A

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

      2. *-rgt-identity N/A

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

      3. distribute-rgt-in N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out N/A

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

      7. +-commutative N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. distribute-lft-in N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sin.f64 N/A

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

      13. +-commutative N/A

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

   5. Simplified 81.4%

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

   6. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt1-in N/A

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

      6. +-commutative N/A

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

      7. lower-*.f64 N/A

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

   8. Simplified 60.2%

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

   9. Taylor expanded in y around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. lower-*.f64 N/A

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

       5. metadata-eval N/A

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

       6. pow-sqr N/A

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

       7. lower-*.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f64 N/A

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

       12. +-commutative N/A

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

       13. distribute-rgt-in N/A

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

       14. *-commutative N/A

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

       15. associate-*l* N/A

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

       16. metadata-eval N/A

           \[\leadsto x \cdot \left(\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \left({x}^{2} \cdot \color{blue}{\frac{-1}{720}} + 1 \cdot \frac{1}{120}\right)\right) \]

       17. metadata-eval N/A

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

       18. metadata-eval N/A

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

       19. lower-fma.f64 N/A

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

       20. unpow2 N/A

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

       21. lower-*.f64 N/A

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

       22. metadata-eval 60.2

           \[\leadsto x \cdot \left(\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{-0.001388888888888889}, 0.008333333333333333\right)\right) \]

   11. Simplified 60.2%

       \[\leadsto x \cdot \color{blue}{\left(\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.008333333333333333\right)\right)} \]

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-sin.f64 97.9

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

   5. Simplified 97.9%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. distribute-lft-in N/A

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

      2. associate-+r+ N/A

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

      3. *-lft-identity N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. distribute-rgt-in N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. distribute-rgt-out N/A

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

      11. +-commutative N/A

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

   5. Simplified 85.0%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. sub-neg N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. metadata-eval N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-*.f64 63.1

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

   8. Simplified 63.1%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin x \cdot \frac{\sinh y}{y} \leq -\infty:\\ \;\;\;\;x \cdot \left(\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.008333333333333333\right)\right)\\ \mathbf{elif}\;\sin x \cdot \frac{\sinh y}{y} \leq 1:\\ \;\;\;\;\sin x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right) \cdot \mathsf{fma}\left(x \cdot x, x \cdot \mathsf{fma}\left(x, x \cdot 0.008333333333333333, -0.16666666666666666\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.5% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. distribute-lft-in N/A

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

      2. associate-+r+ N/A

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

      3. *-lft-identity N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. distribute-rgt-in N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. distribute-rgt-out N/A

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

      11. +-commutative N/A

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

   5. Simplified 95.1%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. sub-neg N/A

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

      9. *-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 70.0

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

   5. Simplified 70.0%

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

4. Final simplification 88.2%

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

Alternative 5: 85.4% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. distribute-rgt-in N/A

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

      2. *-rgt-identity N/A

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

      3. distribute-rgt-in N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out N/A

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

      7. +-commutative N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. distribute-lft-in N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sin.f64 N/A

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

      13. +-commutative N/A

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

   5. Simplified 92.8%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. distribute-lft-in N/A

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

      2. associate-+r+ N/A

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

      3. *-lft-identity N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. distribute-rgt-in N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. distribute-rgt-out N/A

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

      11. +-commutative N/A

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

   5. Simplified 85.0%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. sub-neg N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. metadata-eval N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-*.f64 63.1

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

   8. Simplified 63.1%

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

4. Final simplification 84.7%

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

Alternative 6: 58.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 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

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

   5. Simplified 91.9%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 44.5

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

   8. Simplified 44.5%

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

   9. Taylor expanded in y around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 44.5

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

   11. Simplified 44.5%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. distribute-lft-in N/A

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

      2. associate-+r+ N/A

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

      3. *-lft-identity N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. distribute-rgt-in N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. distribute-rgt-out N/A

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

      11. +-commutative N/A

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

   5. Simplified 93.1%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. sub-neg N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. metadata-eval N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-*.f64 66.1

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

   8. Simplified 66.1%

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

4. Final simplification 57.8%

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

Alternative 7: 56.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin x \cdot \frac{\sinh y}{y} \leq 0.002:\\ \;\;\;\;x \cdot \left(\mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(y \cdot y\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (sin x) (/ (sinh y) y)) 0.002)
   (*
    x
    (*
     (fma x (* x -0.16666666666666666) 1.0)
     (fma y (* y (fma y (* y 0.008333333333333333) 0.16666666666666666)) 1.0)))
   (fma
    (*
     (* y y)
     (fma
      (* y y)
      (fma y (* y 0.0001984126984126984) 0.008333333333333333)
      0.16666666666666666))
    x
    x)))

C

double code(double x, double y) {
	double tmp;
	if ((sin(x) * (sinh(y) / y)) <= 0.002) {
		tmp = x * (fma(x, (x * -0.16666666666666666), 1.0) * fma(y, (y * fma(y, (y * 0.008333333333333333), 0.16666666666666666)), 1.0));
	} else {
		tmp = fma(((y * y) * fma((y * y), fma(y, (y * 0.0001984126984126984), 0.008333333333333333), 0.16666666666666666)), x, x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(sin(x) * Float64(sinh(y) / y)) <= 0.002)
		tmp = Float64(x * Float64(fma(x, Float64(x * -0.16666666666666666), 1.0) * fma(y, Float64(y * fma(y, Float64(y * 0.008333333333333333), 0.16666666666666666)), 1.0)));
	else
		tmp = fma(Float64(Float64(y * y) * fma(Float64(y * y), fma(y, Float64(y * 0.0001984126984126984), 0.008333333333333333), 0.16666666666666666)), x, x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Sin[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], 0.002], N[(x * N[(N[(x * N[(x * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * N[(y * N[(y * N[(y * N[(y * 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * 0.0001984126984126984), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] * x + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < 2e-3

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. distribute-rgt-in N/A

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

      2. *-rgt-identity N/A

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

      3. distribute-rgt-in N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out N/A

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

      7. +-commutative N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. distribute-lft-in N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sin.f64 N/A

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

      13. +-commutative N/A

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

   5. Simplified 91.8%

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

   6. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt1-in N/A

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

      6. +-commutative N/A

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

      7. lower-*.f64 N/A

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

   8. Simplified 63.3%

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

   if 2e-3 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. distribute-lft-in N/A

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

      2. associate-+r+ N/A

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

      3. *-lft-identity N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. distribute-rgt-in N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. distribute-rgt-out N/A

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

      11. +-commutative N/A

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

   5. Simplified 89.0%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-lft-identity N/A

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

      6. lower-fma.f64 N/A

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

   8. Simplified 42.5%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

   10. Applied egg-rr 43.3%

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

Alternative 8: 52.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin x \cdot \frac{\sinh y}{y} \leq 0.002:\\ \;\;\;\;x \cdot \left(\mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right) \cdot \mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (sin x) (/ (sinh y) y)) 0.002)
   (*
    x
    (*
     (fma 0.16666666666666666 (* y y) 1.0)
     (fma x (* x -0.16666666666666666) 1.0)))
   (*
    x
    (fma (* y y) (fma (* y y) 0.008333333333333333 0.16666666666666666) 1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((sin(x) * (sinh(y) / y)) <= 0.002) {
		tmp = x * (fma(0.16666666666666666, (y * y), 1.0) * fma(x, (x * -0.16666666666666666), 1.0));
	} else {
		tmp = x * fma((y * y), fma((y * y), 0.008333333333333333, 0.16666666666666666), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(sin(x) * Float64(sinh(y) / y)) <= 0.002)
		tmp = Float64(x * Float64(fma(0.16666666666666666, Float64(y * y), 1.0) * fma(x, Float64(x * -0.16666666666666666), 1.0)));
	else
		tmp = Float64(x * fma(Float64(y * y), fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Sin[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], 0.002], N[(x * N[(N[(0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x * N[(x * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < 2e-3

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. distribute-rgt-in N/A

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

      2. *-rgt-identity N/A

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

      3. distribute-rgt-in N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out N/A

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

      7. +-commutative N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. distribute-lft-in N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sin.f64 N/A

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

      13. +-commutative N/A

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

   5. Simplified 91.8%

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

   6. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt1-in N/A

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

      6. +-commutative N/A

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

      7. lower-*.f64 N/A

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

   8. Simplified 63.3%

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

   9. Taylor expanded in y around 0

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

       1. associate-+r+ N/A

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

       2. associate-*r* N/A

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

       3. distribute-rgt1-in N/A

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

       4. +-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. +-commutative N/A

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

       7. lower-fma.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 N/A

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

       10. +-commutative N/A

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

       11. *-commutative N/A

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

       12. unpow2 N/A

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

       13. associate-*l* N/A

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

       14. *-commutative N/A

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

       15. lower-fma.f64 N/A

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

       16. *-commutative N/A

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

       17. lower-*.f64 58.9

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

   11. Simplified 58.9%

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

   if 2e-3 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. distribute-rgt-in N/A

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

      2. *-rgt-identity N/A

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

      3. distribute-rgt-in N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out N/A

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

      7. +-commutative N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. distribute-lft-in N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sin.f64 N/A

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

      13. +-commutative N/A

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

   5. Simplified 85.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. distribute-rgt-in N/A

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

      6. associate-+l+ N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*l* N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-*.f64 85.0

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

   7. Applied egg-rr 85.0%

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

   8. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(1 + \left(\frac{1}{120} \cdot {y}^{4} + \frac{1}{6} \cdot {y}^{2}\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(\left(\frac{1}{120} \cdot {y}^{4} + \frac{1}{6} \cdot {y}^{2}\right) + 1\right)} \]

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. associate-*l* N/A

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

      6. distribute-rgt-in N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 42.1

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

   10. Simplified 42.1%

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

Alternative 9: 46.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin x \cdot \frac{\sinh y}{y} \leq -0.01:\\ \;\;\;\;\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(y \cdot y, -0.027777777777777776, -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (sin x) (/ (sinh y) y)) -0.01)
   (* (* x (* x x)) (fma (* y y) -0.027777777777777776 -0.16666666666666666))
   (*
    x
    (fma (* y y) (fma (* y y) 0.008333333333333333 0.16666666666666666) 1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((sin(x) * (sinh(y) / y)) <= -0.01) {
		tmp = (x * (x * x)) * fma((y * y), -0.027777777777777776, -0.16666666666666666);
	} else {
		tmp = x * fma((y * y), fma((y * y), 0.008333333333333333, 0.16666666666666666), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(sin(x) * Float64(sinh(y) / y)) <= -0.01)
		tmp = Float64(Float64(x * Float64(x * x)) * fma(Float64(y * y), -0.027777777777777776, -0.16666666666666666));
	else
		tmp = Float64(x * fma(Float64(y * y), fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Sin[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], -0.01], N[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * -0.027777777777777776 + -0.16666666666666666), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. distribute-rgt-in N/A

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

      2. *-rgt-identity N/A

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

      3. distribute-rgt-in N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out N/A

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

      7. +-commutative N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. distribute-lft-in N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sin.f64 N/A

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

      13. +-commutative N/A

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

   5. Simplified 87.0%

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

   6. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt1-in N/A

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

      6. +-commutative N/A

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

      7. lower-*.f64 N/A

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

   8. Simplified 41.6%

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

   9. Taylor expanded in y around 0

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

       1. associate-+r+ N/A

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

       2. associate-*r* N/A

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

       3. distribute-rgt1-in N/A

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

       4. +-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. +-commutative N/A

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

       7. lower-fma.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 N/A

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

       10. +-commutative N/A

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

       11. *-commutative N/A

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

       12. unpow2 N/A

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

       13. associate-*l* N/A

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

       14. *-commutative N/A

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

       15. lower-fma.f64 N/A

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

       16. *-commutative N/A

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

       17. lower-*.f64 34.8

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

   11. Simplified 34.8%

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

   12. Taylor expanded in x around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. lower-*.f64 N/A

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

       5. cube-mult N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 N/A

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

       10. +-commutative N/A

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

       11. distribute-rgt-in N/A

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

       12. *-commutative N/A

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

       13. associate-*l* N/A

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

       14. metadata-eval N/A

           \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left({y}^{2} \cdot \color{blue}{\frac{-1}{36}} + 1 \cdot \frac{-1}{6}\right) \]

       15. metadata-eval N/A

           \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left({y}^{2} \cdot \frac{-1}{36} + \color{blue}{\frac{-1}{6}}\right) \]

       16. lower-fma.f64 N/A

           \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{-1}{36}, \frac{-1}{6}\right)} \]

       17. unpow2 N/A

           \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{-1}{36}, \frac{-1}{6}\right) \]

       18. lower-*.f64 18.2

           \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, -0.027777777777777776, -0.16666666666666666\right) \]

   14. Simplified 18.2%

       \[\leadsto \color{blue}{\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(y \cdot y, -0.027777777777777776, -0.16666666666666666\right)} \]

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. distribute-rgt-in N/A

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

      2. *-rgt-identity N/A

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

      3. distribute-rgt-in N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out N/A

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

      7. +-commutative N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. distribute-lft-in N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sin.f64 N/A

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

      13. +-commutative N/A

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

   5. Simplified 90.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. distribute-rgt-in N/A

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

      6. associate-+l+ N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*l* N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-*.f64 90.7

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

   7. Applied egg-rr 90.7%

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

   8. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(1 + \left(\frac{1}{120} \cdot {y}^{4} + \frac{1}{6} \cdot {y}^{2}\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(\left(\frac{1}{120} \cdot {y}^{4} + \frac{1}{6} \cdot {y}^{2}\right) + 1\right)} \]

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. associate-*l* N/A

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

      6. distribute-rgt-in N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 63.6

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

   10. Simplified 63.6%

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

Alternative 10: 48.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin x \cdot \frac{\sinh y}{y} \leq -0.01:\\ \;\;\;\;x \cdot \left(\left(y \cdot y\right) \cdot \mathsf{fma}\left(x, x \cdot -0.027777777777777776, 0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.16666666666666666, x \cdot \left(y \cdot y\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (sin x) (/ (sinh y) y)) -0.01)
   (* x (* (* y y) (fma x (* x -0.027777777777777776) 0.16666666666666666)))
   (fma 0.16666666666666666 (* x (* y y)) x)))

C

double code(double x, double y) {
	double tmp;
	if ((sin(x) * (sinh(y) / y)) <= -0.01) {
		tmp = x * ((y * y) * fma(x, (x * -0.027777777777777776), 0.16666666666666666));
	} else {
		tmp = fma(0.16666666666666666, (x * (y * y)), x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(sin(x) * Float64(sinh(y) / y)) <= -0.01)
		tmp = Float64(x * Float64(Float64(y * y) * fma(x, Float64(x * -0.027777777777777776), 0.16666666666666666)));
	else
		tmp = fma(0.16666666666666666, Float64(x * Float64(y * y)), x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Sin[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], -0.01], N[(x * N[(N[(y * y), $MachinePrecision] * N[(x * N[(x * -0.027777777777777776), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.16666666666666666 * N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. distribute-rgt-in N/A

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

      2. *-rgt-identity N/A

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

      3. distribute-rgt-in N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out N/A

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

      7. +-commutative N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. distribute-lft-in N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sin.f64 N/A

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

      13. +-commutative N/A

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

   5. Simplified 87.0%

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

   6. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt1-in N/A

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

      6. +-commutative N/A

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

      7. lower-*.f64 N/A

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

   8. Simplified 41.6%

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

   9. Taylor expanded in y around 0

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

       1. associate-+r+ N/A

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

       2. associate-*r* N/A

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

       3. distribute-rgt1-in N/A

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

       4. +-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. +-commutative N/A

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

       7. lower-fma.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 N/A

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

       10. +-commutative N/A

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

       11. *-commutative N/A

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

       12. unpow2 N/A

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

       13. associate-*l* N/A

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

       14. *-commutative N/A

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

       15. lower-fma.f64 N/A

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

       16. *-commutative N/A

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

       17. lower-*.f64 34.8

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

   11. Simplified 34.8%

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

   12. Taylor expanded in y around inf

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. distribute-lft-in N/A

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

       4. metadata-eval N/A

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

       5. associate-*r* N/A

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

       6. metadata-eval N/A

          \[\leadsto x \cdot \left(\left(\frac{1}{6} + \color{blue}{\frac{-1}{36}} \cdot {x}^{2}\right) \cdot {y}^{2}\right) \]

       7. *-commutative N/A

          \[\leadsto x \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + \frac{-1}{36} \cdot {x}^{2}\right)\right)} \]

       8. lower-*.f64 N/A

          \[\leadsto x \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + \frac{-1}{36} \cdot {x}^{2}\right)\right)} \]

       9. unpow2 N/A

          \[\leadsto x \cdot \left(\color{blue}{\left(y \cdot y\right)} \cdot \left(\frac{1}{6} + \frac{-1}{36} \cdot {x}^{2}\right)\right) \]

       10. lower-*.f64 N/A

           \[\leadsto x \cdot \left(\color{blue}{\left(y \cdot y\right)} \cdot \left(\frac{1}{6} + \frac{-1}{36} \cdot {x}^{2}\right)\right) \]

       11. +-commutative N/A

           \[\leadsto x \cdot \left(\left(y \cdot y\right) \cdot \color{blue}{\left(\frac{-1}{36} \cdot {x}^{2} + \frac{1}{6}\right)}\right) \]

       12. *-commutative N/A

           \[\leadsto x \cdot \left(\left(y \cdot y\right) \cdot \left(\color{blue}{{x}^{2} \cdot \frac{-1}{36}} + \frac{1}{6}\right)\right) \]

       13. unpow2 N/A

           \[\leadsto x \cdot \left(\left(y \cdot y\right) \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{-1}{36} + \frac{1}{6}\right)\right) \]

       14. associate-*l* N/A

           \[\leadsto x \cdot \left(\left(y \cdot y\right) \cdot \left(\color{blue}{x \cdot \left(x \cdot \frac{-1}{36}\right)} + \frac{1}{6}\right)\right) \]

       15. lower-fma.f64 N/A

           \[\leadsto x \cdot \left(\left(y \cdot y\right) \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{-1}{36}, \frac{1}{6}\right)}\right) \]

       16. lower-*.f64 34.7

           \[\leadsto x \cdot \left(\left(y \cdot y\right) \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot -0.027777777777777776}, 0.16666666666666666\right)\right) \]

   14. Simplified 34.7%

       \[\leadsto x \cdot \color{blue}{\left(\left(y \cdot y\right) \cdot \mathsf{fma}\left(x, x \cdot -0.027777777777777776, 0.16666666666666666\right)\right)} \]

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. distribute-lft-in N/A

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

      2. associate-+r+ N/A

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

      3. *-lft-identity N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. distribute-rgt-in N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. distribute-rgt-out N/A

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

      11. +-commutative N/A

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

   5. Simplified 93.1%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-lft-identity N/A

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

      6. lower-fma.f64 N/A

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

   8. Simplified 63.8%

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

   9. Taylor expanded in y around 0

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

   11. Simplified 58.5%

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

4. Final simplification 49.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin x \cdot \frac{\sinh y}{y} \leq -0.01:\\ \;\;\;\;x \cdot \left(\left(y \cdot y\right) \cdot \mathsf{fma}\left(x, x \cdot -0.027777777777777776, 0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.16666666666666666, x \cdot \left(y \cdot y\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 41.0% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-sin.f64 33.4

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

   5. Simplified 33.4%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 15.9

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

   8. Simplified 15.9%

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

   9. Taylor expanded in x around inf

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

       1. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{-1}{6} \cdot {x}^{3}} \]

       2. cube-mult N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 15.6

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

   11. Simplified 15.6%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. distribute-lft-in N/A

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

      2. associate-+r+ N/A

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

      3. *-lft-identity N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. distribute-rgt-in N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. distribute-rgt-out N/A

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

      11. +-commutative N/A

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

   5. Simplified 93.1%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-lft-identity N/A

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

      6. lower-fma.f64 N/A

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

   8. Simplified 63.8%

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

   9. Taylor expanded in y around 0

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

   11. Simplified 58.5%

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

4. Final simplification 41.9%

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

Alternative 12: 41.0% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-sin.f64 33.4

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

   5. Simplified 33.4%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 15.9

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

   8. Simplified 15.9%

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

   9. Taylor expanded in x around inf

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

       1. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{-1}{6} \cdot {x}^{3}} \]

       2. cube-mult N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 15.6

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

   11. Simplified 15.6%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. distribute-rgt-in N/A

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

      2. *-rgt-identity N/A

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

      3. distribute-rgt-in N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out N/A

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

      7. +-commutative N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. distribute-lft-in N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sin.f64 N/A

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

      13. +-commutative N/A

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

   5. Simplified 90.7%

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

   6. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt1-in N/A

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

      6. +-commutative N/A

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

      7. lower-*.f64 N/A

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

   8. Simplified 65.9%

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

   9. Taylor expanded in y around 0

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

       1. associate-+r+ N/A

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

       2. associate-*r* N/A

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

       3. distribute-rgt1-in N/A

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

       4. +-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. +-commutative N/A

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

       7. lower-fma.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 N/A

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

       10. +-commutative N/A

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

       11. *-commutative N/A

           \[\leadsto x \cdot \left(\mathsf{fma}\left(\frac{1}{6}, y \cdot y, 1\right) \cdot \left(\color{blue}{{x}^{2} \cdot \frac{-1}{6}} + 1\right)\right) \]

       12. unpow2 N/A

           \[\leadsto x \cdot \left(\mathsf{fma}\left(\frac{1}{6}, y \cdot y, 1\right) \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{-1}{6} + 1\right)\right) \]

       13. associate-*l* N/A

           \[\leadsto x \cdot \left(\mathsf{fma}\left(\frac{1}{6}, y \cdot y, 1\right) \cdot \left(\color{blue}{x \cdot \left(x \cdot \frac{-1}{6}\right)} + 1\right)\right) \]

       14. *-commutative N/A

           \[\leadsto x \cdot \left(\mathsf{fma}\left(\frac{1}{6}, y \cdot y, 1\right) \cdot \left(x \cdot \color{blue}{\left(\frac{-1}{6} \cdot x\right)} + 1\right)\right) \]

       15. lower-fma.f64 N/A

           \[\leadsto x \cdot \left(\mathsf{fma}\left(\frac{1}{6}, y \cdot y, 1\right) \cdot \color{blue}{\mathsf{fma}\left(x, \frac{-1}{6} \cdot x, 1\right)}\right) \]

       16. *-commutative N/A

           \[\leadsto x \cdot \left(\mathsf{fma}\left(\frac{1}{6}, y \cdot y, 1\right) \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{-1}{6}}, 1\right)\right) \]

       17. lower-*.f64 60.4

           \[\leadsto x \cdot \left(\mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right) \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot -0.16666666666666666}, 1\right)\right) \]

   11. Simplified 60.4%

       \[\leadsto x \cdot \color{blue}{\left(\mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right) \cdot \mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right)\right)} \]

   12. Taylor expanded in x around 0

       \[\leadsto x \cdot \color{blue}{\left(1 + \frac{1}{6} \cdot {y}^{2}\right)} \]
   13. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto x \cdot \color{blue}{\left(\frac{1}{6} \cdot {y}^{2} + 1\right)} \]

       2. lower-fma.f64 N/A

          \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{6}, {y}^{2}, 1\right)} \]

       3. unpow2 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(\frac{1}{6}, \color{blue}{y \cdot y}, 1\right) \]

       4. lower-*.f64 58.5

          \[\leadsto x \cdot \mathsf{fma}\left(0.16666666666666666, \color{blue}{y \cdot y}, 1\right) \]

   14. Simplified 58.5%

       \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sin x \cdot \frac{\sinh y}{y} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* (sin x) (/ (sinh y) y)))

C

double code(double x, double y) {
	return sin(x) * (sinh(y) / y);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = sin(x) * (sinh(y) / y)
end function

Java

public static double code(double x, double y) {
	return Math.sin(x) * (Math.sinh(y) / y);
}

Python

def code(x, y):
	return math.sin(x) * (math.sinh(y) / y)

Julia

function code(x, y)
	return Float64(sin(x) * Float64(sinh(y) / y))
end

MATLAB

function tmp = code(x, y)
	tmp = sin(x) * (sinh(y) / y);
end

Wolfram

code[x_, y_] := N[(N[Sin[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\sin x \cdot \frac{\sinh y}{y} \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 14: 58.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin x \leq -0.01:\\ \;\;\;\;x \cdot \left(\mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right) \cdot \mathsf{fma}\left(x \cdot x, x \cdot \mathsf{fma}\left(x, x \cdot 0.008333333333333333, -0.16666666666666666\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sin x) -0.01)
   (*
    x
    (*
     (fma x (* x -0.16666666666666666) 1.0)
     (fma y (* y (fma y (* y 0.008333333333333333) 0.16666666666666666)) 1.0)))
   (*
    (fma
     y
     (*
      y
      (fma
       (* y y)
       (fma (* y y) 0.0001984126984126984 0.008333333333333333)
       0.16666666666666666))
     1.0)
    (fma
     (* x x)
     (* x (fma x (* x 0.008333333333333333) -0.16666666666666666))
     x))))

C

double code(double x, double y) {
	double tmp;
	if (sin(x) <= -0.01) {
		tmp = x * (fma(x, (x * -0.16666666666666666), 1.0) * fma(y, (y * fma(y, (y * 0.008333333333333333), 0.16666666666666666)), 1.0));
	} else {
		tmp = fma(y, (y * fma((y * y), fma((y * y), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666)), 1.0) * fma((x * x), (x * fma(x, (x * 0.008333333333333333), -0.16666666666666666)), x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (sin(x) <= -0.01)
		tmp = Float64(x * Float64(fma(x, Float64(x * -0.16666666666666666), 1.0) * fma(y, Float64(y * fma(y, Float64(y * 0.008333333333333333), 0.16666666666666666)), 1.0)));
	else
		tmp = Float64(fma(y, Float64(y * fma(Float64(y * y), fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666)), 1.0) * fma(Float64(x * x), Float64(x * fma(x, Float64(x * 0.008333333333333333), -0.16666666666666666)), x));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[Sin[x], $MachinePrecision], -0.01], N[(x * N[(N[(x * N[(x * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * N[(y * N[(y * N[(y * N[(y * 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(x * 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 x) < -0.0100000000000000002

   1. Initial program 100.0%

      \[\sin x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\sin x + {y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{6} \cdot \sin x\right)} \]
   4. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \sin x + \color{blue}{\left(\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right)\right) \cdot {y}^{2} + \left(\frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}\right)} \]

      2. *-rgt-identity N/A

         \[\leadsto \color{blue}{\sin x \cdot 1} + \left(\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right)\right) \cdot {y}^{2} + \left(\frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \sin x \cdot 1 + \color{blue}{{y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{6} \cdot \sin x\right)} \]

      4. *-commutative N/A

         \[\leadsto \sin x \cdot 1 + \color{blue}{\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}} \]

      5. associate-*r* N/A

         \[\leadsto \sin x \cdot 1 + \left(\color{blue}{\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot \sin x} + \frac{1}{6} \cdot \sin x\right) \cdot {y}^{2} \]

      6. distribute-rgt-out N/A

         \[\leadsto \sin x \cdot 1 + \color{blue}{\left(\sin x \cdot \left(\frac{1}{120} \cdot {y}^{2} + \frac{1}{6}\right)\right)} \cdot {y}^{2} \]

      7. +-commutative N/A

         \[\leadsto \sin x \cdot 1 + \left(\sin x \cdot \color{blue}{\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)}\right) \cdot {y}^{2} \]

      8. associate-*l* N/A

         \[\leadsto \sin x \cdot 1 + \color{blue}{\sin x \cdot \left(\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) \cdot {y}^{2}\right)} \]

      9. *-commutative N/A

         \[\leadsto \sin x \cdot 1 + \sin x \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} \]

      10. distribute-lft-in N/A

          \[\leadsto \color{blue}{\sin x \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \color{blue}{\sin x \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} \]

      12. lower-sin.f64 N/A

          \[\leadsto \color{blue}{\sin x} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \sin x \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) + 1\right)} \]

   5. Simplified 93.6%

      \[\leadsto \color{blue}{\sin x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + \left(\frac{-1}{6} \cdot \left({x}^{2} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(1 + \left(\frac{-1}{6} \cdot \left({x}^{2} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto x \cdot \left(1 + \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) + \frac{-1}{6} \cdot \left({x}^{2} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right)}\right) \]

      3. associate-+r+ N/A

         \[\leadsto x \cdot \color{blue}{\left(\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) + \frac{-1}{6} \cdot \left({x}^{2} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right)} \]

      4. associate-*r* N/A

         \[\leadsto x \cdot \left(\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) + \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2}\right) \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)}\right) \]

      5. distribute-rgt1-in N/A

         \[\leadsto x \cdot \color{blue}{\left(\left(\frac{-1}{6} \cdot {x}^{2} + 1\right) \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)} \]

      6. +-commutative N/A

         \[\leadsto x \cdot \left(\color{blue}{\left(1 + \frac{-1}{6} \cdot {x}^{2}\right)} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(\left(1 + \frac{-1}{6} \cdot {x}^{2}\right) \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)} \]

   8. Simplified 26.6%

      \[\leadsto \color{blue}{x \cdot \left(\mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)\right)} \]

   if -0.0100000000000000002 < (sin.f64 x)

   1. Initial program 100.0%

      \[\sin x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\sin x + {y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x + {y}^{2} \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{120} \cdot \sin x\right)\right)} \]
   4. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \sin x + \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x\right) + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{120} \cdot \sin x\right)\right)\right)} \]

      2. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\sin x + {y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x\right)\right) + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{120} \cdot \sin x\right)\right)} \]

      3. *-lft-identity N/A

         \[\leadsto \left(\color{blue}{1 \cdot \sin x} + {y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x\right)\right) + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{120} \cdot \sin x\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \left(1 \cdot \sin x + \color{blue}{\left({y}^{2} \cdot \frac{1}{6}\right) \cdot \sin x}\right) + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{120} \cdot \sin x\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \left(1 \cdot \sin x + \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right)} \cdot \sin x\right) + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{120} \cdot \sin x\right)\right) \]

      6. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\sin x \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)} + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{120} \cdot \sin x\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \sin x \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right) + \color{blue}{\left({y}^{2} \cdot {y}^{2}\right) \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{120} \cdot \sin x\right)} \]

      8. *-commutative N/A

         \[\leadsto \sin x \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right) + \color{blue}{\left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{120} \cdot \sin x\right) \cdot \left({y}^{2} \cdot {y}^{2}\right)} \]

      9. associate-*r* N/A

         \[\leadsto \sin x \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right) + \left(\color{blue}{\left(\frac{1}{5040} \cdot {y}^{2}\right) \cdot \sin x} + \frac{1}{120} \cdot \sin x\right) \cdot \left({y}^{2} \cdot {y}^{2}\right) \]

      10. distribute-rgt-out N/A

          \[\leadsto \sin x \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right) + \color{blue}{\left(\sin x \cdot \left(\frac{1}{5040} \cdot {y}^{2} + \frac{1}{120}\right)\right)} \cdot \left({y}^{2} \cdot {y}^{2}\right) \]

      11. +-commutative N/A

          \[\leadsto \sin x \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right) + \left(\sin x \cdot \color{blue}{\left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)}\right) \cdot \left({y}^{2} \cdot {y}^{2}\right) \]

   5. Simplified 91.2%

      \[\leadsto \color{blue}{\sin x \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right)\right)} \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) + 1\right)}\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(\left({x}^{2} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right) \cdot x + 1 \cdot x\right)} \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      3. *-lft-identity N/A

         \[\leadsto \left(\left({x}^{2} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right) \cdot x + \color{blue}{x}\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      4. associate-*l* N/A

         \[\leadsto \left(\color{blue}{{x}^{2} \cdot \left(\left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) \cdot x\right)} + x\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{2}, \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) \cdot x, x\right)} \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) \cdot x, x\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) \cdot x, x\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{\left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) \cdot x}, x\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{\left(\frac{1}{120} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)} \cdot x, x\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \left(\color{blue}{{x}^{2} \cdot \frac{1}{120}} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right) \cdot x, x\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{120} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right) \cdot x, x\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      12. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \left(\color{blue}{x \cdot \left(x \cdot \frac{1}{120}\right)} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right) \cdot x, x\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \left(x \cdot \left(x \cdot \frac{1}{120}\right) + \color{blue}{\frac{-1}{6}}\right) \cdot x, x\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1}{120}, \frac{-1}{6}\right)} \cdot x, x\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      15. lower-*.f64 68.8

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot 0.008333333333333333}, -0.16666666666666666\right) \cdot x, x\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right) \]

   8. Simplified 68.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.008333333333333333, -0.16666666666666666\right) \cdot x, x\right)} \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin x \leq -0.01:\\ \;\;\;\;x \cdot \left(\mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right) \cdot \mathsf{fma}\left(x \cdot x, x \cdot \mathsf{fma}\left(x, x \cdot 0.008333333333333333, -0.16666666666666666\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 57.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right)\\ \mathbf{if}\;\sin x \leq 0.002:\\ \;\;\;\;\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, t\_0, 0.16666666666666666\right), 1\right) \cdot \mathsf{fma}\left(x \cdot x, x \cdot -0.16666666666666666, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot \left(y \cdot y\right), \mathsf{fma}\left(y, y \cdot t\_0, 0.16666666666666666\right), y\right) \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma (* y y) 0.0001984126984126984 0.008333333333333333)))
   (if (<= (sin x) 0.002)
     (*
      (fma y (* y (fma (* y y) t_0 0.16666666666666666)) 1.0)
      (fma (* x x) (* x -0.16666666666666666) x))
     (* (fma (* y (* y y)) (fma y (* y t_0) 0.16666666666666666) y) (/ x y)))))

C

double code(double x, double y) {
	double t_0 = fma((y * y), 0.0001984126984126984, 0.008333333333333333);
	double tmp;
	if (sin(x) <= 0.002) {
		tmp = fma(y, (y * fma((y * y), t_0, 0.16666666666666666)), 1.0) * fma((x * x), (x * -0.16666666666666666), x);
	} else {
		tmp = fma((y * (y * y)), fma(y, (y * t_0), 0.16666666666666666), y) * (x / y);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333)
	tmp = 0.0
	if (sin(x) <= 0.002)
		tmp = Float64(fma(y, Float64(y * fma(Float64(y * y), t_0, 0.16666666666666666)), 1.0) * fma(Float64(x * x), Float64(x * -0.16666666666666666), x));
	else
		tmp = Float64(fma(Float64(y * Float64(y * y)), fma(y, Float64(y * t_0), 0.16666666666666666), y) * Float64(x / y));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision]}, If[LessEqual[N[Sin[x], $MachinePrecision], 0.002], N[(N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * t$95$0 + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * -0.16666666666666666), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(y * N[(y * t$95$0), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 x) < 2e-3

   1. Initial program 100.0%

      \[\sin x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\sin x + {y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x + {y}^{2} \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{120} \cdot \sin x\right)\right)} \]
   4. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \sin x + \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x\right) + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{120} \cdot \sin x\right)\right)\right)} \]

      2. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\sin x + {y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x\right)\right) + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{120} \cdot \sin x\right)\right)} \]

      3. *-lft-identity N/A

         \[\leadsto \left(\color{blue}{1 \cdot \sin x} + {y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x\right)\right) + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{120} \cdot \sin x\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \left(1 \cdot \sin x + \color{blue}{\left({y}^{2} \cdot \frac{1}{6}\right) \cdot \sin x}\right) + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{120} \cdot \sin x\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \left(1 \cdot \sin x + \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right)} \cdot \sin x\right) + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{120} \cdot \sin x\right)\right) \]

      6. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\sin x \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)} + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{120} \cdot \sin x\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \sin x \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right) + \color{blue}{\left({y}^{2} \cdot {y}^{2}\right) \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{120} \cdot \sin x\right)} \]

      8. *-commutative N/A

         \[\leadsto \sin x \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right) + \color{blue}{\left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{120} \cdot \sin x\right) \cdot \left({y}^{2} \cdot {y}^{2}\right)} \]

      9. associate-*r* N/A

         \[\leadsto \sin x \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right) + \left(\color{blue}{\left(\frac{1}{5040} \cdot {y}^{2}\right) \cdot \sin x} + \frac{1}{120} \cdot \sin x\right) \cdot \left({y}^{2} \cdot {y}^{2}\right) \]

      10. distribute-rgt-out N/A

          \[\leadsto \sin x \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right) + \color{blue}{\left(\sin x \cdot \left(\frac{1}{5040} \cdot {y}^{2} + \frac{1}{120}\right)\right)} \cdot \left({y}^{2} \cdot {y}^{2}\right) \]

      11. +-commutative N/A

          \[\leadsto \sin x \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right) + \left(\sin x \cdot \color{blue}{\left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)}\right) \cdot \left({y}^{2} \cdot {y}^{2}\right) \]

   5. Simplified 92.6%

      \[\leadsto \color{blue}{\sin x \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)} \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(x \cdot \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2} + 1\right)}\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(\left(\frac{-1}{6} \cdot {x}^{2}\right) \cdot x + 1 \cdot x\right)} \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      3. *-lft-identity N/A

         \[\leadsto \left(\left(\frac{-1}{6} \cdot {x}^{2}\right) \cdot x + \color{blue}{x}\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      4. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left({x}^{2} \cdot \frac{-1}{6}\right)} \cdot x + x\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      5. associate-*l* N/A

         \[\leadsto \left(\color{blue}{{x}^{2} \cdot \left(\frac{-1}{6} \cdot x\right)} + x\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{-1}{6} \cdot x, x\right)} \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{6} \cdot x, x\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{6} \cdot x, x\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      9. lower-*.f64 67.9

         \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{-0.16666666666666666 \cdot x}, x\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right) \]

   8. Simplified 67.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, -0.16666666666666666 \cdot x, x\right)} \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right) \]

   if 2e-3 < (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

      \[\leadsto \sin x \cdot \frac{\color{blue}{y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)}}{y} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \sin x \cdot \frac{y \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) + 1\right)}}{y} \]

      2. distribute-lft-in N/A

         \[\leadsto \sin x \cdot \frac{\color{blue}{y \cdot \left({y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right) + y \cdot 1}}{y} \]

      3. *-rgt-identity N/A

         \[\leadsto \sin x \cdot \frac{y \cdot \left({y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right) + \color{blue}{y}}{y} \]

      4. lower-fma.f64 N/A

         \[\leadsto \sin x \cdot \frac{\color{blue}{\mathsf{fma}\left(y, {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right), y\right)}}{y} \]

   5. Simplified 92.7%

      \[\leadsto \sin x \cdot \frac{\color{blue}{\mathsf{fma}\left(y, y \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right)\right), y\right)}}{y} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)} \cdot \frac{\mathsf{fma}\left(y, y \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right)\right), y\right)}{y} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(x \cdot \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2} + 1\right)}\right) \cdot \frac{\mathsf{fma}\left(y, y \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right)\right), y\right)}{y} \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(\left(\frac{-1}{6} \cdot {x}^{2}\right) \cdot x + 1 \cdot x\right)} \cdot \frac{\mathsf{fma}\left(y, y \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right)\right), y\right)}{y} \]

      3. *-lft-identity N/A

         \[\leadsto \left(\left(\frac{-1}{6} \cdot {x}^{2}\right) \cdot x + \color{blue}{x}\right) \cdot \frac{\mathsf{fma}\left(y, y \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right)\right), y\right)}{y} \]

      4. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left({x}^{2} \cdot \frac{-1}{6}\right)} \cdot x + x\right) \cdot \frac{\mathsf{fma}\left(y, y \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right)\right), y\right)}{y} \]

      5. associate-*l* N/A

         \[\leadsto \left(\color{blue}{{x}^{2} \cdot \left(\frac{-1}{6} \cdot x\right)} + x\right) \cdot \frac{\mathsf{fma}\left(y, y \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right)\right), y\right)}{y} \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{-1}{6} \cdot x, x\right)} \cdot \frac{\mathsf{fma}\left(y, y \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right)\right), y\right)}{y} \]

      7. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{6} \cdot x, x\right) \cdot \frac{\mathsf{fma}\left(y, y \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right)\right), y\right)}{y} \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{6} \cdot x, x\right) \cdot \frac{\mathsf{fma}\left(y, y \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right)\right), y\right)}{y} \]

      9. lower-*.f64 31.1

         \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{-0.16666666666666666 \cdot x}, x\right) \cdot \frac{\mathsf{fma}\left(y, y \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right)\right), y\right)}{y} \]

   8. Simplified 31.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, -0.16666666666666666 \cdot x, x\right)} \cdot \frac{\mathsf{fma}\left(y, y \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right)\right), y\right)}{y} \]

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x \cdot \left(y + {y}^{3} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)}{y}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{\color{blue}{\left(y + {y}^{3} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right) \cdot x}}{y} \]

       2. associate-/l* N/A

          \[\leadsto \color{blue}{\left(y + {y}^{3} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right) \cdot \frac{x}{y}} \]

       3. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(y + {y}^{3} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right) \cdot \frac{x}{y}} \]

   11. Simplified 27.7%

       \[\leadsto \color{blue}{\mathsf{fma}\left(y \cdot \left(y \cdot y\right), \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y\right) \cdot \frac{x}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin x \leq 0.002:\\ \;\;\;\;\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right) \cdot \mathsf{fma}\left(x \cdot x, x \cdot -0.16666666666666666, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot \left(y \cdot y\right), \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y\right) \cdot \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 78.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.8 \cdot 10^{-11}:\\ \;\;\;\;\frac{\sinh y}{y} \cdot \mathsf{fma}\left(-0.16666666666666666, x \cdot \left(x \cdot x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1.8e-11)
   (* (/ (sinh y) y) (fma -0.16666666666666666 (* x (* x x)) x))
   (*
    (sin x)
    (fma
     y
     (*
      y
      (fma
       (* y y)
       (fma (* y y) 0.0001984126984126984 0.008333333333333333)
       0.16666666666666666))
     1.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.8e-11) {
		tmp = (sinh(y) / y) * fma(-0.16666666666666666, (x * (x * x)), x);
	} else {
		tmp = sin(x) * fma(y, (y * fma((y * y), fma((y * y), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666)), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.8e-11)
		tmp = Float64(Float64(sinh(y) / y) * fma(-0.16666666666666666, Float64(x * Float64(x * x)), x));
	else
		tmp = Float64(sin(x) * fma(y, Float64(y * fma(Float64(y * y), fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666)), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.8e-11], N[(N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision] * N[(-0.16666666666666666 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], N[(N[Sin[x], $MachinePrecision] * N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.79999999999999992e-11

   1. Initial program 100.0%

      \[\sin x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)} \cdot \frac{\sinh y}{y} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(x \cdot \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2} + 1\right)}\right) \cdot \frac{\sinh y}{y} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{\left(x \cdot \left(\frac{-1}{6} \cdot {x}^{2}\right) + x \cdot 1\right)} \cdot \frac{\sinh y}{y} \]

      3. *-commutative N/A

         \[\leadsto \left(x \cdot \color{blue}{\left({x}^{2} \cdot \frac{-1}{6}\right)} + x \cdot 1\right) \cdot \frac{\sinh y}{y} \]

      4. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(x \cdot {x}^{2}\right) \cdot \frac{-1}{6}} + x \cdot 1\right) \cdot \frac{\sinh y}{y} \]

      5. *-commutative N/A

         \[\leadsto \left(\color{blue}{\frac{-1}{6} \cdot \left(x \cdot {x}^{2}\right)} + x \cdot 1\right) \cdot \frac{\sinh y}{y} \]

      6. *-rgt-identity N/A

         \[\leadsto \left(\frac{-1}{6} \cdot \left(x \cdot {x}^{2}\right) + \color{blue}{x}\right) \cdot \frac{\sinh y}{y} \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1}{6}, x \cdot {x}^{2}, x\right)} \cdot \frac{\sinh y}{y} \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{6}, \color{blue}{x \cdot {x}^{2}}, x\right) \cdot \frac{\sinh y}{y} \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{6}, x \cdot \color{blue}{\left(x \cdot x\right)}, x\right) \cdot \frac{\sinh y}{y} \]

      10. lower-*.f64 77.5

          \[\leadsto \mathsf{fma}\left(-0.16666666666666666, x \cdot \color{blue}{\left(x \cdot x\right)}, x\right) \cdot \frac{\sinh y}{y} \]

   5. Simplified 77.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.16666666666666666, x \cdot \left(x \cdot x\right), x\right)} \cdot \frac{\sinh y}{y} \]

   if 1.79999999999999992e-11 < x

   1. Initial program 100.0%

      \[\sin x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\sin x + {y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x + {y}^{2} \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{120} \cdot \sin x\right)\right)} \]
   4. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \sin x + \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x\right) + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{120} \cdot \sin x\right)\right)\right)} \]

      2. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\sin x + {y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x\right)\right) + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{120} \cdot \sin x\right)\right)} \]

      3. *-lft-identity N/A

         \[\leadsto \left(\color{blue}{1 \cdot \sin x} + {y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x\right)\right) + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{120} \cdot \sin x\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \left(1 \cdot \sin x + \color{blue}{\left({y}^{2} \cdot \frac{1}{6}\right) \cdot \sin x}\right) + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{120} \cdot \sin x\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \left(1 \cdot \sin x + \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right)} \cdot \sin x\right) + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{120} \cdot \sin x\right)\right) \]

      6. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\sin x \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)} + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{120} \cdot \sin x\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \sin x \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right) + \color{blue}{\left({y}^{2} \cdot {y}^{2}\right) \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{120} \cdot \sin x\right)} \]

      8. *-commutative N/A

         \[\leadsto \sin x \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right) + \color{blue}{\left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{120} \cdot \sin x\right) \cdot \left({y}^{2} \cdot {y}^{2}\right)} \]

      9. associate-*r* N/A

         \[\leadsto \sin x \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right) + \left(\color{blue}{\left(\frac{1}{5040} \cdot {y}^{2}\right) \cdot \sin x} + \frac{1}{120} \cdot \sin x\right) \cdot \left({y}^{2} \cdot {y}^{2}\right) \]

      10. distribute-rgt-out N/A

          \[\leadsto \sin x \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right) + \color{blue}{\left(\sin x \cdot \left(\frac{1}{5040} \cdot {y}^{2} + \frac{1}{120}\right)\right)} \cdot \left({y}^{2} \cdot {y}^{2}\right) \]

      11. +-commutative N/A

          \[\leadsto \sin x \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right) + \left(\sin x \cdot \color{blue}{\left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)}\right) \cdot \left({y}^{2} \cdot {y}^{2}\right) \]

   5. Simplified 94.6%

      \[\leadsto \color{blue}{\sin x \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.8 \cdot 10^{-11}:\\ \;\;\;\;\frac{\sinh y}{y} \cdot \mathsf{fma}\left(-0.16666666666666666, x \cdot \left(x \cdot x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 57.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin x \leq -0.01:\\ \;\;\;\;x \cdot \left(\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.008333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(y \cdot y\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sin x) -0.01)
   (*
    x
    (*
     (* (* y y) (* y y))
     (fma (* x x) -0.001388888888888889 0.008333333333333333)))
   (fma
    (*
     (* y y)
     (fma
      (* y y)
      (fma y (* y 0.0001984126984126984) 0.008333333333333333)
      0.16666666666666666))
    x
    x)))

C

double code(double x, double y) {
	double tmp;
	if (sin(x) <= -0.01) {
		tmp = x * (((y * y) * (y * y)) * fma((x * x), -0.001388888888888889, 0.008333333333333333));
	} else {
		tmp = fma(((y * y) * fma((y * y), fma(y, (y * 0.0001984126984126984), 0.008333333333333333), 0.16666666666666666)), x, x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (sin(x) <= -0.01)
		tmp = Float64(x * Float64(Float64(Float64(y * y) * Float64(y * y)) * fma(Float64(x * x), -0.001388888888888889, 0.008333333333333333)));
	else
		tmp = fma(Float64(Float64(y * y) * fma(Float64(y * y), fma(y, Float64(y * 0.0001984126984126984), 0.008333333333333333), 0.16666666666666666)), x, x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[Sin[x], $MachinePrecision], -0.01], N[(x * N[(N[(N[(y * y), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.001388888888888889 + 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * 0.0001984126984126984), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] * x + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 x) < -0.0100000000000000002

   1. Initial program 100.0%

      \[\sin x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\sin x + {y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{6} \cdot \sin x\right)} \]
   4. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \sin x + \color{blue}{\left(\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right)\right) \cdot {y}^{2} + \left(\frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}\right)} \]

      2. *-rgt-identity N/A

         \[\leadsto \color{blue}{\sin x \cdot 1} + \left(\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right)\right) \cdot {y}^{2} + \left(\frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \sin x \cdot 1 + \color{blue}{{y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{6} \cdot \sin x\right)} \]

      4. *-commutative N/A

         \[\leadsto \sin x \cdot 1 + \color{blue}{\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}} \]

      5. associate-*r* N/A

         \[\leadsto \sin x \cdot 1 + \left(\color{blue}{\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot \sin x} + \frac{1}{6} \cdot \sin x\right) \cdot {y}^{2} \]

      6. distribute-rgt-out N/A

         \[\leadsto \sin x \cdot 1 + \color{blue}{\left(\sin x \cdot \left(\frac{1}{120} \cdot {y}^{2} + \frac{1}{6}\right)\right)} \cdot {y}^{2} \]

      7. +-commutative N/A

         \[\leadsto \sin x \cdot 1 + \left(\sin x \cdot \color{blue}{\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)}\right) \cdot {y}^{2} \]

      8. associate-*l* N/A

         \[\leadsto \sin x \cdot 1 + \color{blue}{\sin x \cdot \left(\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) \cdot {y}^{2}\right)} \]

      9. *-commutative N/A

         \[\leadsto \sin x \cdot 1 + \sin x \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} \]

      10. distribute-lft-in N/A

          \[\leadsto \color{blue}{\sin x \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \color{blue}{\sin x \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} \]

      12. lower-sin.f64 N/A

          \[\leadsto \color{blue}{\sin x} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \sin x \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) + 1\right)} \]

   5. Simplified 93.6%

      \[\leadsto \color{blue}{\sin x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + \left(\frac{-1}{6} \cdot \left({x}^{2} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(1 + \left(\frac{-1}{6} \cdot \left({x}^{2} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto x \cdot \left(1 + \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) + \frac{-1}{6} \cdot \left({x}^{2} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right)}\right) \]

      3. associate-+r+ N/A

         \[\leadsto x \cdot \color{blue}{\left(\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) + \frac{-1}{6} \cdot \left({x}^{2} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right)} \]

      4. associate-*r* N/A

         \[\leadsto x \cdot \left(\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) + \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2}\right) \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)}\right) \]

      5. distribute-rgt1-in N/A

         \[\leadsto x \cdot \color{blue}{\left(\left(\frac{-1}{6} \cdot {x}^{2} + 1\right) \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)} \]

      6. +-commutative N/A

         \[\leadsto x \cdot \left(\color{blue}{\left(1 + \frac{-1}{6} \cdot {x}^{2}\right)} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(\left(1 + \frac{-1}{6} \cdot {x}^{2}\right) \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)} \]

   8. Simplified 26.6%

      \[\leadsto \color{blue}{x \cdot \left(\mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)\right)} \]

   9. Taylor expanded in y around inf

      \[\leadsto x \cdot \color{blue}{\left(\frac{1}{120} \cdot \left({y}^{4} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto x \cdot \color{blue}{\left(\left(\frac{1}{120} \cdot {y}^{4}\right) \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)} \]

       2. *-commutative N/A

          \[\leadsto x \cdot \left(\color{blue}{\left({y}^{4} \cdot \frac{1}{120}\right)} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right) \]

       3. associate-*r* N/A

          \[\leadsto x \cdot \color{blue}{\left({y}^{4} \cdot \left(\frac{1}{120} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto x \cdot \color{blue}{\left({y}^{4} \cdot \left(\frac{1}{120} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right)} \]

       5. metadata-eval N/A

          \[\leadsto x \cdot \left({y}^{\color{blue}{\left(2 \cdot 2\right)}} \cdot \left(\frac{1}{120} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right) \]

       6. pow-sqr N/A

          \[\leadsto x \cdot \left(\color{blue}{\left({y}^{2} \cdot {y}^{2}\right)} \cdot \left(\frac{1}{120} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right) \]

       7. lower-*.f64 N/A

          \[\leadsto x \cdot \left(\color{blue}{\left({y}^{2} \cdot {y}^{2}\right)} \cdot \left(\frac{1}{120} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right) \]

       8. unpow2 N/A

          \[\leadsto x \cdot \left(\left(\color{blue}{\left(y \cdot y\right)} \cdot {y}^{2}\right) \cdot \left(\frac{1}{120} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right) \]

       9. lower-*.f64 N/A

          \[\leadsto x \cdot \left(\left(\color{blue}{\left(y \cdot y\right)} \cdot {y}^{2}\right) \cdot \left(\frac{1}{120} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right) \]

       10. unpow2 N/A

           \[\leadsto x \cdot \left(\left(\left(y \cdot y\right) \cdot \color{blue}{\left(y \cdot y\right)}\right) \cdot \left(\frac{1}{120} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right) \]

       11. lower-*.f64 N/A

           \[\leadsto x \cdot \left(\left(\left(y \cdot y\right) \cdot \color{blue}{\left(y \cdot y\right)}\right) \cdot \left(\frac{1}{120} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right) \]

       12. +-commutative N/A

           \[\leadsto x \cdot \left(\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \left(\frac{1}{120} \cdot \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2} + 1\right)}\right)\right) \]

       13. distribute-rgt-in N/A

           \[\leadsto x \cdot \left(\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \color{blue}{\left(\left(\frac{-1}{6} \cdot {x}^{2}\right) \cdot \frac{1}{120} + 1 \cdot \frac{1}{120}\right)}\right) \]

       14. *-commutative N/A

           \[\leadsto x \cdot \left(\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \left(\color{blue}{\left({x}^{2} \cdot \frac{-1}{6}\right)} \cdot \frac{1}{120} + 1 \cdot \frac{1}{120}\right)\right) \]

       15. associate-*l* N/A

           \[\leadsto x \cdot \left(\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \left(\color{blue}{{x}^{2} \cdot \left(\frac{-1}{6} \cdot \frac{1}{120}\right)} + 1 \cdot \frac{1}{120}\right)\right) \]

       16. metadata-eval N/A

           \[\leadsto x \cdot \left(\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \left({x}^{2} \cdot \color{blue}{\frac{-1}{720}} + 1 \cdot \frac{1}{120}\right)\right) \]

       17. metadata-eval N/A

           \[\leadsto x \cdot \left(\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \left({x}^{2} \cdot \color{blue}{\left(\frac{1}{120} \cdot \frac{-1}{6}\right)} + 1 \cdot \frac{1}{120}\right)\right) \]

       18. metadata-eval N/A

           \[\leadsto x \cdot \left(\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \left({x}^{2} \cdot \left(\frac{1}{120} \cdot \frac{-1}{6}\right) + \color{blue}{\frac{1}{120}}\right)\right) \]

       19. lower-fma.f64 N/A

           \[\leadsto x \cdot \left(\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{120} \cdot \frac{-1}{6}, \frac{1}{120}\right)}\right) \]

       20. unpow2 N/A

           \[\leadsto x \cdot \left(\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{120} \cdot \frac{-1}{6}, \frac{1}{120}\right)\right) \]

       21. lower-*.f64 N/A

           \[\leadsto x \cdot \left(\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{120} \cdot \frac{-1}{6}, \frac{1}{120}\right)\right) \]

       22. metadata-eval 26.3

           \[\leadsto x \cdot \left(\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{-0.001388888888888889}, 0.008333333333333333\right)\right) \]

   11. Simplified 26.3%

       \[\leadsto x \cdot \color{blue}{\left(\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.008333333333333333\right)\right)} \]

   if -0.0100000000000000002 < (sin.f64 x)

   1. Initial program 100.0%

      \[\sin x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\sin x + {y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x + {y}^{2} \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{120} \cdot \sin x\right)\right)} \]
   4. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \sin x + \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x\right) + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{120} \cdot \sin x\right)\right)\right)} \]

      2. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\sin x + {y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x\right)\right) + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{120} \cdot \sin x\right)\right)} \]

      3. *-lft-identity N/A

         \[\leadsto \left(\color{blue}{1 \cdot \sin x} + {y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x\right)\right) + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{120} \cdot \sin x\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \left(1 \cdot \sin x + \color{blue}{\left({y}^{2} \cdot \frac{1}{6}\right) \cdot \sin x}\right) + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{120} \cdot \sin x\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \left(1 \cdot \sin x + \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right)} \cdot \sin x\right) + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{120} \cdot \sin x\right)\right) \]

      6. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\sin x \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)} + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{120} \cdot \sin x\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \sin x \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right) + \color{blue}{\left({y}^{2} \cdot {y}^{2}\right) \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{120} \cdot \sin x\right)} \]

      8. *-commutative N/A

         \[\leadsto \sin x \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right) + \color{blue}{\left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{120} \cdot \sin x\right) \cdot \left({y}^{2} \cdot {y}^{2}\right)} \]

      9. associate-*r* N/A

         \[\leadsto \sin x \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right) + \left(\color{blue}{\left(\frac{1}{5040} \cdot {y}^{2}\right) \cdot \sin x} + \frac{1}{120} \cdot \sin x\right) \cdot \left({y}^{2} \cdot {y}^{2}\right) \]

      10. distribute-rgt-out N/A

          \[\leadsto \sin x \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right) + \color{blue}{\left(\sin x \cdot \left(\frac{1}{5040} \cdot {y}^{2} + \frac{1}{120}\right)\right)} \cdot \left({y}^{2} \cdot {y}^{2}\right) \]

      11. +-commutative N/A

          \[\leadsto \sin x \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right) + \left(\sin x \cdot \color{blue}{\left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)}\right) \cdot \left({y}^{2} \cdot {y}^{2}\right) \]

   5. Simplified 91.2%

      \[\leadsto \color{blue}{\sin x \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) + 1\right)} \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right) \cdot x + 1 \cdot x} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot {y}^{2}\right)} \cdot x + 1 \cdot x \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot \left({y}^{2} \cdot x\right)} + 1 \cdot x \]

      5. *-lft-identity N/A

         \[\leadsto \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot \left({y}^{2} \cdot x\right) + \color{blue}{x} \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right), {y}^{2} \cdot x, x\right)} \]

   8. Simplified 65.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), \left(y \cdot y\right) \cdot x, x\right)} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(y \cdot \left(y \cdot \left(\color{blue}{\left(y \cdot y\right)} \cdot \frac{1}{5040} + \frac{1}{120}\right)\right) + \frac{1}{6}\right) \cdot \left(\left(y \cdot y\right) \cdot x\right) + x \]

      2. lift-fma.f64 N/A

         \[\leadsto \left(y \cdot \left(y \cdot \color{blue}{\mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right)}\right) + \frac{1}{6}\right) \cdot \left(\left(y \cdot y\right) \cdot x\right) + x \]

      3. lift-*.f64 N/A

         \[\leadsto \left(y \cdot \color{blue}{\left(y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right)\right)} + \frac{1}{6}\right) \cdot \left(\left(y \cdot y\right) \cdot x\right) + x \]

      4. lift-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right)} \cdot \left(\left(y \cdot y\right) \cdot x\right) + x \]

      5. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right) \cdot \left(\color{blue}{\left(y \cdot y\right)} \cdot x\right) + x \]

      6. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right) \cdot \color{blue}{\left(\left(y \cdot y\right) \cdot x\right)} + x \]

   10. Applied egg-rr 67.3%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\left(y \cdot y\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), x, x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 56.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin x \leq -0.01:\\ \;\;\;\;x \cdot \left(\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.008333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), x \cdot y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sin x) -0.01)
   (*
    x
    (*
     (* (* y y) (* y y))
     (fma (* x x) -0.001388888888888889 0.008333333333333333)))
   (fma
    (*
     y
     (fma
      (* y y)
      (fma y (* y 0.0001984126984126984) 0.008333333333333333)
      0.16666666666666666))
    (* x y)
    x)))

C

double code(double x, double y) {
	double tmp;
	if (sin(x) <= -0.01) {
		tmp = x * (((y * y) * (y * y)) * fma((x * x), -0.001388888888888889, 0.008333333333333333));
	} else {
		tmp = fma((y * fma((y * y), fma(y, (y * 0.0001984126984126984), 0.008333333333333333), 0.16666666666666666)), (x * y), x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (sin(x) <= -0.01)
		tmp = Float64(x * Float64(Float64(Float64(y * y) * Float64(y * y)) * fma(Float64(x * x), -0.001388888888888889, 0.008333333333333333)));
	else
		tmp = fma(Float64(y * fma(Float64(y * y), fma(y, Float64(y * 0.0001984126984126984), 0.008333333333333333), 0.16666666666666666)), Float64(x * y), x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[Sin[x], $MachinePrecision], -0.01], N[(x * N[(N[(N[(y * y), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.001388888888888889 + 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * 0.0001984126984126984), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] * N[(x * y), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 x) < -0.0100000000000000002

   1. Initial program 100.0%

      \[\sin x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\sin x + {y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{6} \cdot \sin x\right)} \]
   4. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \sin x + \color{blue}{\left(\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right)\right) \cdot {y}^{2} + \left(\frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}\right)} \]

      2. *-rgt-identity N/A

         \[\leadsto \color{blue}{\sin x \cdot 1} + \left(\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right)\right) \cdot {y}^{2} + \left(\frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \sin x \cdot 1 + \color{blue}{{y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{6} \cdot \sin x\right)} \]

      4. *-commutative N/A

         \[\leadsto \sin x \cdot 1 + \color{blue}{\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}} \]

      5. associate-*r* N/A

         \[\leadsto \sin x \cdot 1 + \left(\color{blue}{\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot \sin x} + \frac{1}{6} \cdot \sin x\right) \cdot {y}^{2} \]

      6. distribute-rgt-out N/A

         \[\leadsto \sin x \cdot 1 + \color{blue}{\left(\sin x \cdot \left(\frac{1}{120} \cdot {y}^{2} + \frac{1}{6}\right)\right)} \cdot {y}^{2} \]

      7. +-commutative N/A

         \[\leadsto \sin x \cdot 1 + \left(\sin x \cdot \color{blue}{\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)}\right) \cdot {y}^{2} \]

      8. associate-*l* N/A

         \[\leadsto \sin x \cdot 1 + \color{blue}{\sin x \cdot \left(\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) \cdot {y}^{2}\right)} \]

      9. *-commutative N/A

         \[\leadsto \sin x \cdot 1 + \sin x \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} \]

      10. distribute-lft-in N/A

          \[\leadsto \color{blue}{\sin x \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \color{blue}{\sin x \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} \]

      12. lower-sin.f64 N/A

          \[\leadsto \color{blue}{\sin x} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \sin x \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) + 1\right)} \]

   5. Simplified 93.6%

      \[\leadsto \color{blue}{\sin x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + \left(\frac{-1}{6} \cdot \left({x}^{2} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(1 + \left(\frac{-1}{6} \cdot \left({x}^{2} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto x \cdot \left(1 + \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) + \frac{-1}{6} \cdot \left({x}^{2} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right)}\right) \]

      3. associate-+r+ N/A

         \[\leadsto x \cdot \color{blue}{\left(\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) + \frac{-1}{6} \cdot \left({x}^{2} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right)} \]

      4. associate-*r* N/A

         \[\leadsto x \cdot \left(\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) + \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2}\right) \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)}\right) \]

      5. distribute-rgt1-in N/A

         \[\leadsto x \cdot \color{blue}{\left(\left(\frac{-1}{6} \cdot {x}^{2} + 1\right) \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)} \]

      6. +-commutative N/A

         \[\leadsto x \cdot \left(\color{blue}{\left(1 + \frac{-1}{6} \cdot {x}^{2}\right)} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(\left(1 + \frac{-1}{6} \cdot {x}^{2}\right) \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)} \]

   8. Simplified 26.6%

      \[\leadsto \color{blue}{x \cdot \left(\mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)\right)} \]

   9. Taylor expanded in y around inf

      \[\leadsto x \cdot \color{blue}{\left(\frac{1}{120} \cdot \left({y}^{4} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto x \cdot \color{blue}{\left(\left(\frac{1}{120} \cdot {y}^{4}\right) \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)} \]

       2. *-commutative N/A

          \[\leadsto x \cdot \left(\color{blue}{\left({y}^{4} \cdot \frac{1}{120}\right)} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right) \]

       3. associate-*r* N/A

          \[\leadsto x \cdot \color{blue}{\left({y}^{4} \cdot \left(\frac{1}{120} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto x \cdot \color{blue}{\left({y}^{4} \cdot \left(\frac{1}{120} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right)} \]

       5. metadata-eval N/A

          \[\leadsto x \cdot \left({y}^{\color{blue}{\left(2 \cdot 2\right)}} \cdot \left(\frac{1}{120} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right) \]

       6. pow-sqr N/A

          \[\leadsto x \cdot \left(\color{blue}{\left({y}^{2} \cdot {y}^{2}\right)} \cdot \left(\frac{1}{120} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right) \]

       7. lower-*.f64 N/A

          \[\leadsto x \cdot \left(\color{blue}{\left({y}^{2} \cdot {y}^{2}\right)} \cdot \left(\frac{1}{120} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right) \]

       8. unpow2 N/A

          \[\leadsto x \cdot \left(\left(\color{blue}{\left(y \cdot y\right)} \cdot {y}^{2}\right) \cdot \left(\frac{1}{120} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right) \]

       9. lower-*.f64 N/A

          \[\leadsto x \cdot \left(\left(\color{blue}{\left(y \cdot y\right)} \cdot {y}^{2}\right) \cdot \left(\frac{1}{120} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right) \]

       10. unpow2 N/A

           \[\leadsto x \cdot \left(\left(\left(y \cdot y\right) \cdot \color{blue}{\left(y \cdot y\right)}\right) \cdot \left(\frac{1}{120} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right) \]

       11. lower-*.f64 N/A

           \[\leadsto x \cdot \left(\left(\left(y \cdot y\right) \cdot \color{blue}{\left(y \cdot y\right)}\right) \cdot \left(\frac{1}{120} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right) \]

       12. +-commutative N/A

           \[\leadsto x \cdot \left(\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \left(\frac{1}{120} \cdot \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2} + 1\right)}\right)\right) \]

       13. distribute-rgt-in N/A

           \[\leadsto x \cdot \left(\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \color{blue}{\left(\left(\frac{-1}{6} \cdot {x}^{2}\right) \cdot \frac{1}{120} + 1 \cdot \frac{1}{120}\right)}\right) \]

       14. *-commutative N/A

           \[\leadsto x \cdot \left(\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \left(\color{blue}{\left({x}^{2} \cdot \frac{-1}{6}\right)} \cdot \frac{1}{120} + 1 \cdot \frac{1}{120}\right)\right) \]

       15. associate-*l* N/A

           \[\leadsto x \cdot \left(\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \left(\color{blue}{{x}^{2} \cdot \left(\frac{-1}{6} \cdot \frac{1}{120}\right)} + 1 \cdot \frac{1}{120}\right)\right) \]

       16. metadata-eval N/A

           \[\leadsto x \cdot \left(\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \left({x}^{2} \cdot \color{blue}{\frac{-1}{720}} + 1 \cdot \frac{1}{120}\right)\right) \]

       17. metadata-eval N/A

           \[\leadsto x \cdot \left(\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \left({x}^{2} \cdot \color{blue}{\left(\frac{1}{120} \cdot \frac{-1}{6}\right)} + 1 \cdot \frac{1}{120}\right)\right) \]

       18. metadata-eval N/A

           \[\leadsto x \cdot \left(\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \left({x}^{2} \cdot \left(\frac{1}{120} \cdot \frac{-1}{6}\right) + \color{blue}{\frac{1}{120}}\right)\right) \]

       19. lower-fma.f64 N/A

           \[\leadsto x \cdot \left(\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{120} \cdot \frac{-1}{6}, \frac{1}{120}\right)}\right) \]

       20. unpow2 N/A

           \[\leadsto x \cdot \left(\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{120} \cdot \frac{-1}{6}, \frac{1}{120}\right)\right) \]

       21. lower-*.f64 N/A

           \[\leadsto x \cdot \left(\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{120} \cdot \frac{-1}{6}, \frac{1}{120}\right)\right) \]

       22. metadata-eval 26.3

           \[\leadsto x \cdot \left(\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{-0.001388888888888889}, 0.008333333333333333\right)\right) \]

   11. Simplified 26.3%

       \[\leadsto x \cdot \color{blue}{\left(\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.008333333333333333\right)\right)} \]

   if -0.0100000000000000002 < (sin.f64 x)

   1. Initial program 100.0%

      \[\sin x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\sin x + {y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x + {y}^{2} \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{120} \cdot \sin x\right)\right)} \]
   4. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \sin x + \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x\right) + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{120} \cdot \sin x\right)\right)\right)} \]

      2. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\sin x + {y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x\right)\right) + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{120} \cdot \sin x\right)\right)} \]

      3. *-lft-identity N/A

         \[\leadsto \left(\color{blue}{1 \cdot \sin x} + {y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x\right)\right) + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{120} \cdot \sin x\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \left(1 \cdot \sin x + \color{blue}{\left({y}^{2} \cdot \frac{1}{6}\right) \cdot \sin x}\right) + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{120} \cdot \sin x\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \left(1 \cdot \sin x + \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right)} \cdot \sin x\right) + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{120} \cdot \sin x\right)\right) \]

      6. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\sin x \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)} + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{120} \cdot \sin x\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \sin x \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right) + \color{blue}{\left({y}^{2} \cdot {y}^{2}\right) \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{120} \cdot \sin x\right)} \]

      8. *-commutative N/A

         \[\leadsto \sin x \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right) + \color{blue}{\left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{120} \cdot \sin x\right) \cdot \left({y}^{2} \cdot {y}^{2}\right)} \]

      9. associate-*r* N/A

         \[\leadsto \sin x \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right) + \left(\color{blue}{\left(\frac{1}{5040} \cdot {y}^{2}\right) \cdot \sin x} + \frac{1}{120} \cdot \sin x\right) \cdot \left({y}^{2} \cdot {y}^{2}\right) \]

      10. distribute-rgt-out N/A

          \[\leadsto \sin x \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right) + \color{blue}{\left(\sin x \cdot \left(\frac{1}{5040} \cdot {y}^{2} + \frac{1}{120}\right)\right)} \cdot \left({y}^{2} \cdot {y}^{2}\right) \]

      11. +-commutative N/A

          \[\leadsto \sin x \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right) + \left(\sin x \cdot \color{blue}{\left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)}\right) \cdot \left({y}^{2} \cdot {y}^{2}\right) \]

   5. Simplified 91.2%

      \[\leadsto \color{blue}{\sin x \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) + 1\right)} \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right) \cdot x + 1 \cdot x} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot {y}^{2}\right)} \cdot x + 1 \cdot x \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot \left({y}^{2} \cdot x\right)} + 1 \cdot x \]

      5. *-lft-identity N/A

         \[\leadsto \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot \left({y}^{2} \cdot x\right) + \color{blue}{x} \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right), {y}^{2} \cdot x, x\right)} \]

   8. Simplified 65.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), \left(y \cdot y\right) \cdot x, x\right)} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(y \cdot \left(y \cdot \left(\color{blue}{\left(y \cdot y\right)} \cdot \frac{1}{5040} + \frac{1}{120}\right)\right) + \frac{1}{6}\right) \cdot \left(\left(y \cdot y\right) \cdot x\right) + x \]

      2. lift-fma.f64 N/A

         \[\leadsto \left(y \cdot \left(y \cdot \color{blue}{\mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right)}\right) + \frac{1}{6}\right) \cdot \left(\left(y \cdot y\right) \cdot x\right) + x \]

      3. lift-*.f64 N/A

         \[\leadsto \left(y \cdot \color{blue}{\left(y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right)\right)} + \frac{1}{6}\right) \cdot \left(\left(y \cdot y\right) \cdot x\right) + x \]

      4. lift-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right)} \cdot \left(\left(y \cdot y\right) \cdot x\right) + x \]

      5. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right) \cdot \left(\color{blue}{\left(y \cdot y\right)} \cdot x\right) + x \]

      6. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right) \cdot \color{blue}{\left(\left(y \cdot y\right) \cdot x\right)} + x \]

   10. Applied egg-rr 66.8%

       \[\leadsto \color{blue}{\mathsf{fma}\left(y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y \cdot x, x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin x \leq -0.01:\\ \;\;\;\;x \cdot \left(\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.008333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), x \cdot y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 77.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sinh y}{y} \cdot \mathsf{fma}\left(-0.16666666666666666, x \cdot \left(x \cdot x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \mathsf{fma}\left(y \cdot \left(y \cdot \left(y \cdot 0.008333333333333333\right)\right), y, \mathsf{fma}\left(y, y \cdot 0.16666666666666666, 1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 5e-7)
   (* (/ (sinh y) y) (fma -0.16666666666666666 (* x (* x x)) x))
   (*
    (sin x)
    (fma
     (* y (* y (* y 0.008333333333333333)))
     y
     (fma y (* y 0.16666666666666666) 1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 5e-7) {
		tmp = (sinh(y) / y) * fma(-0.16666666666666666, (x * (x * x)), x);
	} else {
		tmp = sin(x) * fma((y * (y * (y * 0.008333333333333333))), y, fma(y, (y * 0.16666666666666666), 1.0));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 5e-7)
		tmp = Float64(Float64(sinh(y) / y) * fma(-0.16666666666666666, Float64(x * Float64(x * x)), x));
	else
		tmp = Float64(sin(x) * fma(Float64(y * Float64(y * Float64(y * 0.008333333333333333))), y, fma(y, Float64(y * 0.16666666666666666), 1.0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 5e-7], N[(N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision] * N[(-0.16666666666666666 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], N[(N[Sin[x], $MachinePrecision] * N[(N[(y * N[(y * N[(y * 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y + N[(y * N[(y * 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.99999999999999977e-7

   1. Initial program 100.0%

      \[\sin x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)} \cdot \frac{\sinh y}{y} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(x \cdot \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2} + 1\right)}\right) \cdot \frac{\sinh y}{y} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{\left(x \cdot \left(\frac{-1}{6} \cdot {x}^{2}\right) + x \cdot 1\right)} \cdot \frac{\sinh y}{y} \]

      3. *-commutative N/A

         \[\leadsto \left(x \cdot \color{blue}{\left({x}^{2} \cdot \frac{-1}{6}\right)} + x \cdot 1\right) \cdot \frac{\sinh y}{y} \]

      4. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(x \cdot {x}^{2}\right) \cdot \frac{-1}{6}} + x \cdot 1\right) \cdot \frac{\sinh y}{y} \]

      5. *-commutative N/A

         \[\leadsto \left(\color{blue}{\frac{-1}{6} \cdot \left(x \cdot {x}^{2}\right)} + x \cdot 1\right) \cdot \frac{\sinh y}{y} \]

      6. *-rgt-identity N/A

         \[\leadsto \left(\frac{-1}{6} \cdot \left(x \cdot {x}^{2}\right) + \color{blue}{x}\right) \cdot \frac{\sinh y}{y} \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1}{6}, x \cdot {x}^{2}, x\right)} \cdot \frac{\sinh y}{y} \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{6}, \color{blue}{x \cdot {x}^{2}}, x\right) \cdot \frac{\sinh y}{y} \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{6}, x \cdot \color{blue}{\left(x \cdot x\right)}, x\right) \cdot \frac{\sinh y}{y} \]

      10. lower-*.f64 77.6

          \[\leadsto \mathsf{fma}\left(-0.16666666666666666, x \cdot \color{blue}{\left(x \cdot x\right)}, x\right) \cdot \frac{\sinh y}{y} \]

   5. Simplified 77.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.16666666666666666, x \cdot \left(x \cdot x\right), x\right)} \cdot \frac{\sinh y}{y} \]

   if 4.99999999999999977e-7 < x

   1. Initial program 100.0%

      \[\sin x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\sin x + {y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{6} \cdot \sin x\right)} \]
   4. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \sin x + \color{blue}{\left(\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right)\right) \cdot {y}^{2} + \left(\frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}\right)} \]

      2. *-rgt-identity N/A

         \[\leadsto \color{blue}{\sin x \cdot 1} + \left(\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right)\right) \cdot {y}^{2} + \left(\frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \sin x \cdot 1 + \color{blue}{{y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{6} \cdot \sin x\right)} \]

      4. *-commutative N/A

         \[\leadsto \sin x \cdot 1 + \color{blue}{\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}} \]

      5. associate-*r* N/A

         \[\leadsto \sin x \cdot 1 + \left(\color{blue}{\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot \sin x} + \frac{1}{6} \cdot \sin x\right) \cdot {y}^{2} \]

      6. distribute-rgt-out N/A

         \[\leadsto \sin x \cdot 1 + \color{blue}{\left(\sin x \cdot \left(\frac{1}{120} \cdot {y}^{2} + \frac{1}{6}\right)\right)} \cdot {y}^{2} \]

      7. +-commutative N/A

         \[\leadsto \sin x \cdot 1 + \left(\sin x \cdot \color{blue}{\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)}\right) \cdot {y}^{2} \]

      8. associate-*l* N/A

         \[\leadsto \sin x \cdot 1 + \color{blue}{\sin x \cdot \left(\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) \cdot {y}^{2}\right)} \]

      9. *-commutative N/A

         \[\leadsto \sin x \cdot 1 + \sin x \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} \]

      10. distribute-lft-in N/A

          \[\leadsto \color{blue}{\sin x \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \color{blue}{\sin x \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} \]

      12. lower-sin.f64 N/A

          \[\leadsto \color{blue}{\sin x} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \sin x \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) + 1\right)} \]

   5. Simplified 91.8%

      \[\leadsto \color{blue}{\sin x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sin x \cdot \left(\color{blue}{\left(y \cdot y\right)} \cdot \left(y \cdot \left(y \cdot \frac{1}{120}\right) + \frac{1}{6}\right) + 1\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \sin x \cdot \left(\left(y \cdot y\right) \cdot \left(y \cdot \color{blue}{\left(y \cdot \frac{1}{120}\right)} + \frac{1}{6}\right) + 1\right) \]

      3. lift-fma.f64 N/A

         \[\leadsto \sin x \cdot \left(\left(y \cdot y\right) \cdot \color{blue}{\mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right)} + 1\right) \]

      4. lift-fma.f64 N/A

         \[\leadsto \sin x \cdot \left(\left(y \cdot y\right) \cdot \color{blue}{\left(y \cdot \left(y \cdot \frac{1}{120}\right) + \frac{1}{6}\right)} + 1\right) \]

      5. distribute-rgt-in N/A

         \[\leadsto \sin x \cdot \left(\color{blue}{\left(\left(y \cdot \left(y \cdot \frac{1}{120}\right)\right) \cdot \left(y \cdot y\right) + \frac{1}{6} \cdot \left(y \cdot y\right)\right)} + 1\right) \]

      6. associate-+l+ N/A

         \[\leadsto \sin x \cdot \color{blue}{\left(\left(y \cdot \left(y \cdot \frac{1}{120}\right)\right) \cdot \left(y \cdot y\right) + \left(\frac{1}{6} \cdot \left(y \cdot y\right) + 1\right)\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \sin x \cdot \left(\left(y \cdot \left(y \cdot \frac{1}{120}\right)\right) \cdot \color{blue}{\left(y \cdot y\right)} + \left(\frac{1}{6} \cdot \left(y \cdot y\right) + 1\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \sin x \cdot \left(\color{blue}{\left(\left(y \cdot \left(y \cdot \frac{1}{120}\right)\right) \cdot y\right) \cdot y} + \left(\frac{1}{6} \cdot \left(y \cdot y\right) + 1\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \sin x \cdot \left(\left(\left(y \cdot \left(y \cdot \frac{1}{120}\right)\right) \cdot y\right) \cdot y + \left(\color{blue}{\left(y \cdot y\right) \cdot \frac{1}{6}} + 1\right)\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \sin x \cdot \color{blue}{\mathsf{fma}\left(\left(y \cdot \left(y \cdot \frac{1}{120}\right)\right) \cdot y, y, \left(y \cdot y\right) \cdot \frac{1}{6} + 1\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \sin x \cdot \mathsf{fma}\left(\color{blue}{\left(y \cdot \left(y \cdot \frac{1}{120}\right)\right) \cdot y}, y, \left(y \cdot y\right) \cdot \frac{1}{6} + 1\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \sin x \cdot \mathsf{fma}\left(\color{blue}{\left(y \cdot \left(y \cdot \frac{1}{120}\right)\right)} \cdot y, y, \left(y \cdot y\right) \cdot \frac{1}{6} + 1\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \sin x \cdot \mathsf{fma}\left(\left(y \cdot \left(y \cdot \frac{1}{120}\right)\right) \cdot y, y, \color{blue}{\left(y \cdot y\right)} \cdot \frac{1}{6} + 1\right) \]

      14. associate-*l* N/A

          \[\leadsto \sin x \cdot \mathsf{fma}\left(\left(y \cdot \left(y \cdot \frac{1}{120}\right)\right) \cdot y, y, \color{blue}{y \cdot \left(y \cdot \frac{1}{6}\right)} + 1\right) \]

      15. lower-fma.f64 N/A

          \[\leadsto \sin x \cdot \mathsf{fma}\left(\left(y \cdot \left(y \cdot \frac{1}{120}\right)\right) \cdot y, y, \color{blue}{\mathsf{fma}\left(y, y \cdot \frac{1}{6}, 1\right)}\right) \]

      16. lower-*.f64 91.8

          \[\leadsto \sin x \cdot \mathsf{fma}\left(\left(y \cdot \left(y \cdot 0.008333333333333333\right)\right) \cdot y, y, \mathsf{fma}\left(y, \color{blue}{y \cdot 0.16666666666666666}, 1\right)\right) \]

   7. Applied egg-rr 91.8%

      \[\leadsto \sin x \cdot \color{blue}{\mathsf{fma}\left(\left(y \cdot \left(y \cdot 0.008333333333333333\right)\right) \cdot y, y, \mathsf{fma}\left(y, y \cdot 0.16666666666666666, 1\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sinh y}{y} \cdot \mathsf{fma}\left(-0.16666666666666666, x \cdot \left(x \cdot x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \mathsf{fma}\left(y \cdot \left(y \cdot \left(y \cdot 0.008333333333333333\right)\right), y, \mathsf{fma}\left(y, y \cdot 0.16666666666666666, 1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 55.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin x \leq -0.01:\\ \;\;\;\;x \cdot \left(\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.008333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot \left(y \cdot \left(\left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right), x \cdot \left(y \cdot y\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sin x) -0.01)
   (*
    x
    (*
     (* (* y y) (* y y))
     (fma (* x x) -0.001388888888888889 0.008333333333333333)))
   (fma (* y (* y (* (* y y) 0.0001984126984126984))) (* x (* y y)) x)))

C

double code(double x, double y) {
	double tmp;
	if (sin(x) <= -0.01) {
		tmp = x * (((y * y) * (y * y)) * fma((x * x), -0.001388888888888889, 0.008333333333333333));
	} else {
		tmp = fma((y * (y * ((y * y) * 0.0001984126984126984))), (x * (y * y)), x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (sin(x) <= -0.01)
		tmp = Float64(x * Float64(Float64(Float64(y * y) * Float64(y * y)) * fma(Float64(x * x), -0.001388888888888889, 0.008333333333333333)));
	else
		tmp = fma(Float64(y * Float64(y * Float64(Float64(y * y) * 0.0001984126984126984))), Float64(x * Float64(y * y)), x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[Sin[x], $MachinePrecision], -0.01], N[(x * N[(N[(N[(y * y), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.001388888888888889 + 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 x) < -0.0100000000000000002

   1. Initial program 100.0%

      \[\sin x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\sin x + {y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{6} \cdot \sin x\right)} \]
   4. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \sin x + \color{blue}{\left(\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right)\right) \cdot {y}^{2} + \left(\frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}\right)} \]

      2. *-rgt-identity N/A

         \[\leadsto \color{blue}{\sin x \cdot 1} + \left(\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right)\right) \cdot {y}^{2} + \left(\frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \sin x \cdot 1 + \color{blue}{{y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{6} \cdot \sin x\right)} \]

      4. *-commutative N/A

         \[\leadsto \sin x \cdot 1 + \color{blue}{\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}} \]

      5. associate-*r* N/A

         \[\leadsto \sin x \cdot 1 + \left(\color{blue}{\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot \sin x} + \frac{1}{6} \cdot \sin x\right) \cdot {y}^{2} \]

      6. distribute-rgt-out N/A

         \[\leadsto \sin x \cdot 1 + \color{blue}{\left(\sin x \cdot \left(\frac{1}{120} \cdot {y}^{2} + \frac{1}{6}\right)\right)} \cdot {y}^{2} \]

      7. +-commutative N/A

         \[\leadsto \sin x \cdot 1 + \left(\sin x \cdot \color{blue}{\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)}\right) \cdot {y}^{2} \]

      8. associate-*l* N/A

         \[\leadsto \sin x \cdot 1 + \color{blue}{\sin x \cdot \left(\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) \cdot {y}^{2}\right)} \]

      9. *-commutative N/A

         \[\leadsto \sin x \cdot 1 + \sin x \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} \]

      10. distribute-lft-in N/A

          \[\leadsto \color{blue}{\sin x \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \color{blue}{\sin x \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} \]

      12. lower-sin.f64 N/A

          \[\leadsto \color{blue}{\sin x} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \sin x \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) + 1\right)} \]

   5. Simplified 93.6%

      \[\leadsto \color{blue}{\sin x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + \left(\frac{-1}{6} \cdot \left({x}^{2} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(1 + \left(\frac{-1}{6} \cdot \left({x}^{2} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto x \cdot \left(1 + \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) + \frac{-1}{6} \cdot \left({x}^{2} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right)}\right) \]

      3. associate-+r+ N/A

         \[\leadsto x \cdot \color{blue}{\left(\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) + \frac{-1}{6} \cdot \left({x}^{2} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right)} \]

      4. associate-*r* N/A

         \[\leadsto x \cdot \left(\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) + \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2}\right) \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)}\right) \]

      5. distribute-rgt1-in N/A

         \[\leadsto x \cdot \color{blue}{\left(\left(\frac{-1}{6} \cdot {x}^{2} + 1\right) \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)} \]

      6. +-commutative N/A

         \[\leadsto x \cdot \left(\color{blue}{\left(1 + \frac{-1}{6} \cdot {x}^{2}\right)} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(\left(1 + \frac{-1}{6} \cdot {x}^{2}\right) \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)} \]

   8. Simplified 26.6%

      \[\leadsto \color{blue}{x \cdot \left(\mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)\right)} \]

   9. Taylor expanded in y around inf

      \[\leadsto x \cdot \color{blue}{\left(\frac{1}{120} \cdot \left({y}^{4} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto x \cdot \color{blue}{\left(\left(\frac{1}{120} \cdot {y}^{4}\right) \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)} \]

       2. *-commutative N/A

          \[\leadsto x \cdot \left(\color{blue}{\left({y}^{4} \cdot \frac{1}{120}\right)} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right) \]

       3. associate-*r* N/A

          \[\leadsto x \cdot \color{blue}{\left({y}^{4} \cdot \left(\frac{1}{120} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto x \cdot \color{blue}{\left({y}^{4} \cdot \left(\frac{1}{120} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right)} \]

       5. metadata-eval N/A

          \[\leadsto x \cdot \left({y}^{\color{blue}{\left(2 \cdot 2\right)}} \cdot \left(\frac{1}{120} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right) \]

       6. pow-sqr N/A

          \[\leadsto x \cdot \left(\color{blue}{\left({y}^{2} \cdot {y}^{2}\right)} \cdot \left(\frac{1}{120} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right) \]

       7. lower-*.f64 N/A

          \[\leadsto x \cdot \left(\color{blue}{\left({y}^{2} \cdot {y}^{2}\right)} \cdot \left(\frac{1}{120} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right) \]

       8. unpow2 N/A

          \[\leadsto x \cdot \left(\left(\color{blue}{\left(y \cdot y\right)} \cdot {y}^{2}\right) \cdot \left(\frac{1}{120} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right) \]

       9. lower-*.f64 N/A

          \[\leadsto x \cdot \left(\left(\color{blue}{\left(y \cdot y\right)} \cdot {y}^{2}\right) \cdot \left(\frac{1}{120} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right) \]

       10. unpow2 N/A

           \[\leadsto x \cdot \left(\left(\left(y \cdot y\right) \cdot \color{blue}{\left(y \cdot y\right)}\right) \cdot \left(\frac{1}{120} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right) \]

       11. lower-*.f64 N/A

           \[\leadsto x \cdot \left(\left(\left(y \cdot y\right) \cdot \color{blue}{\left(y \cdot y\right)}\right) \cdot \left(\frac{1}{120} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right) \]

       12. +-commutative N/A

           \[\leadsto x \cdot \left(\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \left(\frac{1}{120} \cdot \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2} + 1\right)}\right)\right) \]

       13. distribute-rgt-in N/A

           \[\leadsto x \cdot \left(\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \color{blue}{\left(\left(\frac{-1}{6} \cdot {x}^{2}\right) \cdot \frac{1}{120} + 1 \cdot \frac{1}{120}\right)}\right) \]

       14. *-commutative N/A

           \[\leadsto x \cdot \left(\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \left(\color{blue}{\left({x}^{2} \cdot \frac{-1}{6}\right)} \cdot \frac{1}{120} + 1 \cdot \frac{1}{120}\right)\right) \]

       15. associate-*l* N/A

           \[\leadsto x \cdot \left(\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \left(\color{blue}{{x}^{2} \cdot \left(\frac{-1}{6} \cdot \frac{1}{120}\right)} + 1 \cdot \frac{1}{120}\right)\right) \]

       16. metadata-eval N/A

           \[\leadsto x \cdot \left(\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \left({x}^{2} \cdot \color{blue}{\frac{-1}{720}} + 1 \cdot \frac{1}{120}\right)\right) \]

       17. metadata-eval N/A

           \[\leadsto x \cdot \left(\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \left({x}^{2} \cdot \color{blue}{\left(\frac{1}{120} \cdot \frac{-1}{6}\right)} + 1 \cdot \frac{1}{120}\right)\right) \]

       18. metadata-eval N/A

           \[\leadsto x \cdot \left(\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \left({x}^{2} \cdot \left(\frac{1}{120} \cdot \frac{-1}{6}\right) + \color{blue}{\frac{1}{120}}\right)\right) \]

       19. lower-fma.f64 N/A

           \[\leadsto x \cdot \left(\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{120} \cdot \frac{-1}{6}, \frac{1}{120}\right)}\right) \]

       20. unpow2 N/A

           \[\leadsto x \cdot \left(\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{120} \cdot \frac{-1}{6}, \frac{1}{120}\right)\right) \]

       21. lower-*.f64 N/A

           \[\leadsto x \cdot \left(\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{120} \cdot \frac{-1}{6}, \frac{1}{120}\right)\right) \]

       22. metadata-eval 26.3

           \[\leadsto x \cdot \left(\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{-0.001388888888888889}, 0.008333333333333333\right)\right) \]

   11. Simplified 26.3%

       \[\leadsto x \cdot \color{blue}{\left(\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.008333333333333333\right)\right)} \]

   if -0.0100000000000000002 < (sin.f64 x)

   1. Initial program 100.0%

      \[\sin x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\sin x + {y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x + {y}^{2} \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{120} \cdot \sin x\right)\right)} \]
   4. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \sin x + \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x\right) + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{120} \cdot \sin x\right)\right)\right)} \]

      2. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\sin x + {y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x\right)\right) + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{120} \cdot \sin x\right)\right)} \]

      3. *-lft-identity N/A

         \[\leadsto \left(\color{blue}{1 \cdot \sin x} + {y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x\right)\right) + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{120} \cdot \sin x\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \left(1 \cdot \sin x + \color{blue}{\left({y}^{2} \cdot \frac{1}{6}\right) \cdot \sin x}\right) + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{120} \cdot \sin x\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \left(1 \cdot \sin x + \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right)} \cdot \sin x\right) + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{120} \cdot \sin x\right)\right) \]

      6. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\sin x \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)} + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{120} \cdot \sin x\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \sin x \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right) + \color{blue}{\left({y}^{2} \cdot {y}^{2}\right) \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{120} \cdot \sin x\right)} \]

      8. *-commutative N/A

         \[\leadsto \sin x \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right) + \color{blue}{\left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{120} \cdot \sin x\right) \cdot \left({y}^{2} \cdot {y}^{2}\right)} \]

      9. associate-*r* N/A

         \[\leadsto \sin x \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right) + \left(\color{blue}{\left(\frac{1}{5040} \cdot {y}^{2}\right) \cdot \sin x} + \frac{1}{120} \cdot \sin x\right) \cdot \left({y}^{2} \cdot {y}^{2}\right) \]

      10. distribute-rgt-out N/A

          \[\leadsto \sin x \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right) + \color{blue}{\left(\sin x \cdot \left(\frac{1}{5040} \cdot {y}^{2} + \frac{1}{120}\right)\right)} \cdot \left({y}^{2} \cdot {y}^{2}\right) \]

      11. +-commutative N/A

          \[\leadsto \sin x \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right) + \left(\sin x \cdot \color{blue}{\left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)}\right) \cdot \left({y}^{2} \cdot {y}^{2}\right) \]

   5. Simplified 91.2%

      \[\leadsto \color{blue}{\sin x \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) + 1\right)} \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right) \cdot x + 1 \cdot x} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot {y}^{2}\right)} \cdot x + 1 \cdot x \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot \left({y}^{2} \cdot x\right)} + 1 \cdot x \]

      5. *-lft-identity N/A

         \[\leadsto \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot \left({y}^{2} \cdot x\right) + \color{blue}{x} \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right), {y}^{2} \cdot x, x\right)} \]

   8. Simplified 65.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), \left(y \cdot y\right) \cdot x, x\right)} \]

   9. Taylor expanded in y around inf

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{5040} \cdot {y}^{4}}, \left(y \cdot y\right) \cdot x, x\right) \]
   10. Step-by-step derivation

       1. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{5040} \cdot {y}^{\color{blue}{\left(2 \cdot 2\right)}}, \left(y \cdot y\right) \cdot x, x\right) \]

       2. pow-sqr N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{5040} \cdot \color{blue}{\left({y}^{2} \cdot {y}^{2}\right)}, \left(y \cdot y\right) \cdot x, x\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{5040} \cdot {y}^{2}\right) \cdot {y}^{2}}, \left(y \cdot y\right) \cdot x, x\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{{y}^{2} \cdot \left(\frac{1}{5040} \cdot {y}^{2}\right)}, \left(y \cdot y\right) \cdot x, x\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(y \cdot y\right)} \cdot \left(\frac{1}{5040} \cdot {y}^{2}\right), \left(y \cdot y\right) \cdot x, x\right) \]

       6. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{y \cdot \left(y \cdot \left(\frac{1}{5040} \cdot {y}^{2}\right)\right)}, \left(y \cdot y\right) \cdot x, x\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(y \cdot \color{blue}{\left(\left(\frac{1}{5040} \cdot {y}^{2}\right) \cdot y\right)}, \left(y \cdot y\right) \cdot x, x\right) \]

       8. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(y \cdot \color{blue}{\left(\frac{1}{5040} \cdot \left({y}^{2} \cdot y\right)\right)}, \left(y \cdot y\right) \cdot x, x\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(y \cdot \left(\frac{1}{5040} \cdot \left(\color{blue}{\left(y \cdot y\right)} \cdot y\right)\right), \left(y \cdot y\right) \cdot x, x\right) \]

       10. unpow3 N/A

           \[\leadsto \mathsf{fma}\left(y \cdot \left(\frac{1}{5040} \cdot \color{blue}{{y}^{3}}\right), \left(y \cdot y\right) \cdot x, x\right) \]

       11. lower-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(\color{blue}{y \cdot \left(\frac{1}{5040} \cdot {y}^{3}\right)}, \left(y \cdot y\right) \cdot x, x\right) \]

       12. unpow3 N/A

           \[\leadsto \mathsf{fma}\left(y \cdot \left(\frac{1}{5040} \cdot \color{blue}{\left(\left(y \cdot y\right) \cdot y\right)}\right), \left(y \cdot y\right) \cdot x, x\right) \]

       13. unpow2 N/A

           \[\leadsto \mathsf{fma}\left(y \cdot \left(\frac{1}{5040} \cdot \left(\color{blue}{{y}^{2}} \cdot y\right)\right), \left(y \cdot y\right) \cdot x, x\right) \]

       14. associate-*r* N/A

           \[\leadsto \mathsf{fma}\left(y \cdot \color{blue}{\left(\left(\frac{1}{5040} \cdot {y}^{2}\right) \cdot y\right)}, \left(y \cdot y\right) \cdot x, x\right) \]

       15. *-commutative N/A

           \[\leadsto \mathsf{fma}\left(y \cdot \color{blue}{\left(y \cdot \left(\frac{1}{5040} \cdot {y}^{2}\right)\right)}, \left(y \cdot y\right) \cdot x, x\right) \]

       16. lower-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(y \cdot \color{blue}{\left(y \cdot \left(\frac{1}{5040} \cdot {y}^{2}\right)\right)}, \left(y \cdot y\right) \cdot x, x\right) \]

       17. *-commutative N/A

           \[\leadsto \mathsf{fma}\left(y \cdot \left(y \cdot \color{blue}{\left({y}^{2} \cdot \frac{1}{5040}\right)}\right), \left(y \cdot y\right) \cdot x, x\right) \]

       18. lower-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(y \cdot \left(y \cdot \color{blue}{\left({y}^{2} \cdot \frac{1}{5040}\right)}\right), \left(y \cdot y\right) \cdot x, x\right) \]

       19. unpow2 N/A

           \[\leadsto \mathsf{fma}\left(y \cdot \left(y \cdot \left(\color{blue}{\left(y \cdot y\right)} \cdot \frac{1}{5040}\right)\right), \left(y \cdot y\right) \cdot x, x\right) \]

       20. lower-*.f64 65.3

           \[\leadsto \mathsf{fma}\left(y \cdot \left(y \cdot \left(\color{blue}{\left(y \cdot y\right)} \cdot 0.0001984126984126984\right)\right), \left(y \cdot y\right) \cdot x, x\right) \]

   11. Simplified 65.3%

       \[\leadsto \mathsf{fma}\left(\color{blue}{y \cdot \left(y \cdot \left(\left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right)}, \left(y \cdot y\right) \cdot x, x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin x \leq -0.01:\\ \;\;\;\;x \cdot \left(\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.008333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot \left(y \cdot \left(\left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right), x \cdot \left(y \cdot y\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 55.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin x \leq -0.01:\\ \;\;\;\;x \cdot \left(\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.008333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sin x) -0.01)
   (*
    x
    (*
     (* (* y y) (* y y))
     (fma (* x x) -0.001388888888888889 0.008333333333333333)))
   (*
    x
    (fma (* y y) (fma (* y y) 0.008333333333333333 0.16666666666666666) 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (sin(x) <= -0.01) {
		tmp = x * (((y * y) * (y * y)) * fma((x * x), -0.001388888888888889, 0.008333333333333333));
	} else {
		tmp = x * fma((y * y), fma((y * y), 0.008333333333333333, 0.16666666666666666), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (sin(x) <= -0.01)
		tmp = Float64(x * Float64(Float64(Float64(y * y) * Float64(y * y)) * fma(Float64(x * x), -0.001388888888888889, 0.008333333333333333)));
	else
		tmp = Float64(x * fma(Float64(y * y), fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[Sin[x], $MachinePrecision], -0.01], N[(x * N[(N[(N[(y * y), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.001388888888888889 + 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 x) < -0.0100000000000000002

   1. Initial program 100.0%

      \[\sin x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\sin x + {y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{6} \cdot \sin x\right)} \]
   4. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \sin x + \color{blue}{\left(\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right)\right) \cdot {y}^{2} + \left(\frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}\right)} \]

      2. *-rgt-identity N/A

         \[\leadsto \color{blue}{\sin x \cdot 1} + \left(\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right)\right) \cdot {y}^{2} + \left(\frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \sin x \cdot 1 + \color{blue}{{y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{6} \cdot \sin x\right)} \]

      4. *-commutative N/A

         \[\leadsto \sin x \cdot 1 + \color{blue}{\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}} \]

      5. associate-*r* N/A

         \[\leadsto \sin x \cdot 1 + \left(\color{blue}{\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot \sin x} + \frac{1}{6} \cdot \sin x\right) \cdot {y}^{2} \]

      6. distribute-rgt-out N/A

         \[\leadsto \sin x \cdot 1 + \color{blue}{\left(\sin x \cdot \left(\frac{1}{120} \cdot {y}^{2} + \frac{1}{6}\right)\right)} \cdot {y}^{2} \]

      7. +-commutative N/A

         \[\leadsto \sin x \cdot 1 + \left(\sin x \cdot \color{blue}{\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)}\right) \cdot {y}^{2} \]

      8. associate-*l* N/A

         \[\leadsto \sin x \cdot 1 + \color{blue}{\sin x \cdot \left(\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) \cdot {y}^{2}\right)} \]

      9. *-commutative N/A

         \[\leadsto \sin x \cdot 1 + \sin x \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} \]

      10. distribute-lft-in N/A

          \[\leadsto \color{blue}{\sin x \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \color{blue}{\sin x \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} \]

      12. lower-sin.f64 N/A

          \[\leadsto \color{blue}{\sin x} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \sin x \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) + 1\right)} \]

   5. Simplified 93.6%

      \[\leadsto \color{blue}{\sin x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + \left(\frac{-1}{6} \cdot \left({x}^{2} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(1 + \left(\frac{-1}{6} \cdot \left({x}^{2} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto x \cdot \left(1 + \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) + \frac{-1}{6} \cdot \left({x}^{2} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right)}\right) \]

      3. associate-+r+ N/A

         \[\leadsto x \cdot \color{blue}{\left(\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) + \frac{-1}{6} \cdot \left({x}^{2} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right)} \]

      4. associate-*r* N/A

         \[\leadsto x \cdot \left(\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) + \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2}\right) \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)}\right) \]

      5. distribute-rgt1-in N/A

         \[\leadsto x \cdot \color{blue}{\left(\left(\frac{-1}{6} \cdot {x}^{2} + 1\right) \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)} \]

      6. +-commutative N/A

         \[\leadsto x \cdot \left(\color{blue}{\left(1 + \frac{-1}{6} \cdot {x}^{2}\right)} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(\left(1 + \frac{-1}{6} \cdot {x}^{2}\right) \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)} \]

   8. Simplified 26.6%

      \[\leadsto \color{blue}{x \cdot \left(\mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)\right)} \]

   9. Taylor expanded in y around inf

      \[\leadsto x \cdot \color{blue}{\left(\frac{1}{120} \cdot \left({y}^{4} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto x \cdot \color{blue}{\left(\left(\frac{1}{120} \cdot {y}^{4}\right) \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)} \]

       2. *-commutative N/A

          \[\leadsto x \cdot \left(\color{blue}{\left({y}^{4} \cdot \frac{1}{120}\right)} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right) \]

       3. associate-*r* N/A

          \[\leadsto x \cdot \color{blue}{\left({y}^{4} \cdot \left(\frac{1}{120} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto x \cdot \color{blue}{\left({y}^{4} \cdot \left(\frac{1}{120} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right)} \]

       5. metadata-eval N/A

          \[\leadsto x \cdot \left({y}^{\color{blue}{\left(2 \cdot 2\right)}} \cdot \left(\frac{1}{120} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right) \]

       6. pow-sqr N/A

          \[\leadsto x \cdot \left(\color{blue}{\left({y}^{2} \cdot {y}^{2}\right)} \cdot \left(\frac{1}{120} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right) \]

       7. lower-*.f64 N/A

          \[\leadsto x \cdot \left(\color{blue}{\left({y}^{2} \cdot {y}^{2}\right)} \cdot \left(\frac{1}{120} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right) \]

       8. unpow2 N/A

          \[\leadsto x \cdot \left(\left(\color{blue}{\left(y \cdot y\right)} \cdot {y}^{2}\right) \cdot \left(\frac{1}{120} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right) \]

       9. lower-*.f64 N/A

          \[\leadsto x \cdot \left(\left(\color{blue}{\left(y \cdot y\right)} \cdot {y}^{2}\right) \cdot \left(\frac{1}{120} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right) \]

       10. unpow2 N/A

           \[\leadsto x \cdot \left(\left(\left(y \cdot y\right) \cdot \color{blue}{\left(y \cdot y\right)}\right) \cdot \left(\frac{1}{120} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right) \]

       11. lower-*.f64 N/A

           \[\leadsto x \cdot \left(\left(\left(y \cdot y\right) \cdot \color{blue}{\left(y \cdot y\right)}\right) \cdot \left(\frac{1}{120} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right) \]

       12. +-commutative N/A

           \[\leadsto x \cdot \left(\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \left(\frac{1}{120} \cdot \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2} + 1\right)}\right)\right) \]

       13. distribute-rgt-in N/A

           \[\leadsto x \cdot \left(\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \color{blue}{\left(\left(\frac{-1}{6} \cdot {x}^{2}\right) \cdot \frac{1}{120} + 1 \cdot \frac{1}{120}\right)}\right) \]

       14. *-commutative N/A

           \[\leadsto x \cdot \left(\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \left(\color{blue}{\left({x}^{2} \cdot \frac{-1}{6}\right)} \cdot \frac{1}{120} + 1 \cdot \frac{1}{120}\right)\right) \]

       15. associate-*l* N/A

           \[\leadsto x \cdot \left(\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \left(\color{blue}{{x}^{2} \cdot \left(\frac{-1}{6} \cdot \frac{1}{120}\right)} + 1 \cdot \frac{1}{120}\right)\right) \]

       16. metadata-eval N/A

           \[\leadsto x \cdot \left(\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \left({x}^{2} \cdot \color{blue}{\frac{-1}{720}} + 1 \cdot \frac{1}{120}\right)\right) \]

       17. metadata-eval N/A

           \[\leadsto x \cdot \left(\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \left({x}^{2} \cdot \color{blue}{\left(\frac{1}{120} \cdot \frac{-1}{6}\right)} + 1 \cdot \frac{1}{120}\right)\right) \]

       18. metadata-eval N/A

           \[\leadsto x \cdot \left(\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \left({x}^{2} \cdot \left(\frac{1}{120} \cdot \frac{-1}{6}\right) + \color{blue}{\frac{1}{120}}\right)\right) \]

       19. lower-fma.f64 N/A

           \[\leadsto x \cdot \left(\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{120} \cdot \frac{-1}{6}, \frac{1}{120}\right)}\right) \]

       20. unpow2 N/A

           \[\leadsto x \cdot \left(\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{120} \cdot \frac{-1}{6}, \frac{1}{120}\right)\right) \]

       21. lower-*.f64 N/A

           \[\leadsto x \cdot \left(\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{120} \cdot \frac{-1}{6}, \frac{1}{120}\right)\right) \]

       22. metadata-eval 26.3

           \[\leadsto x \cdot \left(\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{-0.001388888888888889}, 0.008333333333333333\right)\right) \]

   11. Simplified 26.3%

       \[\leadsto x \cdot \color{blue}{\left(\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.008333333333333333\right)\right)} \]

   if -0.0100000000000000002 < (sin.f64 x)

   1. Initial program 100.0%

      \[\sin x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\sin x + {y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{6} \cdot \sin x\right)} \]
   4. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \sin x + \color{blue}{\left(\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right)\right) \cdot {y}^{2} + \left(\frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}\right)} \]

      2. *-rgt-identity N/A

         \[\leadsto \color{blue}{\sin x \cdot 1} + \left(\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right)\right) \cdot {y}^{2} + \left(\frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \sin x \cdot 1 + \color{blue}{{y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{6} \cdot \sin x\right)} \]

      4. *-commutative N/A

         \[\leadsto \sin x \cdot 1 + \color{blue}{\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}} \]

      5. associate-*r* N/A

         \[\leadsto \sin x \cdot 1 + \left(\color{blue}{\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot \sin x} + \frac{1}{6} \cdot \sin x\right) \cdot {y}^{2} \]

      6. distribute-rgt-out N/A

         \[\leadsto \sin x \cdot 1 + \color{blue}{\left(\sin x \cdot \left(\frac{1}{120} \cdot {y}^{2} + \frac{1}{6}\right)\right)} \cdot {y}^{2} \]

      7. +-commutative N/A

         \[\leadsto \sin x \cdot 1 + \left(\sin x \cdot \color{blue}{\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)}\right) \cdot {y}^{2} \]

      8. associate-*l* N/A

         \[\leadsto \sin x \cdot 1 + \color{blue}{\sin x \cdot \left(\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) \cdot {y}^{2}\right)} \]

      9. *-commutative N/A

         \[\leadsto \sin x \cdot 1 + \sin x \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} \]

      10. distribute-lft-in N/A

          \[\leadsto \color{blue}{\sin x \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \color{blue}{\sin x \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} \]

      12. lower-sin.f64 N/A

          \[\leadsto \color{blue}{\sin x} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \sin x \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) + 1\right)} \]

   5. Simplified 87.7%

      \[\leadsto \color{blue}{\sin x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sin x \cdot \left(\color{blue}{\left(y \cdot y\right)} \cdot \left(y \cdot \left(y \cdot \frac{1}{120}\right) + \frac{1}{6}\right) + 1\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \sin x \cdot \left(\left(y \cdot y\right) \cdot \left(y \cdot \color{blue}{\left(y \cdot \frac{1}{120}\right)} + \frac{1}{6}\right) + 1\right) \]

      3. lift-fma.f64 N/A

         \[\leadsto \sin x \cdot \left(\left(y \cdot y\right) \cdot \color{blue}{\mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right)} + 1\right) \]

      4. lift-fma.f64 N/A

         \[\leadsto \sin x \cdot \left(\left(y \cdot y\right) \cdot \color{blue}{\left(y \cdot \left(y \cdot \frac{1}{120}\right) + \frac{1}{6}\right)} + 1\right) \]

      5. distribute-rgt-in N/A

         \[\leadsto \sin x \cdot \left(\color{blue}{\left(\left(y \cdot \left(y \cdot \frac{1}{120}\right)\right) \cdot \left(y \cdot y\right) + \frac{1}{6} \cdot \left(y \cdot y\right)\right)} + 1\right) \]

      6. associate-+l+ N/A

         \[\leadsto \sin x \cdot \color{blue}{\left(\left(y \cdot \left(y \cdot \frac{1}{120}\right)\right) \cdot \left(y \cdot y\right) + \left(\frac{1}{6} \cdot \left(y \cdot y\right) + 1\right)\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \sin x \cdot \left(\left(y \cdot \left(y \cdot \frac{1}{120}\right)\right) \cdot \color{blue}{\left(y \cdot y\right)} + \left(\frac{1}{6} \cdot \left(y \cdot y\right) + 1\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \sin x \cdot \left(\color{blue}{\left(\left(y \cdot \left(y \cdot \frac{1}{120}\right)\right) \cdot y\right) \cdot y} + \left(\frac{1}{6} \cdot \left(y \cdot y\right) + 1\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \sin x \cdot \left(\left(\left(y \cdot \left(y \cdot \frac{1}{120}\right)\right) \cdot y\right) \cdot y + \left(\color{blue}{\left(y \cdot y\right) \cdot \frac{1}{6}} + 1\right)\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \sin x \cdot \color{blue}{\mathsf{fma}\left(\left(y \cdot \left(y \cdot \frac{1}{120}\right)\right) \cdot y, y, \left(y \cdot y\right) \cdot \frac{1}{6} + 1\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \sin x \cdot \mathsf{fma}\left(\color{blue}{\left(y \cdot \left(y \cdot \frac{1}{120}\right)\right) \cdot y}, y, \left(y \cdot y\right) \cdot \frac{1}{6} + 1\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \sin x \cdot \mathsf{fma}\left(\color{blue}{\left(y \cdot \left(y \cdot \frac{1}{120}\right)\right)} \cdot y, y, \left(y \cdot y\right) \cdot \frac{1}{6} + 1\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \sin x \cdot \mathsf{fma}\left(\left(y \cdot \left(y \cdot \frac{1}{120}\right)\right) \cdot y, y, \color{blue}{\left(y \cdot y\right)} \cdot \frac{1}{6} + 1\right) \]

      14. associate-*l* N/A

          \[\leadsto \sin x \cdot \mathsf{fma}\left(\left(y \cdot \left(y \cdot \frac{1}{120}\right)\right) \cdot y, y, \color{blue}{y \cdot \left(y \cdot \frac{1}{6}\right)} + 1\right) \]

      15. lower-fma.f64 N/A

          \[\leadsto \sin x \cdot \mathsf{fma}\left(\left(y \cdot \left(y \cdot \frac{1}{120}\right)\right) \cdot y, y, \color{blue}{\mathsf{fma}\left(y, y \cdot \frac{1}{6}, 1\right)}\right) \]

      16. lower-*.f64 87.7

          \[\leadsto \sin x \cdot \mathsf{fma}\left(\left(y \cdot \left(y \cdot 0.008333333333333333\right)\right) \cdot y, y, \mathsf{fma}\left(y, \color{blue}{y \cdot 0.16666666666666666}, 1\right)\right) \]

   7. Applied egg-rr 87.7%

      \[\leadsto \sin x \cdot \color{blue}{\mathsf{fma}\left(\left(y \cdot \left(y \cdot 0.008333333333333333\right)\right) \cdot y, y, \mathsf{fma}\left(y, y \cdot 0.16666666666666666, 1\right)\right)} \]

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + \left(\frac{1}{120} \cdot {y}^{4} + \frac{1}{6} \cdot {y}^{2}\right)\right)} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(1 + \left(\frac{1}{120} \cdot {y}^{4} + \frac{1}{6} \cdot {y}^{2}\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(\left(\frac{1}{120} \cdot {y}^{4} + \frac{1}{6} \cdot {y}^{2}\right) + 1\right)} \]

      3. metadata-eval N/A

         \[\leadsto x \cdot \left(\left(\frac{1}{120} \cdot {y}^{\color{blue}{\left(2 \cdot 2\right)}} + \frac{1}{6} \cdot {y}^{2}\right) + 1\right) \]

      4. pow-sqr N/A

         \[\leadsto x \cdot \left(\left(\frac{1}{120} \cdot \color{blue}{\left({y}^{2} \cdot {y}^{2}\right)} + \frac{1}{6} \cdot {y}^{2}\right) + 1\right) \]

      5. associate-*l* N/A

         \[\leadsto x \cdot \left(\left(\color{blue}{\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot {y}^{2}} + \frac{1}{6} \cdot {y}^{2}\right) + 1\right) \]

      6. distribute-rgt-in N/A

         \[\leadsto x \cdot \left(\color{blue}{{y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2} + \frac{1}{6}\right)} + 1\right) \]

      7. +-commutative N/A

         \[\leadsto x \cdot \left({y}^{2} \cdot \color{blue}{\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)} + 1\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{1}{6} + \frac{1}{120} \cdot {y}^{2}, 1\right)} \]

      9. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{6} + \frac{1}{120} \cdot {y}^{2}, 1\right) \]

      10. lower-*.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{6} + \frac{1}{120} \cdot {y}^{2}, 1\right) \]

      11. +-commutative N/A

          \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{\frac{1}{120} \cdot {y}^{2} + \frac{1}{6}}, 1\right) \]

      12. *-commutative N/A

          \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{{y}^{2} \cdot \frac{1}{120}} + \frac{1}{6}, 1\right) \]

      13. lower-fma.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{1}{120}, \frac{1}{6}\right)}, 1\right) \]

      14. unpow2 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      15. lower-*.f64 65.2

          \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, 0.008333333333333333, 0.16666666666666666\right), 1\right) \]

   10. Simplified 65.2%

       \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 22: 77.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sinh y}{y} \cdot \mathsf{fma}\left(-0.16666666666666666, x \cdot \left(x \cdot x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 5e-7)
   (* (/ (sinh y) y) (fma -0.16666666666666666 (* x (* x x)) x))
   (*
    (sin x)
    (fma (* y y) (fma y (* y 0.008333333333333333) 0.16666666666666666) 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 5e-7) {
		tmp = (sinh(y) / y) * fma(-0.16666666666666666, (x * (x * x)), x);
	} else {
		tmp = sin(x) * fma((y * y), fma(y, (y * 0.008333333333333333), 0.16666666666666666), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 5e-7)
		tmp = Float64(Float64(sinh(y) / y) * fma(-0.16666666666666666, Float64(x * Float64(x * x)), x));
	else
		tmp = Float64(sin(x) * fma(Float64(y * y), fma(y, Float64(y * 0.008333333333333333), 0.16666666666666666), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 5e-7], N[(N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision] * N[(-0.16666666666666666 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], N[(N[Sin[x], $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.99999999999999977e-7

   1. Initial program 100.0%

      \[\sin x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)} \cdot \frac{\sinh y}{y} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(x \cdot \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2} + 1\right)}\right) \cdot \frac{\sinh y}{y} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{\left(x \cdot \left(\frac{-1}{6} \cdot {x}^{2}\right) + x \cdot 1\right)} \cdot \frac{\sinh y}{y} \]

      3. *-commutative N/A

         \[\leadsto \left(x \cdot \color{blue}{\left({x}^{2} \cdot \frac{-1}{6}\right)} + x \cdot 1\right) \cdot \frac{\sinh y}{y} \]

      4. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(x \cdot {x}^{2}\right) \cdot \frac{-1}{6}} + x \cdot 1\right) \cdot \frac{\sinh y}{y} \]

      5. *-commutative N/A

         \[\leadsto \left(\color{blue}{\frac{-1}{6} \cdot \left(x \cdot {x}^{2}\right)} + x \cdot 1\right) \cdot \frac{\sinh y}{y} \]

      6. *-rgt-identity N/A

         \[\leadsto \left(\frac{-1}{6} \cdot \left(x \cdot {x}^{2}\right) + \color{blue}{x}\right) \cdot \frac{\sinh y}{y} \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1}{6}, x \cdot {x}^{2}, x\right)} \cdot \frac{\sinh y}{y} \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{6}, \color{blue}{x \cdot {x}^{2}}, x\right) \cdot \frac{\sinh y}{y} \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{6}, x \cdot \color{blue}{\left(x \cdot x\right)}, x\right) \cdot \frac{\sinh y}{y} \]

      10. lower-*.f64 77.6

          \[\leadsto \mathsf{fma}\left(-0.16666666666666666, x \cdot \color{blue}{\left(x \cdot x\right)}, x\right) \cdot \frac{\sinh y}{y} \]

   5. Simplified 77.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.16666666666666666, x \cdot \left(x \cdot x\right), x\right)} \cdot \frac{\sinh y}{y} \]

   if 4.99999999999999977e-7 < x

   1. Initial program 100.0%

      \[\sin x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\sin x + {y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{6} \cdot \sin x\right)} \]
   4. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \sin x + \color{blue}{\left(\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right)\right) \cdot {y}^{2} + \left(\frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}\right)} \]

      2. *-rgt-identity N/A

         \[\leadsto \color{blue}{\sin x \cdot 1} + \left(\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right)\right) \cdot {y}^{2} + \left(\frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \sin x \cdot 1 + \color{blue}{{y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{6} \cdot \sin x\right)} \]

      4. *-commutative N/A

         \[\leadsto \sin x \cdot 1 + \color{blue}{\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}} \]

      5. associate-*r* N/A

         \[\leadsto \sin x \cdot 1 + \left(\color{blue}{\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot \sin x} + \frac{1}{6} \cdot \sin x\right) \cdot {y}^{2} \]

      6. distribute-rgt-out N/A

         \[\leadsto \sin x \cdot 1 + \color{blue}{\left(\sin x \cdot \left(\frac{1}{120} \cdot {y}^{2} + \frac{1}{6}\right)\right)} \cdot {y}^{2} \]

      7. +-commutative N/A

         \[\leadsto \sin x \cdot 1 + \left(\sin x \cdot \color{blue}{\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)}\right) \cdot {y}^{2} \]

      8. associate-*l* N/A

         \[\leadsto \sin x \cdot 1 + \color{blue}{\sin x \cdot \left(\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) \cdot {y}^{2}\right)} \]

      9. *-commutative N/A

         \[\leadsto \sin x \cdot 1 + \sin x \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} \]

      10. distribute-lft-in N/A

          \[\leadsto \color{blue}{\sin x \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \color{blue}{\sin x \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} \]

      12. lower-sin.f64 N/A

          \[\leadsto \color{blue}{\sin x} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \sin x \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) + 1\right)} \]

   5. Simplified 91.8%

      \[\leadsto \color{blue}{\sin x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sinh y}{y} \cdot \mathsf{fma}\left(-0.16666666666666666, x \cdot \left(x \cdot x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 48.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin x \leq -0.01:\\ \;\;\;\;\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(y \cdot y, -0.027777777777777776, -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.16666666666666666, x \cdot \left(y \cdot y\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sin x) -0.01)
   (* (* x (* x x)) (fma (* y y) -0.027777777777777776 -0.16666666666666666))
   (fma 0.16666666666666666 (* x (* y y)) x)))

C

double code(double x, double y) {
	double tmp;
	if (sin(x) <= -0.01) {
		tmp = (x * (x * x)) * fma((y * y), -0.027777777777777776, -0.16666666666666666);
	} else {
		tmp = fma(0.16666666666666666, (x * (y * y)), x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (sin(x) <= -0.01)
		tmp = Float64(Float64(x * Float64(x * x)) * fma(Float64(y * y), -0.027777777777777776, -0.16666666666666666));
	else
		tmp = fma(0.16666666666666666, Float64(x * Float64(y * y)), x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[Sin[x], $MachinePrecision], -0.01], N[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * -0.027777777777777776 + -0.16666666666666666), $MachinePrecision]), $MachinePrecision], N[(0.16666666666666666 * N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 x) < -0.0100000000000000002

   1. Initial program 100.0%

      \[\sin x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\sin x + {y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{6} \cdot \sin x\right)} \]
   4. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \sin x + \color{blue}{\left(\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right)\right) \cdot {y}^{2} + \left(\frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}\right)} \]

      2. *-rgt-identity N/A

         \[\leadsto \color{blue}{\sin x \cdot 1} + \left(\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right)\right) \cdot {y}^{2} + \left(\frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \sin x \cdot 1 + \color{blue}{{y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{6} \cdot \sin x\right)} \]

      4. *-commutative N/A

         \[\leadsto \sin x \cdot 1 + \color{blue}{\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}} \]

      5. associate-*r* N/A

         \[\leadsto \sin x \cdot 1 + \left(\color{blue}{\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot \sin x} + \frac{1}{6} \cdot \sin x\right) \cdot {y}^{2} \]

      6. distribute-rgt-out N/A

         \[\leadsto \sin x \cdot 1 + \color{blue}{\left(\sin x \cdot \left(\frac{1}{120} \cdot {y}^{2} + \frac{1}{6}\right)\right)} \cdot {y}^{2} \]

      7. +-commutative N/A

         \[\leadsto \sin x \cdot 1 + \left(\sin x \cdot \color{blue}{\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)}\right) \cdot {y}^{2} \]

      8. associate-*l* N/A

         \[\leadsto \sin x \cdot 1 + \color{blue}{\sin x \cdot \left(\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) \cdot {y}^{2}\right)} \]

      9. *-commutative N/A

         \[\leadsto \sin x \cdot 1 + \sin x \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} \]

      10. distribute-lft-in N/A

          \[\leadsto \color{blue}{\sin x \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \color{blue}{\sin x \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} \]

      12. lower-sin.f64 N/A

          \[\leadsto \color{blue}{\sin x} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \sin x \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) + 1\right)} \]

   5. Simplified 93.6%

      \[\leadsto \color{blue}{\sin x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + \left(\frac{-1}{6} \cdot \left({x}^{2} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(1 + \left(\frac{-1}{6} \cdot \left({x}^{2} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto x \cdot \left(1 + \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) + \frac{-1}{6} \cdot \left({x}^{2} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right)}\right) \]

      3. associate-+r+ N/A

         \[\leadsto x \cdot \color{blue}{\left(\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) + \frac{-1}{6} \cdot \left({x}^{2} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right)} \]

      4. associate-*r* N/A

         \[\leadsto x \cdot \left(\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) + \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2}\right) \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)}\right) \]

      5. distribute-rgt1-in N/A

         \[\leadsto x \cdot \color{blue}{\left(\left(\frac{-1}{6} \cdot {x}^{2} + 1\right) \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)} \]

      6. +-commutative N/A

         \[\leadsto x \cdot \left(\color{blue}{\left(1 + \frac{-1}{6} \cdot {x}^{2}\right)} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(\left(1 + \frac{-1}{6} \cdot {x}^{2}\right) \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)} \]

   8. Simplified 26.6%

      \[\leadsto \color{blue}{x \cdot \left(\mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)\right)} \]

   9. Taylor expanded in y around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + \left(\frac{-1}{6} \cdot {x}^{2} + \frac{1}{6} \cdot \left({y}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right)\right)} \]
   10. Step-by-step derivation

       1. associate-+r+ N/A

          \[\leadsto x \cdot \color{blue}{\left(\left(1 + \frac{-1}{6} \cdot {x}^{2}\right) + \frac{1}{6} \cdot \left({y}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right)} \]

       2. associate-*r* N/A

          \[\leadsto x \cdot \left(\left(1 + \frac{-1}{6} \cdot {x}^{2}\right) + \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)}\right) \]

       3. distribute-rgt1-in N/A

          \[\leadsto x \cdot \color{blue}{\left(\left(\frac{1}{6} \cdot {y}^{2} + 1\right) \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)} \]

       4. +-commutative N/A

          \[\leadsto x \cdot \left(\color{blue}{\left(1 + \frac{1}{6} \cdot {y}^{2}\right)} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right) \]

       5. lower-*.f64 N/A

          \[\leadsto x \cdot \color{blue}{\left(\left(1 + \frac{1}{6} \cdot {y}^{2}\right) \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)} \]

       6. +-commutative N/A

          \[\leadsto x \cdot \left(\color{blue}{\left(\frac{1}{6} \cdot {y}^{2} + 1\right)} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right) \]

       7. lower-fma.f64 N/A

          \[\leadsto x \cdot \left(\color{blue}{\mathsf{fma}\left(\frac{1}{6}, {y}^{2}, 1\right)} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right) \]

       8. unpow2 N/A

          \[\leadsto x \cdot \left(\mathsf{fma}\left(\frac{1}{6}, \color{blue}{y \cdot y}, 1\right) \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right) \]

       9. lower-*.f64 N/A

          \[\leadsto x \cdot \left(\mathsf{fma}\left(\frac{1}{6}, \color{blue}{y \cdot y}, 1\right) \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right) \]

       10. +-commutative N/A

           \[\leadsto x \cdot \left(\mathsf{fma}\left(\frac{1}{6}, y \cdot y, 1\right) \cdot \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2} + 1\right)}\right) \]

       11. *-commutative N/A

           \[\leadsto x \cdot \left(\mathsf{fma}\left(\frac{1}{6}, y \cdot y, 1\right) \cdot \left(\color{blue}{{x}^{2} \cdot \frac{-1}{6}} + 1\right)\right) \]

       12. unpow2 N/A

           \[\leadsto x \cdot \left(\mathsf{fma}\left(\frac{1}{6}, y \cdot y, 1\right) \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{-1}{6} + 1\right)\right) \]

       13. associate-*l* N/A

           \[\leadsto x \cdot \left(\mathsf{fma}\left(\frac{1}{6}, y \cdot y, 1\right) \cdot \left(\color{blue}{x \cdot \left(x \cdot \frac{-1}{6}\right)} + 1\right)\right) \]

       14. *-commutative N/A

           \[\leadsto x \cdot \left(\mathsf{fma}\left(\frac{1}{6}, y \cdot y, 1\right) \cdot \left(x \cdot \color{blue}{\left(\frac{-1}{6} \cdot x\right)} + 1\right)\right) \]

       15. lower-fma.f64 N/A

           \[\leadsto x \cdot \left(\mathsf{fma}\left(\frac{1}{6}, y \cdot y, 1\right) \cdot \color{blue}{\mathsf{fma}\left(x, \frac{-1}{6} \cdot x, 1\right)}\right) \]

       16. *-commutative N/A

           \[\leadsto x \cdot \left(\mathsf{fma}\left(\frac{1}{6}, y \cdot y, 1\right) \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{-1}{6}}, 1\right)\right) \]

       17. lower-*.f64 26.6

           \[\leadsto x \cdot \left(\mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right) \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot -0.16666666666666666}, 1\right)\right) \]

   11. Simplified 26.6%

       \[\leadsto x \cdot \color{blue}{\left(\mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right) \cdot \mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right)\right)} \]

   12. Taylor expanded in x around inf

       \[\leadsto \color{blue}{\frac{-1}{6} \cdot \left({x}^{3} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)} \]
   13. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{-1}{6} \cdot {x}^{3}\right) \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)} \]

       2. *-commutative N/A

          \[\leadsto \color{blue}{\left({x}^{3} \cdot \frac{-1}{6}\right)} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right) \]

       3. associate-*r* N/A

          \[\leadsto \color{blue}{{x}^{3} \cdot \left(\frac{-1}{6} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto \color{blue}{{x}^{3} \cdot \left(\frac{-1}{6} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)} \]

       5. cube-mult N/A

          \[\leadsto \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot \left(\frac{-1}{6} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \left(x \cdot \color{blue}{{x}^{2}}\right) \cdot \left(\frac{-1}{6} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right) \]

       7. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(x \cdot {x}^{2}\right)} \cdot \left(\frac{-1}{6} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(\frac{-1}{6} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right) \]

       9. lower-*.f64 N/A

          \[\leadsto \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(\frac{-1}{6} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right) \]

       10. +-commutative N/A

           \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{-1}{6} \cdot \color{blue}{\left(\frac{1}{6} \cdot {y}^{2} + 1\right)}\right) \]

       11. distribute-rgt-in N/A

           \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(\left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \frac{-1}{6} + 1 \cdot \frac{-1}{6}\right)} \]

       12. *-commutative N/A

           \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\color{blue}{\left({y}^{2} \cdot \frac{1}{6}\right)} \cdot \frac{-1}{6} + 1 \cdot \frac{-1}{6}\right) \]

       13. associate-*l* N/A

           \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\color{blue}{{y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{-1}{6}\right)} + 1 \cdot \frac{-1}{6}\right) \]

       14. metadata-eval N/A

           \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left({y}^{2} \cdot \color{blue}{\frac{-1}{36}} + 1 \cdot \frac{-1}{6}\right) \]

       15. metadata-eval N/A

           \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left({y}^{2} \cdot \frac{-1}{36} + \color{blue}{\frac{-1}{6}}\right) \]

       16. lower-fma.f64 N/A

           \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{-1}{36}, \frac{-1}{6}\right)} \]

       17. unpow2 N/A

           \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{-1}{36}, \frac{-1}{6}\right) \]

       18. lower-*.f64 26.6

           \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, -0.027777777777777776, -0.16666666666666666\right) \]

   14. Simplified 26.6%

       \[\leadsto \color{blue}{\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(y \cdot y, -0.027777777777777776, -0.16666666666666666\right)} \]

   if -0.0100000000000000002 < (sin.f64 x)

   1. Initial program 100.0%

      \[\sin x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\sin x + {y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x + {y}^{2} \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{120} \cdot \sin x\right)\right)} \]
   4. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \sin x + \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x\right) + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{120} \cdot \sin x\right)\right)\right)} \]

      2. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\sin x + {y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x\right)\right) + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{120} \cdot \sin x\right)\right)} \]

      3. *-lft-identity N/A

         \[\leadsto \left(\color{blue}{1 \cdot \sin x} + {y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x\right)\right) + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{120} \cdot \sin x\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \left(1 \cdot \sin x + \color{blue}{\left({y}^{2} \cdot \frac{1}{6}\right) \cdot \sin x}\right) + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{120} \cdot \sin x\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \left(1 \cdot \sin x + \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right)} \cdot \sin x\right) + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{120} \cdot \sin x\right)\right) \]

      6. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\sin x \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)} + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{120} \cdot \sin x\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \sin x \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right) + \color{blue}{\left({y}^{2} \cdot {y}^{2}\right) \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{120} \cdot \sin x\right)} \]

      8. *-commutative N/A

         \[\leadsto \sin x \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right) + \color{blue}{\left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{120} \cdot \sin x\right) \cdot \left({y}^{2} \cdot {y}^{2}\right)} \]

      9. associate-*r* N/A

         \[\leadsto \sin x \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right) + \left(\color{blue}{\left(\frac{1}{5040} \cdot {y}^{2}\right) \cdot \sin x} + \frac{1}{120} \cdot \sin x\right) \cdot \left({y}^{2} \cdot {y}^{2}\right) \]

      10. distribute-rgt-out N/A

          \[\leadsto \sin x \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right) + \color{blue}{\left(\sin x \cdot \left(\frac{1}{5040} \cdot {y}^{2} + \frac{1}{120}\right)\right)} \cdot \left({y}^{2} \cdot {y}^{2}\right) \]

      11. +-commutative N/A

          \[\leadsto \sin x \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right) + \left(\sin x \cdot \color{blue}{\left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)}\right) \cdot \left({y}^{2} \cdot {y}^{2}\right) \]

   5. Simplified 91.2%

      \[\leadsto \color{blue}{\sin x \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) + 1\right)} \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right) \cdot x + 1 \cdot x} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot {y}^{2}\right)} \cdot x + 1 \cdot x \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot \left({y}^{2} \cdot x\right)} + 1 \cdot x \]

      5. *-lft-identity N/A

         \[\leadsto \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot \left({y}^{2} \cdot x\right) + \color{blue}{x} \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right), {y}^{2} \cdot x, x\right)} \]

   8. Simplified 65.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), \left(y \cdot y\right) \cdot x, x\right)} \]

   9. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{6}}, \left(y \cdot y\right) \cdot x, x\right) \]
   10. Step-by-step derivation

   11. Simplified 57.4%

       \[\leadsto \mathsf{fma}\left(\color{blue}{0.16666666666666666}, \left(y \cdot y\right) \cdot x, x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin x \leq -0.01:\\ \;\;\;\;\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(y \cdot y, -0.027777777777777776, -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.16666666666666666, x \cdot \left(y \cdot y\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 10.5% accurate, 13.6× speedup

Math

\[\begin{array}{l} \\ -0.16666666666666666 \cdot \left(x \cdot \left(x \cdot x\right)\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* -0.16666666666666666 (* x (* x x))))

C

double code(double x, double y) {
	return -0.16666666666666666 * (x * (x * x));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (-0.16666666666666666d0) * (x * (x * x))
end function

Java

public static double code(double x, double y) {
	return -0.16666666666666666 * (x * (x * x));
}

Python

def code(x, y):
	return -0.16666666666666666 * (x * (x * x))

Julia

function code(x, y)
	return Float64(-0.16666666666666666 * Float64(x * Float64(x * x)))
end

MATLAB

function tmp = code(x, y)
	tmp = -0.16666666666666666 * (x * (x * x));
end

Wolfram

code[x_, y_] := N[(-0.16666666666666666 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\sin x \cdot \frac{\sinh y}{y} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\sin x} \]
4. Step-by-step derivation

   1. lower-sin.f64 47.0

      \[\leadsto \color{blue}{\sin x} \]

5. Simplified 47.0%

   \[\leadsto \color{blue}{\sin x} \]

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto x \cdot \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2} + 1\right)} \]

   2. distribute-rgt-in N/A

      \[\leadsto \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2}\right) \cdot x + 1 \cdot x} \]

   3. *-lft-identity N/A

      \[\leadsto \left(\frac{-1}{6} \cdot {x}^{2}\right) \cdot x + \color{blue}{x} \]

   4. *-commutative N/A

      \[\leadsto \color{blue}{\left({x}^{2} \cdot \frac{-1}{6}\right)} \cdot x + x \]

   5. associate-*l* N/A

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{-1}{6} \cdot x\right)} + x \]

   6. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{-1}{6} \cdot x, x\right)} \]

   7. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{6} \cdot x, x\right) \]

   8. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{6} \cdot x, x\right) \]

   9. lower-*.f64 34.1

      \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{-0.16666666666666666 \cdot x}, x\right) \]

8. Simplified 34.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, -0.16666666666666666 \cdot x, x\right)} \]

9. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{-1}{6} \cdot {x}^{3}} \]
10. Step-by-step derivation

    1. lower-*.f64 N/A

       \[\leadsto \color{blue}{\frac{-1}{6} \cdot {x}^{3}} \]

    2. cube-mult N/A

       \[\leadsto \frac{-1}{6} \cdot \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \]

    3. unpow2 N/A

       \[\leadsto \frac{-1}{6} \cdot \left(x \cdot \color{blue}{{x}^{2}}\right) \]

    4. lower-*.f64 N/A

       \[\leadsto \frac{-1}{6} \cdot \color{blue}{\left(x \cdot {x}^{2}\right)} \]

    5. unpow2 N/A

       \[\leadsto \frac{-1}{6} \cdot \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \]

    6. lower-*.f64 12.4

       \[\leadsto -0.16666666666666666 \cdot \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \]

11. Simplified 12.4%

    \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(x \cdot \left(x \cdot x\right)\right)} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024208 
(FPCore (x y)
  :name "Linear.Quaternion:$ccos from linear-1.19.1.3"
  :precision binary64
  (* (sin x) (/ (sinh y) y)))