Linear.Quaternion:$ccos from linear-1.19.1.3

Percentage Accurate: 100.0% → 100.0%

Time: 15.2s

Alternatives: 26

Speedup: 1.0×

• Could not converge on a ground truth

  1. x = 3.7606687305471174e+31
  2. y = 7.899857492210644e+188

• 50.2% 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. clear-num N/A

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

   2. un-div-inv N/A

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

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

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

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

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

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

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

   6. sinh-lowering-sinh.f64 100.0

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

4. Applied egg-rr 100.0%

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

Alternative 2: 77.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y}{y}\\ t_1 := \sin x \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;y \cdot \left(y \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, -3.306878306878307 \cdot 10^{-5}, 0.001388888888888889\right), -0.027777777777777776\right), 0.16666666666666666\right)\right)\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+20}:\\ \;\;\;\;\sin x \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot 0.0001984126984126984, y, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sinh y) y)) (t_1 (* (sin x) t_0)))
   (if (<= t_1 (- INFINITY))
     (*
      y
      (*
       y
       (*
        x
        (fma
         (* x x)
         (fma
          x
          (* x (fma (* x x) -3.306878306878307e-5 0.001388888888888889))
          -0.027777777777777776)
         0.16666666666666666))))
     (if (<= t_1 1e+20)
       (*
        (sin x)
        (fma
         y
         (*
          y
          (fma
           (* y y)
           (fma (* y 0.0001984126984126984) y 0.008333333333333333)
           0.16666666666666666))
         1.0))
       (* x t_0)))))

C

double code(double x, double y) {
	double t_0 = sinh(y) / y;
	double t_1 = sin(x) * t_0;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = y * (y * (x * fma((x * x), fma(x, (x * fma((x * x), -3.306878306878307e-5, 0.001388888888888889)), -0.027777777777777776), 0.16666666666666666)));
	} else if (t_1 <= 1e+20) {
		tmp = sin(x) * fma(y, (y * fma((y * y), fma((y * 0.0001984126984126984), y, 0.008333333333333333), 0.16666666666666666)), 1.0);
	} else {
		tmp = x * t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sinh(y) / y)
	t_1 = Float64(sin(x) * t_0)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(y * Float64(y * Float64(x * fma(Float64(x * x), fma(x, Float64(x * fma(Float64(x * x), -3.306878306878307e-5, 0.001388888888888889)), -0.027777777777777776), 0.16666666666666666))));
	elseif (t_1 <= 1e+20)
		tmp = Float64(sin(x) * fma(y, Float64(y * fma(Float64(y * y), fma(Float64(y * 0.0001984126984126984), y, 0.008333333333333333), 0.16666666666666666)), 1.0));
	else
		tmp = Float64(x * t_0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[x], $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(y * N[(y * N[(x * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * -3.306878306878307e-5 + 0.001388888888888889), $MachinePrecision]), $MachinePrecision] + -0.027777777777777776), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+20], N[(N[Sin[x], $MachinePrecision] * N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * N[(N[(y * 0.0001984126984126984), $MachinePrecision] * y + 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(x * t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\sin x + \frac{1}{6} \cdot \left({y}^{2} \cdot \sin x\right)} \]
   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. *-lowering-*.f64 N/A

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

      5. sin-lowering-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. accelerator-lowering-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. *-lowering-*.f64 50.9

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

   5. Simplified 50.9%

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

   6. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

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

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

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

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

      8. *-commutative N/A

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

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

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

      10. sin-lowering-sin.f64 45.1

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

   8. Simplified 45.1%

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

   9. Taylor expanded in x around 0

      \[\leadsto y \cdot \left(y \cdot \color{blue}{\left(x \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{30240} \cdot {x}^{2}\right) - \frac{1}{36}\right)\right)\right)}\right) \]
   10. Step-by-step derivation

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

          \[\leadsto y \cdot \left(y \cdot \color{blue}{\left(x \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{30240} \cdot {x}^{2}\right) - \frac{1}{36}\right)\right)\right)}\right) \]

       2. +-commutative N/A

          \[\leadsto y \cdot \left(y \cdot \left(x \cdot \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{30240} \cdot {x}^{2}\right) - \frac{1}{36}\right) + \frac{1}{6}\right)}\right)\right) \]

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

          \[\leadsto y \cdot \left(y \cdot \left(x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, {x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{30240} \cdot {x}^{2}\right) - \frac{1}{36}, \frac{1}{6}\right)}\right)\right) \]

       4. unpow2 N/A

          \[\leadsto y \cdot \left(y \cdot \left(x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, {x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{30240} \cdot {x}^{2}\right) - \frac{1}{36}, \frac{1}{6}\right)\right)\right) \]

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

          \[\leadsto y \cdot \left(y \cdot \left(x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, {x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{30240} \cdot {x}^{2}\right) - \frac{1}{36}, \frac{1}{6}\right)\right)\right) \]

       6. sub-neg N/A

          \[\leadsto y \cdot \left(y \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{30240} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{36}\right)\right)}, \frac{1}{6}\right)\right)\right) \]

       7. unpow2 N/A

          \[\leadsto y \cdot \left(y \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{720} + \frac{-1}{30240} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{36}\right)\right), \frac{1}{6}\right)\right)\right) \]

       8. associate-*l* N/A

          \[\leadsto y \cdot \left(y \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(x \cdot \left(\frac{1}{720} + \frac{-1}{30240} \cdot {x}^{2}\right)\right)} + \left(\mathsf{neg}\left(\frac{1}{36}\right)\right), \frac{1}{6}\right)\right)\right) \]

       9. metadata-eval N/A

          \[\leadsto y \cdot \left(y \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, x \cdot \left(x \cdot \left(\frac{1}{720} + \frac{-1}{30240} \cdot {x}^{2}\right)\right) + \color{blue}{\frac{-1}{36}}, \frac{1}{6}\right)\right)\right) \]

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

           \[\leadsto y \cdot \left(y \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, x \cdot \left(\frac{1}{720} + \frac{-1}{30240} \cdot {x}^{2}\right), \frac{-1}{36}\right)}, \frac{1}{6}\right)\right)\right) \]

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

           \[\leadsto y \cdot \left(y \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{720} + \frac{-1}{30240} \cdot {x}^{2}\right)}, \frac{-1}{36}\right), \frac{1}{6}\right)\right)\right) \]

       12. +-commutative N/A

           \[\leadsto y \cdot \left(y \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\left(\frac{-1}{30240} \cdot {x}^{2} + \frac{1}{720}\right)}, \frac{-1}{36}\right), \frac{1}{6}\right)\right)\right) \]

       13. *-commutative N/A

           \[\leadsto y \cdot \left(y \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{-1}{30240}} + \frac{1}{720}\right), \frac{-1}{36}\right), \frac{1}{6}\right)\right)\right) \]

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

           \[\leadsto y \cdot \left(y \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{-1}{30240}, \frac{1}{720}\right)}, \frac{-1}{36}\right), \frac{1}{6}\right)\right)\right) \]

       15. unpow2 N/A

           \[\leadsto y \cdot \left(y \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{30240}, \frac{1}{720}\right), \frac{-1}{36}\right), \frac{1}{6}\right)\right)\right) \]

       16. *-lowering-*.f64 37.4

           \[\leadsto y \cdot \left(y \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, -3.306878306878307 \cdot 10^{-5}, 0.001388888888888889\right), -0.027777777777777776\right), 0.16666666666666666\right)\right)\right) \]

   11. Simplified 37.4%

       \[\leadsto y \cdot \left(y \cdot \color{blue}{\left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, -3.306878306878307 \cdot 10^{-5}, 0.001388888888888889\right), -0.027777777777777776\right), 0.16666666666666666\right)\right)}\right) \]

   if -inf.0 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < 1e20

   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 98.5%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

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

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

      4. *-lowering-*.f64 98.5

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

   7. Applied egg-rr 98.5%

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

   if 1e20 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 76.0%

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

Alternative 3: 77.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y}{y}\\ t_1 := \sin x \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;y \cdot \left(y \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, -3.306878306878307 \cdot 10^{-5}, 0.001388888888888889\right), -0.027777777777777776\right), 0.16666666666666666\right)\right)\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+20}:\\ \;\;\;\;\sin x \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sinh y) y)) (t_1 (* (sin x) t_0)))
   (if (<= t_1 (- INFINITY))
     (*
      y
      (*
       y
       (*
        x
        (fma
         (* x x)
         (fma
          x
          (* x (fma (* x x) -3.306878306878307e-5 0.001388888888888889))
          -0.027777777777777776)
         0.16666666666666666))))
     (if (<= t_1 1e+20)
       (*
        (sin x)
        (fma
         y
         (* y (fma (* y y) 0.008333333333333333 0.16666666666666666))
         1.0))
       (* x t_0)))))

C

double code(double x, double y) {
	double t_0 = sinh(y) / y;
	double t_1 = sin(x) * t_0;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = y * (y * (x * fma((x * x), fma(x, (x * fma((x * x), -3.306878306878307e-5, 0.001388888888888889)), -0.027777777777777776), 0.16666666666666666)));
	} else if (t_1 <= 1e+20) {
		tmp = sin(x) * fma(y, (y * fma((y * y), 0.008333333333333333, 0.16666666666666666)), 1.0);
	} else {
		tmp = x * t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sinh(y) / y)
	t_1 = Float64(sin(x) * t_0)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(y * Float64(y * Float64(x * fma(Float64(x * x), fma(x, Float64(x * fma(Float64(x * x), -3.306878306878307e-5, 0.001388888888888889)), -0.027777777777777776), 0.16666666666666666))));
	elseif (t_1 <= 1e+20)
		tmp = Float64(sin(x) * fma(y, Float64(y * fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666)), 1.0));
	else
		tmp = Float64(x * t_0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[x], $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(y * N[(y * N[(x * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * -3.306878306878307e-5 + 0.001388888888888889), $MachinePrecision]), $MachinePrecision] + -0.027777777777777776), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+20], N[(N[Sin[x], $MachinePrecision] * N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(x * t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\sin x + \frac{1}{6} \cdot \left({y}^{2} \cdot \sin x\right)} \]
   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. *-lowering-*.f64 N/A

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

      5. sin-lowering-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. accelerator-lowering-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. *-lowering-*.f64 50.9

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

   5. Simplified 50.9%

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

   6. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

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

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

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

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

      8. *-commutative N/A

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

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

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

      10. sin-lowering-sin.f64 45.1

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

   8. Simplified 45.1%

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

   9. Taylor expanded in x around 0

      \[\leadsto y \cdot \left(y \cdot \color{blue}{\left(x \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{30240} \cdot {x}^{2}\right) - \frac{1}{36}\right)\right)\right)}\right) \]
   10. Step-by-step derivation

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

          \[\leadsto y \cdot \left(y \cdot \color{blue}{\left(x \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{30240} \cdot {x}^{2}\right) - \frac{1}{36}\right)\right)\right)}\right) \]

       2. +-commutative N/A

          \[\leadsto y \cdot \left(y \cdot \left(x \cdot \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{30240} \cdot {x}^{2}\right) - \frac{1}{36}\right) + \frac{1}{6}\right)}\right)\right) \]

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

          \[\leadsto y \cdot \left(y \cdot \left(x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, {x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{30240} \cdot {x}^{2}\right) - \frac{1}{36}, \frac{1}{6}\right)}\right)\right) \]

       4. unpow2 N/A

          \[\leadsto y \cdot \left(y \cdot \left(x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, {x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{30240} \cdot {x}^{2}\right) - \frac{1}{36}, \frac{1}{6}\right)\right)\right) \]

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

          \[\leadsto y \cdot \left(y \cdot \left(x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, {x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{30240} \cdot {x}^{2}\right) - \frac{1}{36}, \frac{1}{6}\right)\right)\right) \]

       6. sub-neg N/A

          \[\leadsto y \cdot \left(y \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{30240} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{36}\right)\right)}, \frac{1}{6}\right)\right)\right) \]

       7. unpow2 N/A

          \[\leadsto y \cdot \left(y \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{720} + \frac{-1}{30240} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{36}\right)\right), \frac{1}{6}\right)\right)\right) \]

       8. associate-*l* N/A

          \[\leadsto y \cdot \left(y \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(x \cdot \left(\frac{1}{720} + \frac{-1}{30240} \cdot {x}^{2}\right)\right)} + \left(\mathsf{neg}\left(\frac{1}{36}\right)\right), \frac{1}{6}\right)\right)\right) \]

       9. metadata-eval N/A

          \[\leadsto y \cdot \left(y \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, x \cdot \left(x \cdot \left(\frac{1}{720} + \frac{-1}{30240} \cdot {x}^{2}\right)\right) + \color{blue}{\frac{-1}{36}}, \frac{1}{6}\right)\right)\right) \]

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

           \[\leadsto y \cdot \left(y \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, x \cdot \left(\frac{1}{720} + \frac{-1}{30240} \cdot {x}^{2}\right), \frac{-1}{36}\right)}, \frac{1}{6}\right)\right)\right) \]

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

           \[\leadsto y \cdot \left(y \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{720} + \frac{-1}{30240} \cdot {x}^{2}\right)}, \frac{-1}{36}\right), \frac{1}{6}\right)\right)\right) \]

       12. +-commutative N/A

           \[\leadsto y \cdot \left(y \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\left(\frac{-1}{30240} \cdot {x}^{2} + \frac{1}{720}\right)}, \frac{-1}{36}\right), \frac{1}{6}\right)\right)\right) \]

       13. *-commutative N/A

           \[\leadsto y \cdot \left(y \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{-1}{30240}} + \frac{1}{720}\right), \frac{-1}{36}\right), \frac{1}{6}\right)\right)\right) \]

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

           \[\leadsto y \cdot \left(y \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{-1}{30240}, \frac{1}{720}\right)}, \frac{-1}{36}\right), \frac{1}{6}\right)\right)\right) \]

       15. unpow2 N/A

           \[\leadsto y \cdot \left(y \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{30240}, \frac{1}{720}\right), \frac{-1}{36}\right), \frac{1}{6}\right)\right)\right) \]

       16. *-lowering-*.f64 37.4

           \[\leadsto y \cdot \left(y \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, -3.306878306878307 \cdot 10^{-5}, 0.001388888888888889\right), -0.027777777777777776\right), 0.16666666666666666\right)\right)\right) \]

   11. Simplified 37.4%

       \[\leadsto y \cdot \left(y \cdot \color{blue}{\left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, -3.306878306878307 \cdot 10^{-5}, 0.001388888888888889\right), -0.027777777777777776\right), 0.16666666666666666\right)\right)}\right) \]

   if -inf.0 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < 1e20

   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 98.5%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-+l+ N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. distribute-lft1-in N/A

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

      9. +-commutative N/A

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

      10. distribute-rgt-out N/A

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

      11. +-commutative N/A

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

      12. associate-+r+ N/A

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

   8. Simplified 98.5%

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

   if 1e20 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 76.0%

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

Alternative 4: 77.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y}{y}\\ t_1 := \sin x \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;y \cdot \left(y \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, -3.306878306878307 \cdot 10^{-5}, 0.001388888888888889\right), -0.027777777777777776\right), 0.16666666666666666\right)\right)\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+20}:\\ \;\;\;\;\sin x \cdot \mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sinh y) y)) (t_1 (* (sin x) t_0)))
   (if (<= t_1 (- INFINITY))
     (*
      y
      (*
       y
       (*
        x
        (fma
         (* x x)
         (fma
          x
          (* x (fma (* x x) -3.306878306878307e-5 0.001388888888888889))
          -0.027777777777777776)
         0.16666666666666666))))
     (if (<= t_1 1e+20)
       (* (sin x) (fma 0.16666666666666666 (* y y) 1.0))
       (* x t_0)))))

C

double code(double x, double y) {
	double t_0 = sinh(y) / y;
	double t_1 = sin(x) * t_0;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = y * (y * (x * fma((x * x), fma(x, (x * fma((x * x), -3.306878306878307e-5, 0.001388888888888889)), -0.027777777777777776), 0.16666666666666666)));
	} else if (t_1 <= 1e+20) {
		tmp = sin(x) * fma(0.16666666666666666, (y * y), 1.0);
	} else {
		tmp = x * t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sinh(y) / y)
	t_1 = Float64(sin(x) * t_0)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(y * Float64(y * Float64(x * fma(Float64(x * x), fma(x, Float64(x * fma(Float64(x * x), -3.306878306878307e-5, 0.001388888888888889)), -0.027777777777777776), 0.16666666666666666))));
	elseif (t_1 <= 1e+20)
		tmp = Float64(sin(x) * fma(0.16666666666666666, Float64(y * y), 1.0));
	else
		tmp = Float64(x * t_0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[x], $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(y * N[(y * N[(x * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * -3.306878306878307e-5 + 0.001388888888888889), $MachinePrecision]), $MachinePrecision] + -0.027777777777777776), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+20], N[(N[Sin[x], $MachinePrecision] * N[(0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(x * t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\sin x + \frac{1}{6} \cdot \left({y}^{2} \cdot \sin x\right)} \]
   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. *-lowering-*.f64 N/A

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

      5. sin-lowering-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. accelerator-lowering-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. *-lowering-*.f64 50.9

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

   5. Simplified 50.9%

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

   6. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

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

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

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

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

      8. *-commutative N/A

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

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

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

      10. sin-lowering-sin.f64 45.1

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

   8. Simplified 45.1%

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

   9. Taylor expanded in x around 0

      \[\leadsto y \cdot \left(y \cdot \color{blue}{\left(x \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{30240} \cdot {x}^{2}\right) - \frac{1}{36}\right)\right)\right)}\right) \]
   10. Step-by-step derivation

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

          \[\leadsto y \cdot \left(y \cdot \color{blue}{\left(x \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{30240} \cdot {x}^{2}\right) - \frac{1}{36}\right)\right)\right)}\right) \]

       2. +-commutative N/A

          \[\leadsto y \cdot \left(y \cdot \left(x \cdot \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{30240} \cdot {x}^{2}\right) - \frac{1}{36}\right) + \frac{1}{6}\right)}\right)\right) \]

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

          \[\leadsto y \cdot \left(y \cdot \left(x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, {x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{30240} \cdot {x}^{2}\right) - \frac{1}{36}, \frac{1}{6}\right)}\right)\right) \]

       4. unpow2 N/A

          \[\leadsto y \cdot \left(y \cdot \left(x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, {x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{30240} \cdot {x}^{2}\right) - \frac{1}{36}, \frac{1}{6}\right)\right)\right) \]

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

          \[\leadsto y \cdot \left(y \cdot \left(x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, {x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{30240} \cdot {x}^{2}\right) - \frac{1}{36}, \frac{1}{6}\right)\right)\right) \]

       6. sub-neg N/A

          \[\leadsto y \cdot \left(y \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{30240} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{36}\right)\right)}, \frac{1}{6}\right)\right)\right) \]

       7. unpow2 N/A

          \[\leadsto y \cdot \left(y \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{720} + \frac{-1}{30240} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{36}\right)\right), \frac{1}{6}\right)\right)\right) \]

       8. associate-*l* N/A

          \[\leadsto y \cdot \left(y \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(x \cdot \left(\frac{1}{720} + \frac{-1}{30240} \cdot {x}^{2}\right)\right)} + \left(\mathsf{neg}\left(\frac{1}{36}\right)\right), \frac{1}{6}\right)\right)\right) \]

       9. metadata-eval N/A

          \[\leadsto y \cdot \left(y \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, x \cdot \left(x \cdot \left(\frac{1}{720} + \frac{-1}{30240} \cdot {x}^{2}\right)\right) + \color{blue}{\frac{-1}{36}}, \frac{1}{6}\right)\right)\right) \]

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

           \[\leadsto y \cdot \left(y \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, x \cdot \left(\frac{1}{720} + \frac{-1}{30240} \cdot {x}^{2}\right), \frac{-1}{36}\right)}, \frac{1}{6}\right)\right)\right) \]

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

           \[\leadsto y \cdot \left(y \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{720} + \frac{-1}{30240} \cdot {x}^{2}\right)}, \frac{-1}{36}\right), \frac{1}{6}\right)\right)\right) \]

       12. +-commutative N/A

           \[\leadsto y \cdot \left(y \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\left(\frac{-1}{30240} \cdot {x}^{2} + \frac{1}{720}\right)}, \frac{-1}{36}\right), \frac{1}{6}\right)\right)\right) \]

       13. *-commutative N/A

           \[\leadsto y \cdot \left(y \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{-1}{30240}} + \frac{1}{720}\right), \frac{-1}{36}\right), \frac{1}{6}\right)\right)\right) \]

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

           \[\leadsto y \cdot \left(y \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{-1}{30240}, \frac{1}{720}\right)}, \frac{-1}{36}\right), \frac{1}{6}\right)\right)\right) \]

       15. unpow2 N/A

           \[\leadsto y \cdot \left(y \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{30240}, \frac{1}{720}\right), \frac{-1}{36}\right), \frac{1}{6}\right)\right)\right) \]

       16. *-lowering-*.f64 37.4

           \[\leadsto y \cdot \left(y \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, -3.306878306878307 \cdot 10^{-5}, 0.001388888888888889\right), -0.027777777777777776\right), 0.16666666666666666\right)\right)\right) \]

   11. Simplified 37.4%

       \[\leadsto y \cdot \left(y \cdot \color{blue}{\left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, -3.306878306878307 \cdot 10^{-5}, 0.001388888888888889\right), -0.027777777777777776\right), 0.16666666666666666\right)\right)}\right) \]

   if -inf.0 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < 1e20

   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. *-lowering-*.f64 N/A

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

      5. sin-lowering-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. accelerator-lowering-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. *-lowering-*.f64 98.1

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

   5. Simplified 98.1%

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

   if 1e20 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 76.0%

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

Alternative 5: 76.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y}{y}\\ t_1 := \sin x \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;y \cdot \left(y \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, -3.306878306878307 \cdot 10^{-5}, 0.001388888888888889\right), -0.027777777777777776\right), 0.16666666666666666\right)\right)\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+20}:\\ \;\;\;\;\sin x\\ \mathbf{else}:\\ \;\;\;\;x \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sinh y) y)) (t_1 (* (sin x) t_0)))
   (if (<= t_1 (- INFINITY))
     (*
      y
      (*
       y
       (*
        x
        (fma
         (* x x)
         (fma
          x
          (* x (fma (* x x) -3.306878306878307e-5 0.001388888888888889))
          -0.027777777777777776)
         0.16666666666666666))))
     (if (<= t_1 1e+20) (sin x) (* x t_0)))))

C

double code(double x, double y) {
	double t_0 = sinh(y) / y;
	double t_1 = sin(x) * t_0;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = y * (y * (x * fma((x * x), fma(x, (x * fma((x * x), -3.306878306878307e-5, 0.001388888888888889)), -0.027777777777777776), 0.16666666666666666)));
	} else if (t_1 <= 1e+20) {
		tmp = sin(x);
	} else {
		tmp = x * t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sinh(y) / y)
	t_1 = Float64(sin(x) * t_0)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(y * Float64(y * Float64(x * fma(Float64(x * x), fma(x, Float64(x * fma(Float64(x * x), -3.306878306878307e-5, 0.001388888888888889)), -0.027777777777777776), 0.16666666666666666))));
	elseif (t_1 <= 1e+20)
		tmp = sin(x);
	else
		tmp = Float64(x * t_0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[x], $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(y * N[(y * N[(x * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * -3.306878306878307e-5 + 0.001388888888888889), $MachinePrecision]), $MachinePrecision] + -0.027777777777777776), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+20], N[Sin[x], $MachinePrecision], N[(x * t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\sin x + \frac{1}{6} \cdot \left({y}^{2} \cdot \sin x\right)} \]
   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. *-lowering-*.f64 N/A

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

      5. sin-lowering-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. accelerator-lowering-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. *-lowering-*.f64 50.9

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

   5. Simplified 50.9%

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

   6. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

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

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

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

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

      8. *-commutative N/A

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

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

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

      10. sin-lowering-sin.f64 45.1

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

   8. Simplified 45.1%

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

   9. Taylor expanded in x around 0

      \[\leadsto y \cdot \left(y \cdot \color{blue}{\left(x \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{30240} \cdot {x}^{2}\right) - \frac{1}{36}\right)\right)\right)}\right) \]
   10. Step-by-step derivation

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

          \[\leadsto y \cdot \left(y \cdot \color{blue}{\left(x \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{30240} \cdot {x}^{2}\right) - \frac{1}{36}\right)\right)\right)}\right) \]

       2. +-commutative N/A

          \[\leadsto y \cdot \left(y \cdot \left(x \cdot \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{30240} \cdot {x}^{2}\right) - \frac{1}{36}\right) + \frac{1}{6}\right)}\right)\right) \]

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

          \[\leadsto y \cdot \left(y \cdot \left(x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, {x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{30240} \cdot {x}^{2}\right) - \frac{1}{36}, \frac{1}{6}\right)}\right)\right) \]

       4. unpow2 N/A

          \[\leadsto y \cdot \left(y \cdot \left(x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, {x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{30240} \cdot {x}^{2}\right) - \frac{1}{36}, \frac{1}{6}\right)\right)\right) \]

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

          \[\leadsto y \cdot \left(y \cdot \left(x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, {x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{30240} \cdot {x}^{2}\right) - \frac{1}{36}, \frac{1}{6}\right)\right)\right) \]

       6. sub-neg N/A

          \[\leadsto y \cdot \left(y \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{30240} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{36}\right)\right)}, \frac{1}{6}\right)\right)\right) \]

       7. unpow2 N/A

          \[\leadsto y \cdot \left(y \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{720} + \frac{-1}{30240} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{36}\right)\right), \frac{1}{6}\right)\right)\right) \]

       8. associate-*l* N/A

          \[\leadsto y \cdot \left(y \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(x \cdot \left(\frac{1}{720} + \frac{-1}{30240} \cdot {x}^{2}\right)\right)} + \left(\mathsf{neg}\left(\frac{1}{36}\right)\right), \frac{1}{6}\right)\right)\right) \]

       9. metadata-eval N/A

          \[\leadsto y \cdot \left(y \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, x \cdot \left(x \cdot \left(\frac{1}{720} + \frac{-1}{30240} \cdot {x}^{2}\right)\right) + \color{blue}{\frac{-1}{36}}, \frac{1}{6}\right)\right)\right) \]

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

           \[\leadsto y \cdot \left(y \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, x \cdot \left(\frac{1}{720} + \frac{-1}{30240} \cdot {x}^{2}\right), \frac{-1}{36}\right)}, \frac{1}{6}\right)\right)\right) \]

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

           \[\leadsto y \cdot \left(y \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{720} + \frac{-1}{30240} \cdot {x}^{2}\right)}, \frac{-1}{36}\right), \frac{1}{6}\right)\right)\right) \]

       12. +-commutative N/A

           \[\leadsto y \cdot \left(y \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\left(\frac{-1}{30240} \cdot {x}^{2} + \frac{1}{720}\right)}, \frac{-1}{36}\right), \frac{1}{6}\right)\right)\right) \]

       13. *-commutative N/A

           \[\leadsto y \cdot \left(y \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{-1}{30240}} + \frac{1}{720}\right), \frac{-1}{36}\right), \frac{1}{6}\right)\right)\right) \]

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

           \[\leadsto y \cdot \left(y \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{-1}{30240}, \frac{1}{720}\right)}, \frac{-1}{36}\right), \frac{1}{6}\right)\right)\right) \]

       15. unpow2 N/A

           \[\leadsto y \cdot \left(y \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{30240}, \frac{1}{720}\right), \frac{-1}{36}\right), \frac{1}{6}\right)\right)\right) \]

       16. *-lowering-*.f64 37.4

           \[\leadsto y \cdot \left(y \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, -3.306878306878307 \cdot 10^{-5}, 0.001388888888888889\right), -0.027777777777777776\right), 0.16666666666666666\right)\right)\right) \]

   11. Simplified 37.4%

       \[\leadsto y \cdot \left(y \cdot \color{blue}{\left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, -3.306878306878307 \cdot 10^{-5}, 0.001388888888888889\right), -0.027777777777777776\right), 0.16666666666666666\right)\right)}\right) \]

   if -inf.0 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < 1e20

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. sin-lowering-sin.f64 97.5

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

   5. Simplified 97.5%

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

   if 1e20 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 76.0%

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

Alternative 6: 74.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin x \cdot \frac{\sinh y}{y}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;y \cdot \left(y \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, -3.306878306878307 \cdot 10^{-5}, 0.001388888888888889\right), -0.027777777777777776\right), 0.16666666666666666\right)\right)\right)\\ \mathbf{elif}\;t\_0 \leq 10^{+20}:\\ \;\;\;\;\sin x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{y}{x \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sin x) (/ (sinh y) y))))
   (if (<= t_0 (- INFINITY))
     (*
      y
      (*
       y
       (*
        x
        (fma
         (* x x)
         (fma
          x
          (* x (fma (* x x) -3.306878306878307e-5 0.001388888888888889))
          -0.027777777777777776)
         0.16666666666666666))))
     (if (<= t_0 1e+20)
       (sin x)
       (/
        1.0
        (/
         y
         (*
          x
          (fma
           (fma
            (* y y)
            (fma y (* y 0.0001984126984126984) 0.008333333333333333)
            0.16666666666666666)
           (* y (* y y))
           y))))))))

C

double code(double x, double y) {
	double t_0 = sin(x) * (sinh(y) / y);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = y * (y * (x * fma((x * x), fma(x, (x * fma((x * x), -3.306878306878307e-5, 0.001388888888888889)), -0.027777777777777776), 0.16666666666666666)));
	} else if (t_0 <= 1e+20) {
		tmp = sin(x);
	} else {
		tmp = 1.0 / (y / (x * fma(fma((y * y), fma(y, (y * 0.0001984126984126984), 0.008333333333333333), 0.16666666666666666), (y * (y * y)), y)));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(x) * Float64(sinh(y) / y))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(y * Float64(y * Float64(x * fma(Float64(x * x), fma(x, Float64(x * fma(Float64(x * x), -3.306878306878307e-5, 0.001388888888888889)), -0.027777777777777776), 0.16666666666666666))));
	elseif (t_0 <= 1e+20)
		tmp = sin(x);
	else
		tmp = Float64(1.0 / Float64(y / Float64(x * fma(fma(Float64(y * y), fma(y, Float64(y * 0.0001984126984126984), 0.008333333333333333), 0.16666666666666666), Float64(y * Float64(y * y)), y))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(y * N[(y * N[(x * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * -3.306878306878307e-5 + 0.001388888888888889), $MachinePrecision]), $MachinePrecision] + -0.027777777777777776), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+20], N[Sin[x], $MachinePrecision], N[(1.0 / N[(y / N[(x * N[(N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * 0.0001984126984126984), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]), $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

      \[\leadsto \color{blue}{\sin x + \frac{1}{6} \cdot \left({y}^{2} \cdot \sin x\right)} \]
   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. *-lowering-*.f64 N/A

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

      5. sin-lowering-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. accelerator-lowering-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. *-lowering-*.f64 50.9

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

   5. Simplified 50.9%

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

   6. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

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

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

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

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

      8. *-commutative N/A

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

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

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

      10. sin-lowering-sin.f64 45.1

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

   8. Simplified 45.1%

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

   9. Taylor expanded in x around 0

      \[\leadsto y \cdot \left(y \cdot \color{blue}{\left(x \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{30240} \cdot {x}^{2}\right) - \frac{1}{36}\right)\right)\right)}\right) \]
   10. Step-by-step derivation

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

          \[\leadsto y \cdot \left(y \cdot \color{blue}{\left(x \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{30240} \cdot {x}^{2}\right) - \frac{1}{36}\right)\right)\right)}\right) \]

       2. +-commutative N/A

          \[\leadsto y \cdot \left(y \cdot \left(x \cdot \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{30240} \cdot {x}^{2}\right) - \frac{1}{36}\right) + \frac{1}{6}\right)}\right)\right) \]

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

          \[\leadsto y \cdot \left(y \cdot \left(x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, {x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{30240} \cdot {x}^{2}\right) - \frac{1}{36}, \frac{1}{6}\right)}\right)\right) \]

       4. unpow2 N/A

          \[\leadsto y \cdot \left(y \cdot \left(x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, {x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{30240} \cdot {x}^{2}\right) - \frac{1}{36}, \frac{1}{6}\right)\right)\right) \]

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

          \[\leadsto y \cdot \left(y \cdot \left(x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, {x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{30240} \cdot {x}^{2}\right) - \frac{1}{36}, \frac{1}{6}\right)\right)\right) \]

       6. sub-neg N/A

          \[\leadsto y \cdot \left(y \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{30240} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{36}\right)\right)}, \frac{1}{6}\right)\right)\right) \]

       7. unpow2 N/A

          \[\leadsto y \cdot \left(y \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{720} + \frac{-1}{30240} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{36}\right)\right), \frac{1}{6}\right)\right)\right) \]

       8. associate-*l* N/A

          \[\leadsto y \cdot \left(y \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(x \cdot \left(\frac{1}{720} + \frac{-1}{30240} \cdot {x}^{2}\right)\right)} + \left(\mathsf{neg}\left(\frac{1}{36}\right)\right), \frac{1}{6}\right)\right)\right) \]

       9. metadata-eval N/A

          \[\leadsto y \cdot \left(y \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, x \cdot \left(x \cdot \left(\frac{1}{720} + \frac{-1}{30240} \cdot {x}^{2}\right)\right) + \color{blue}{\frac{-1}{36}}, \frac{1}{6}\right)\right)\right) \]

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

           \[\leadsto y \cdot \left(y \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, x \cdot \left(\frac{1}{720} + \frac{-1}{30240} \cdot {x}^{2}\right), \frac{-1}{36}\right)}, \frac{1}{6}\right)\right)\right) \]

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

           \[\leadsto y \cdot \left(y \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{720} + \frac{-1}{30240} \cdot {x}^{2}\right)}, \frac{-1}{36}\right), \frac{1}{6}\right)\right)\right) \]

       12. +-commutative N/A

           \[\leadsto y \cdot \left(y \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\left(\frac{-1}{30240} \cdot {x}^{2} + \frac{1}{720}\right)}, \frac{-1}{36}\right), \frac{1}{6}\right)\right)\right) \]

       13. *-commutative N/A

           \[\leadsto y \cdot \left(y \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{-1}{30240}} + \frac{1}{720}\right), \frac{-1}{36}\right), \frac{1}{6}\right)\right)\right) \]

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

           \[\leadsto y \cdot \left(y \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{-1}{30240}, \frac{1}{720}\right)}, \frac{-1}{36}\right), \frac{1}{6}\right)\right)\right) \]

       15. unpow2 N/A

           \[\leadsto y \cdot \left(y \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{30240}, \frac{1}{720}\right), \frac{-1}{36}\right), \frac{1}{6}\right)\right)\right) \]

       16. *-lowering-*.f64 37.4

           \[\leadsto y \cdot \left(y \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, -3.306878306878307 \cdot 10^{-5}, 0.001388888888888889\right), -0.027777777777777776\right), 0.16666666666666666\right)\right)\right) \]

   11. Simplified 37.4%

       \[\leadsto y \cdot \left(y \cdot \color{blue}{\left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, -3.306878306878307 \cdot 10^{-5}, 0.001388888888888889\right), -0.027777777777777776\right), 0.16666666666666666\right)\right)}\right) \]

   if -inf.0 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < 1e20

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. sin-lowering-sin.f64 97.5

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

   5. Simplified 97.5%

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

   if 1e20 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-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. accelerator-lowering-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. *-lowering-*.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. *-lowering-*.f64 80.0

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

   5. Simplified 80.0%

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

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. pow-plus N/A

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

      7. metadata-eval N/A

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

      8. cube-unmult N/A

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

      9. unpow2 N/A

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

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

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

   8. Simplified 73.6%

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

   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. /-lowering-/.f64 N/A

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

   11. Simplified 65.9%

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

       1. clear-num N/A

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

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

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

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

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

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

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

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

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

       6. associate-*r* N/A

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

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

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

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

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

       9. associate-*l* N/A

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

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

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

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

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

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

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

       13. *-lowering-*.f64 65.9

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

   13. Applied egg-rr 65.9%

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

Alternative 7: 40.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin x \cdot \frac{\sinh y}{y}\\ \mathbf{if}\;t\_0 \leq -0.005:\\ \;\;\;\;-0.16666666666666666 \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \mathbf{elif}\;t\_0 \leq 0.2:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot \left(x \cdot \left(y \cdot y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sin x) (/ (sinh y) y))))
   (if (<= t_0 -0.005)
     (* -0.16666666666666666 (* x (* x x)))
     (if (<= t_0 0.2) x (* 0.16666666666666666 (* x (* y y)))))))

C

double code(double x, double y) {
	double t_0 = sin(x) * (sinh(y) / y);
	double tmp;
	if (t_0 <= -0.005) {
		tmp = -0.16666666666666666 * (x * (x * x));
	} else if (t_0 <= 0.2) {
		tmp = x;
	} else {
		tmp = 0.16666666666666666 * (x * (y * y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin(x) * (sinh(y) / y)
    if (t_0 <= (-0.005d0)) then
        tmp = (-0.16666666666666666d0) * (x * (x * x))
    else if (t_0 <= 0.2d0) then
        tmp = x
    else
        tmp = 0.16666666666666666d0 * (x * (y * y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sin(x) * (Math.sinh(y) / y);
	double tmp;
	if (t_0 <= -0.005) {
		tmp = -0.16666666666666666 * (x * (x * x));
	} else if (t_0 <= 0.2) {
		tmp = x;
	} else {
		tmp = 0.16666666666666666 * (x * (y * y));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sin(x) * (math.sinh(y) / y)
	tmp = 0
	if t_0 <= -0.005:
		tmp = -0.16666666666666666 * (x * (x * x))
	elif t_0 <= 0.2:
		tmp = x
	else:
		tmp = 0.16666666666666666 * (x * (y * y))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sin(x) * Float64(sinh(y) / y))
	tmp = 0.0
	if (t_0 <= -0.005)
		tmp = Float64(-0.16666666666666666 * Float64(x * Float64(x * x)));
	elseif (t_0 <= 0.2)
		tmp = x;
	else
		tmp = Float64(0.16666666666666666 * Float64(x * Float64(y * y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sin(x) * (sinh(y) / y);
	tmp = 0.0;
	if (t_0 <= -0.005)
		tmp = -0.16666666666666666 * (x * (x * x));
	elseif (t_0 <= 0.2)
		tmp = x;
	else
		tmp = 0.16666666666666666 * (x * (y * y));
	end
	tmp_2 = 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, -0.005], N[(-0.16666666666666666 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.2], x, N[(0.16666666666666666 * N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < -0.0050000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. sin-lowering-sin.f64 34.2

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

   5. Simplified 34.2%

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

   6. Taylor expanded in x around 0

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. +-commutative N/A

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

      4. associate-*l* N/A

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

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

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

      6. unpow2 N/A

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

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

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

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

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

      9. sub-neg N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. metadata-eval N/A

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

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

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

      16. *-commutative N/A

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

      17. *-lowering-*.f64 25.0

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

   8. Simplified 25.0%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. distribute-lft-in N/A

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

       3. *-rgt-identity N/A

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

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

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

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

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

       6. unpow2 N/A

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

       7. *-lowering-*.f64 14.2

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

   11. Simplified 14.2%

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

   12. Taylor expanded in x around inf

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

       1. *-lowering-*.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. *-lowering-*.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. *-lowering-*.f64 13.3

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

   14. Simplified 13.3%

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

   if -0.0050000000000000001 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < 0.20000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. sin-lowering-sin.f64 98.9

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

   5. Simplified 98.9%

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

   6. Taylor expanded in x around 0

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

   8. Simplified 90.3%

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

   if 0.20000000000000001 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

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

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

      5. sin-lowering-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. accelerator-lowering-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. *-lowering-*.f64 61.0

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

   5. Simplified 61.0%

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

   6. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

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

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

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

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

      8. *-commutative N/A

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

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

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

      10. sin-lowering-sin.f64 27.6

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

   8. Simplified 27.6%

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

   9. Taylor expanded in x around 0

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

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

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

       2. *-commutative N/A

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

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

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

       4. unpow2 N/A

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

       5. *-lowering-*.f64 32.4

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

   11. Simplified 32.4%

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

4. Final simplification 38.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin x \cdot \frac{\sinh y}{y} \leq -0.005:\\ \;\;\;\;-0.16666666666666666 \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \mathbf{elif}\;\sin x \cdot \frac{\sinh y}{y} \leq 0.2:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot \left(x \cdot \left(y \cdot y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 57.3% accurate, 0.8× speedup

Math

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

FPCore

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

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), fma((y * y), 0.008333333333333333, 0.16666666666666666), 1.0);
	} else {
		tmp = (x * fma((y * (y * ((y * y) * 0.0001984126984126984))), (y * (y * y)), y)) / y;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(sin(x) * Float64(sinh(y) / y)) <= 0.01)
		tmp = Float64(fma(x, Float64(x * Float64(x * -0.16666666666666666)), x) * fma(Float64(y * y), fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666), 1.0));
	else
		tmp = Float64(Float64(x * fma(Float64(y * Float64(y * Float64(Float64(y * y) * 0.0001984126984126984))), Float64(y * Float64(y * y)), y)) / y);
	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 * N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision] / y), $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 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. accelerator-lowering-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. *-lowering-*.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. *-lowering-*.f64 61.9

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

   5. Simplified 61.9%

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

   6. Taylor expanded in y around 0

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

      1. associate-+r+ N/A

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

      2. distribute-lft-in N/A

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

      3. associate-+r+ N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt1-in N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

   8. Simplified 55.2%

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

   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 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. accelerator-lowering-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. *-lowering-*.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. *-lowering-*.f64 61.2

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

   5. Simplified 61.2%

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

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. pow-plus N/A

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

      7. metadata-eval N/A

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

      8. cube-unmult N/A

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

      9. unpow2 N/A

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

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

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

   8. Simplified 56.3%

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

   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. /-lowering-/.f64 N/A

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

   11. Simplified 50.8%

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

   12. Taylor expanded in y around inf

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

       1. metadata-eval N/A

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

       2. pow-sqr N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. unpow2 N/A

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

       6. associate-*l* N/A

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

       7. *-commutative N/A

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

       8. associate-*r* N/A

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

       9. unpow2 N/A

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

       10. unpow3 N/A

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

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

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

       12. unpow3 N/A

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

       13. unpow2 N/A

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

       14. associate-*r* N/A

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

       15. *-commutative N/A

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

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

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

       17. *-commutative N/A

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

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

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

       19. unpow2 N/A

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

       20. *-lowering-*.f64 50.8

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

   14. Simplified 50.8%

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

Alternative 9: 53.5% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(sin(x) * Float64(sinh(y) / y)) <= 0.01)
		tmp = Float64(fma(0.16666666666666666, Float64(y * y), 1.0) * fma(x, Float64(x * Float64(x * -0.16666666666666666)), x));
	else
		tmp = Float64(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[N[(N[Sin[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], 0.01], N[(N[(0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x * N[(x * N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], N[(x * 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 (*.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

      \[\leadsto \color{blue}{\sin x + \frac{1}{6} \cdot \left({y}^{2} \cdot \sin x\right)} \]
   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. *-lowering-*.f64 N/A

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

      5. sin-lowering-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. accelerator-lowering-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. *-lowering-*.f64 77.9

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

   5. Simplified 77.9%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

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

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

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

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

      9. *-lowering-*.f64 52.5

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

   8. Simplified 52.5%

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

   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 83.6%

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

   6. Taylor expanded in x around 0

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

   8. Simplified 48.9%

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

4. Final simplification 51.1%

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

Alternative 10: 58.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin x \leq -0.005:\\ \;\;\;\;y \cdot \left(y \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, -3.306878306878307 \cdot 10^{-5}, 0.001388888888888889\right), -0.027777777777777776\right), 0.16666666666666666\right)\right)\right)\\ \mathbf{elif}\;\sin x \leq 5 \cdot 10^{-8}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right) \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.008333333333333333, -0.16666666666666666\right), 1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sin x) -0.005)
   (*
    y
    (*
     y
     (*
      x
      (fma
       (* x x)
       (fma
        x
        (* x (fma (* x x) -3.306878306878307e-5 0.001388888888888889))
        -0.027777777777777776)
       0.16666666666666666))))
   (if (<= (sin x) 5e-8)
     (*
      x
      (fma
       y
       (*
        y
        (fma
         (* y y)
         (fma (* y y) 0.0001984126984126984 0.008333333333333333)
         0.16666666666666666))
       1.0))
     (*
      x
      (*
       (fma 0.16666666666666666 (* y y) 1.0)
       (fma
        x
        (* x (fma x (* x 0.008333333333333333) -0.16666666666666666))
        1.0))))))

C

double code(double x, double y) {
	double tmp;
	if (sin(x) <= -0.005) {
		tmp = y * (y * (x * fma((x * x), fma(x, (x * fma((x * x), -3.306878306878307e-5, 0.001388888888888889)), -0.027777777777777776), 0.16666666666666666)));
	} else if (sin(x) <= 5e-8) {
		tmp = x * fma(y, (y * fma((y * y), fma((y * y), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666)), 1.0);
	} else {
		tmp = x * (fma(0.16666666666666666, (y * y), 1.0) * fma(x, (x * fma(x, (x * 0.008333333333333333), -0.16666666666666666)), 1.0));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (sin(x) <= -0.005)
		tmp = Float64(y * Float64(y * Float64(x * fma(Float64(x * x), fma(x, Float64(x * fma(Float64(x * x), -3.306878306878307e-5, 0.001388888888888889)), -0.027777777777777776), 0.16666666666666666))));
	elseif (sin(x) <= 5e-8)
		tmp = Float64(x * fma(y, Float64(y * fma(Float64(y * y), fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666)), 1.0));
	else
		tmp = Float64(x * Float64(fma(0.16666666666666666, Float64(y * y), 1.0) * fma(x, Float64(x * fma(x, Float64(x * 0.008333333333333333), -0.16666666666666666)), 1.0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[Sin[x], $MachinePrecision], -0.005], N[(y * N[(y * N[(x * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * -3.306878306878307e-5 + 0.001388888888888889), $MachinePrecision]), $MachinePrecision] + -0.027777777777777776), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[x], $MachinePrecision], 5e-8], N[(x * 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[(x * N[(N[(0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x * N[(x * N[(x * N[(x * 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 x) < -0.0050000000000000001

   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. *-lowering-*.f64 N/A

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

      5. sin-lowering-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. accelerator-lowering-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. *-lowering-*.f64 72.5

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

   5. Simplified 72.5%

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

   6. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

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

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

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

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

      8. *-commutative N/A

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

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

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

      10. sin-lowering-sin.f64 32.6

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

   8. Simplified 32.6%

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

   9. Taylor expanded in x around 0

      \[\leadsto y \cdot \left(y \cdot \color{blue}{\left(x \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{30240} \cdot {x}^{2}\right) - \frac{1}{36}\right)\right)\right)}\right) \]
   10. Step-by-step derivation

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

          \[\leadsto y \cdot \left(y \cdot \color{blue}{\left(x \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{30240} \cdot {x}^{2}\right) - \frac{1}{36}\right)\right)\right)}\right) \]

       2. +-commutative N/A

          \[\leadsto y \cdot \left(y \cdot \left(x \cdot \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{30240} \cdot {x}^{2}\right) - \frac{1}{36}\right) + \frac{1}{6}\right)}\right)\right) \]

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

          \[\leadsto y \cdot \left(y \cdot \left(x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, {x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{30240} \cdot {x}^{2}\right) - \frac{1}{36}, \frac{1}{6}\right)}\right)\right) \]

       4. unpow2 N/A

          \[\leadsto y \cdot \left(y \cdot \left(x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, {x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{30240} \cdot {x}^{2}\right) - \frac{1}{36}, \frac{1}{6}\right)\right)\right) \]

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

          \[\leadsto y \cdot \left(y \cdot \left(x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, {x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{30240} \cdot {x}^{2}\right) - \frac{1}{36}, \frac{1}{6}\right)\right)\right) \]

       6. sub-neg N/A

          \[\leadsto y \cdot \left(y \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{30240} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{36}\right)\right)}, \frac{1}{6}\right)\right)\right) \]

       7. unpow2 N/A

          \[\leadsto y \cdot \left(y \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{720} + \frac{-1}{30240} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{36}\right)\right), \frac{1}{6}\right)\right)\right) \]

       8. associate-*l* N/A

          \[\leadsto y \cdot \left(y \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(x \cdot \left(\frac{1}{720} + \frac{-1}{30240} \cdot {x}^{2}\right)\right)} + \left(\mathsf{neg}\left(\frac{1}{36}\right)\right), \frac{1}{6}\right)\right)\right) \]

       9. metadata-eval N/A

          \[\leadsto y \cdot \left(y \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, x \cdot \left(x \cdot \left(\frac{1}{720} + \frac{-1}{30240} \cdot {x}^{2}\right)\right) + \color{blue}{\frac{-1}{36}}, \frac{1}{6}\right)\right)\right) \]

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

           \[\leadsto y \cdot \left(y \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, x \cdot \left(\frac{1}{720} + \frac{-1}{30240} \cdot {x}^{2}\right), \frac{-1}{36}\right)}, \frac{1}{6}\right)\right)\right) \]

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

           \[\leadsto y \cdot \left(y \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{720} + \frac{-1}{30240} \cdot {x}^{2}\right)}, \frac{-1}{36}\right), \frac{1}{6}\right)\right)\right) \]

       12. +-commutative N/A

           \[\leadsto y \cdot \left(y \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\left(\frac{-1}{30240} \cdot {x}^{2} + \frac{1}{720}\right)}, \frac{-1}{36}\right), \frac{1}{6}\right)\right)\right) \]

       13. *-commutative N/A

           \[\leadsto y \cdot \left(y \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{-1}{30240}} + \frac{1}{720}\right), \frac{-1}{36}\right), \frac{1}{6}\right)\right)\right) \]

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

           \[\leadsto y \cdot \left(y \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{-1}{30240}, \frac{1}{720}\right)}, \frac{-1}{36}\right), \frac{1}{6}\right)\right)\right) \]

       15. unpow2 N/A

           \[\leadsto y \cdot \left(y \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{30240}, \frac{1}{720}\right), \frac{-1}{36}\right), \frac{1}{6}\right)\right)\right) \]

       16. *-lowering-*.f64 25.0

           \[\leadsto y \cdot \left(y \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, -3.306878306878307 \cdot 10^{-5}, 0.001388888888888889\right), -0.027777777777777776\right), 0.16666666666666666\right)\right)\right) \]

   11. Simplified 25.0%

       \[\leadsto y \cdot \left(y \cdot \color{blue}{\left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, -3.306878306878307 \cdot 10^{-5}, 0.001388888888888889\right), -0.027777777777777776\right), 0.16666666666666666\right)\right)}\right) \]

   if -0.0050000000000000001 < (sin.f64 x) < 4.9999999999999998e-8

   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 89.2%

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

   6. Taylor expanded in x around 0

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

   8. Simplified 89.2%

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

   if 4.9999999999999998e-8 < (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

      \[\leadsto \color{blue}{\sin x + \frac{1}{6} \cdot \left({y}^{2} \cdot \sin x\right)} \]
   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. *-lowering-*.f64 N/A

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

      5. sin-lowering-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. accelerator-lowering-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. *-lowering-*.f64 75.0

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

   5. Simplified 75.0%

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

   6. Taylor expanded in x around 0

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

   8. Simplified 30.4%

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

Alternative 11: 58.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin x \leq -0.005:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot \left(x \cdot -0.16666666666666666\right), x\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), 1\right)\\ \mathbf{elif}\;\sin x \leq 5 \cdot 10^{-8}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right) \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.008333333333333333, -0.16666666666666666\right), 1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sin x) -0.005)
   (*
    (fma x (* x (* x -0.16666666666666666)) x)
    (fma (* y y) (fma (* y y) 0.008333333333333333 0.16666666666666666) 1.0))
   (if (<= (sin x) 5e-8)
     (*
      x
      (fma
       y
       (*
        y
        (fma
         (* y y)
         (fma (* y y) 0.0001984126984126984 0.008333333333333333)
         0.16666666666666666))
       1.0))
     (*
      x
      (*
       (fma 0.16666666666666666 (* y y) 1.0)
       (fma
        x
        (* x (fma x (* x 0.008333333333333333) -0.16666666666666666))
        1.0))))))

C

double code(double x, double y) {
	double tmp;
	if (sin(x) <= -0.005) {
		tmp = fma(x, (x * (x * -0.16666666666666666)), x) * fma((y * y), fma((y * y), 0.008333333333333333, 0.16666666666666666), 1.0);
	} else if (sin(x) <= 5e-8) {
		tmp = x * fma(y, (y * fma((y * y), fma((y * y), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666)), 1.0);
	} else {
		tmp = x * (fma(0.16666666666666666, (y * y), 1.0) * fma(x, (x * fma(x, (x * 0.008333333333333333), -0.16666666666666666)), 1.0));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (sin(x) <= -0.005)
		tmp = Float64(fma(x, Float64(x * Float64(x * -0.16666666666666666)), x) * fma(Float64(y * y), fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666), 1.0));
	elseif (sin(x) <= 5e-8)
		tmp = Float64(x * fma(y, Float64(y * fma(Float64(y * y), fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666)), 1.0));
	else
		tmp = Float64(x * Float64(fma(0.16666666666666666, Float64(y * y), 1.0) * fma(x, Float64(x * fma(x, Float64(x * 0.008333333333333333), -0.16666666666666666)), 1.0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[Sin[x], $MachinePrecision], -0.005], N[(N[(x * N[(x * N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[x], $MachinePrecision], 5e-8], N[(x * 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[(x * N[(N[(0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x * N[(x * N[(x * N[(x * 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 x) < -0.0050000000000000001

   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. accelerator-lowering-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. *-lowering-*.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. *-lowering-*.f64 26.0

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

   5. Simplified 26.0%

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

   6. Taylor expanded in y around 0

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

      1. associate-+r+ N/A

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

      2. distribute-lft-in N/A

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

      3. associate-+r+ N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt1-in N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

   8. Simplified 23.6%

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

   if -0.0050000000000000001 < (sin.f64 x) < 4.9999999999999998e-8

   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 89.2%

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

   6. Taylor expanded in x around 0

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

   8. Simplified 89.2%

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

   if 4.9999999999999998e-8 < (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

      \[\leadsto \color{blue}{\sin x + \frac{1}{6} \cdot \left({y}^{2} \cdot \sin x\right)} \]
   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. *-lowering-*.f64 N/A

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

      5. sin-lowering-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. accelerator-lowering-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. *-lowering-*.f64 75.0

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

   5. Simplified 75.0%

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

   6. Taylor expanded in x around 0

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

   8. Simplified 30.4%

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

Alternative 12: 57.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{if}\;\sin x \leq -0.005:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(x, x \cdot \left(x \cdot -0.16666666666666666\right), x\right)\\ \mathbf{elif}\;\sin x \leq 5 \cdot 10^{-8}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(t\_0 \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.008333333333333333, -0.16666666666666666\right), 1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma 0.16666666666666666 (* y y) 1.0)))
   (if (<= (sin x) -0.005)
     (* t_0 (fma x (* x (* x -0.16666666666666666)) x))
     (if (<= (sin x) 5e-8)
       (*
        x
        (fma
         y
         (*
          y
          (fma
           (* y y)
           (fma (* y y) 0.0001984126984126984 0.008333333333333333)
           0.16666666666666666))
         1.0))
       (*
        x
        (*
         t_0
         (fma
          x
          (* x (fma x (* x 0.008333333333333333) -0.16666666666666666))
          1.0)))))))

C

double code(double x, double y) {
	double t_0 = fma(0.16666666666666666, (y * y), 1.0);
	double tmp;
	if (sin(x) <= -0.005) {
		tmp = t_0 * fma(x, (x * (x * -0.16666666666666666)), x);
	} else if (sin(x) <= 5e-8) {
		tmp = x * fma(y, (y * fma((y * y), fma((y * y), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666)), 1.0);
	} else {
		tmp = x * (t_0 * fma(x, (x * fma(x, (x * 0.008333333333333333), -0.16666666666666666)), 1.0));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(0.16666666666666666, Float64(y * y), 1.0)
	tmp = 0.0
	if (sin(x) <= -0.005)
		tmp = Float64(t_0 * fma(x, Float64(x * Float64(x * -0.16666666666666666)), x));
	elseif (sin(x) <= 5e-8)
		tmp = Float64(x * fma(y, Float64(y * fma(Float64(y * y), fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666)), 1.0));
	else
		tmp = Float64(x * Float64(t_0 * fma(x, Float64(x * fma(x, Float64(x * 0.008333333333333333), -0.16666666666666666)), 1.0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[N[Sin[x], $MachinePrecision], -0.005], N[(t$95$0 * N[(x * N[(x * N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[x], $MachinePrecision], 5e-8], N[(x * 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[(x * N[(t$95$0 * N[(x * N[(x * N[(x * N[(x * 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 x) < -0.0050000000000000001

   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. *-lowering-*.f64 N/A

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

      5. sin-lowering-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. accelerator-lowering-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. *-lowering-*.f64 72.5

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

   5. Simplified 72.5%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

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

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

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

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

      9. *-lowering-*.f64 23.6

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

   8. Simplified 23.6%

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

   if -0.0050000000000000001 < (sin.f64 x) < 4.9999999999999998e-8

   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 89.2%

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

   6. Taylor expanded in x around 0

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

   8. Simplified 89.2%

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

   if 4.9999999999999998e-8 < (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

      \[\leadsto \color{blue}{\sin x + \frac{1}{6} \cdot \left({y}^{2} \cdot \sin x\right)} \]
   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. *-lowering-*.f64 N/A

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

      5. sin-lowering-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. accelerator-lowering-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. *-lowering-*.f64 75.0

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

   5. Simplified 75.0%

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

   6. Taylor expanded in x around 0

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

   8. Simplified 30.4%

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

4. Final simplification 54.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin x \leq -0.005:\\ \;\;\;\;\mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right) \cdot \mathsf{fma}\left(x, x \cdot \left(x \cdot -0.16666666666666666\right), x\right)\\ \mathbf{elif}\;\sin x \leq 5 \cdot 10^{-8}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right) \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.008333333333333333, -0.16666666666666666\right), 1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 55.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin x \leq -0.005:\\ \;\;\;\;\mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right) \cdot \mathsf{fma}\left(x, x \cdot \left(x \cdot -0.16666666666666666\right), x\right)\\ \mathbf{elif}\;\sin x \leq 0.028:\\ \;\;\;\;x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.001388888888888889, -0.027777777777777776\right), 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sin x) -0.005)
   (*
    (fma 0.16666666666666666 (* y y) 1.0)
    (fma x (* x (* x -0.16666666666666666)) x))
   (if (<= (sin x) 0.028)
     (*
      x
      (fma (* y y) (fma (* y y) 0.008333333333333333 0.16666666666666666) 1.0))
     (*
      y
      (*
       y
       (*
        x
        (fma
         (* x x)
         (fma (* x x) 0.001388888888888889 -0.027777777777777776)
         0.16666666666666666)))))))

C

double code(double x, double y) {
	double tmp;
	if (sin(x) <= -0.005) {
		tmp = fma(0.16666666666666666, (y * y), 1.0) * fma(x, (x * (x * -0.16666666666666666)), x);
	} else if (sin(x) <= 0.028) {
		tmp = x * fma((y * y), fma((y * y), 0.008333333333333333, 0.16666666666666666), 1.0);
	} else {
		tmp = y * (y * (x * fma((x * x), fma((x * x), 0.001388888888888889, -0.027777777777777776), 0.16666666666666666)));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (sin(x) <= -0.005)
		tmp = Float64(fma(0.16666666666666666, Float64(y * y), 1.0) * fma(x, Float64(x * Float64(x * -0.16666666666666666)), x));
	elseif (sin(x) <= 0.028)
		tmp = Float64(x * fma(Float64(y * y), fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666), 1.0));
	else
		tmp = Float64(y * Float64(y * Float64(x * fma(Float64(x * x), fma(Float64(x * x), 0.001388888888888889, -0.027777777777777776), 0.16666666666666666))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[Sin[x], $MachinePrecision], -0.005], N[(N[(0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x * N[(x * N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[x], $MachinePrecision], 0.028], N[(x * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(y * N[(y * N[(x * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.001388888888888889 + -0.027777777777777776), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 x) < -0.0050000000000000001

   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. *-lowering-*.f64 N/A

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

      5. sin-lowering-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. accelerator-lowering-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. *-lowering-*.f64 72.5

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

   5. Simplified 72.5%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

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

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

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

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

      9. *-lowering-*.f64 23.6

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

   8. Simplified 23.6%

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

   if -0.0050000000000000001 < (sin.f64 x) < 0.0280000000000000006

   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. accelerator-lowering-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. *-lowering-*.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. *-lowering-*.f64 98.9

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

   5. Simplified 98.9%

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

   6. Taylor expanded in y around 0

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

      1. associate-+r+ N/A

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

      2. distribute-lft-in N/A

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

      3. associate-+r+ N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt1-in N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

   8. Simplified 82.0%

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

   9. Taylor expanded in x around 0

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

   11. Simplified 81.2%

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

   if 0.0280000000000000006 < (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

      \[\leadsto \color{blue}{\sin x + \frac{1}{6} \cdot \left({y}^{2} \cdot \sin x\right)} \]
   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. *-lowering-*.f64 N/A

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

      5. sin-lowering-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. accelerator-lowering-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. *-lowering-*.f64 73.2

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

   5. Simplified 73.2%

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

   6. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

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

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

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

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

      8. *-commutative N/A

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

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

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

      10. sin-lowering-sin.f64 33.1

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

   8. Simplified 33.1%

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

   9. Taylor expanded in x around 0

      \[\leadsto y \cdot \left(y \cdot \color{blue}{\left(x \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{720} \cdot {x}^{2} - \frac{1}{36}\right)\right)\right)}\right) \]
   10. Step-by-step derivation

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

          \[\leadsto y \cdot \left(y \cdot \color{blue}{\left(x \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{720} \cdot {x}^{2} - \frac{1}{36}\right)\right)\right)}\right) \]

       2. +-commutative N/A

          \[\leadsto y \cdot \left(y \cdot \left(x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{720} \cdot {x}^{2} - \frac{1}{36}\right) + \frac{1}{6}\right)}\right)\right) \]

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

          \[\leadsto y \cdot \left(y \cdot \left(x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{720} \cdot {x}^{2} - \frac{1}{36}, \frac{1}{6}\right)}\right)\right) \]

       4. unpow2 N/A

          \[\leadsto y \cdot \left(y \cdot \left(x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{720} \cdot {x}^{2} - \frac{1}{36}, \frac{1}{6}\right)\right)\right) \]

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

          \[\leadsto y \cdot \left(y \cdot \left(x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{720} \cdot {x}^{2} - \frac{1}{36}, \frac{1}{6}\right)\right)\right) \]

       6. sub-neg N/A

          \[\leadsto y \cdot \left(y \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{720} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{36}\right)\right)}, \frac{1}{6}\right)\right)\right) \]

       7. *-commutative N/A

          \[\leadsto y \cdot \left(y \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{1}{720}} + \left(\mathsf{neg}\left(\frac{1}{36}\right)\right), \frac{1}{6}\right)\right)\right) \]

       8. metadata-eval N/A

          \[\leadsto y \cdot \left(y \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, {x}^{2} \cdot \frac{1}{720} + \color{blue}{\frac{-1}{36}}, \frac{1}{6}\right)\right)\right) \]

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

          \[\leadsto y \cdot \left(y \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{720}, \frac{-1}{36}\right)}, \frac{1}{6}\right)\right)\right) \]

       10. unpow2 N/A

           \[\leadsto y \cdot \left(y \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{720}, \frac{-1}{36}\right), \frac{1}{6}\right)\right)\right) \]

       11. *-lowering-*.f64 27.5

           \[\leadsto y \cdot \left(y \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.001388888888888889, -0.027777777777777776\right), 0.16666666666666666\right)\right)\right) \]

   11. Simplified 27.5%

       \[\leadsto y \cdot \left(y \cdot \color{blue}{\left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.001388888888888889, -0.027777777777777776\right), 0.16666666666666666\right)\right)}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 51.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin x \leq -0.005:\\ \;\;\;\;\mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right) \cdot \mathsf{fma}\left(x, x \cdot \left(x \cdot -0.16666666666666666\right), x\right)\\ \mathbf{elif}\;\sin x \leq 0.028:\\ \;\;\;\;x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.001388888888888889, -0.027777777777777776\right), 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 54.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin x \leq -0.005:\\ \;\;\;\;\mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right) \cdot \mathsf{fma}\left(x, x \cdot \left(x \cdot -0.16666666666666666\right), x\right)\\ \mathbf{elif}\;\sin x \leq 5 \cdot 10^{-8}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, x \cdot \mathsf{fma}\left(x, x \cdot 0.008333333333333333, -0.16666666666666666\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sin x) -0.005)
   (*
    (fma 0.16666666666666666 (* y y) 1.0)
    (fma x (* x (* x -0.16666666666666666)) x))
   (if (<= (sin x) 5e-8)
     (*
      x
      (fma (* y y) (fma (* y y) 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) <= -0.005) {
		tmp = fma(0.16666666666666666, (y * y), 1.0) * fma(x, (x * (x * -0.16666666666666666)), x);
	} else if (sin(x) <= 5e-8) {
		tmp = x * fma((y * y), fma((y * y), 0.008333333333333333, 0.16666666666666666), 1.0);
	} else {
		tmp = 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) <= -0.005)
		tmp = Float64(fma(0.16666666666666666, Float64(y * y), 1.0) * fma(x, Float64(x * Float64(x * -0.16666666666666666)), x));
	elseif (sin(x) <= 5e-8)
		tmp = Float64(x * fma(Float64(y * y), fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666), 1.0));
	else
		tmp = 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], -0.005], N[(N[(0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x * N[(x * N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[x], $MachinePrecision], 5e-8], N[(x * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(x * 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 x) < -0.0050000000000000001

   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. *-lowering-*.f64 N/A

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

      5. sin-lowering-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. accelerator-lowering-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. *-lowering-*.f64 72.5

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

   5. Simplified 72.5%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

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

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

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

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

      9. *-lowering-*.f64 23.6

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

   8. Simplified 23.6%

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

   if -0.0050000000000000001 < (sin.f64 x) < 4.9999999999999998e-8

   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. accelerator-lowering-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. *-lowering-*.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. *-lowering-*.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around 0

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

      1. associate-+r+ N/A

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

      2. distribute-lft-in N/A

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

      3. associate-+r+ N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt1-in N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

   8. Simplified 82.6%

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

   9. Taylor expanded in x around 0

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

   11. Simplified 82.6%

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

   if 4.9999999999999998e-8 < (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

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

      1. sin-lowering-sin.f64 47.5

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

   5. Simplified 47.5%

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

   6. Taylor expanded in x around 0

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. +-commutative N/A

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

      4. associate-*l* N/A

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

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

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

      6. unpow2 N/A

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

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

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

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

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

      9. sub-neg N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. metadata-eval N/A

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

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

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

      16. *-commutative N/A

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

      17. *-lowering-*.f64 27.2

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

   8. Simplified 27.2%

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

4. Final simplification 51.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin x \leq -0.005:\\ \;\;\;\;\mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right) \cdot \mathsf{fma}\left(x, x \cdot \left(x \cdot -0.16666666666666666\right), x\right)\\ \mathbf{elif}\;\sin x \leq 5 \cdot 10^{-8}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, x \cdot \mathsf{fma}\left(x, x \cdot 0.008333333333333333, -0.16666666666666666\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 54.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin x \leq -0.005:\\ \;\;\;\;\left(-0.16666666666666666 \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \mathsf{fma}\left(y, y \cdot 0.16666666666666666, 1\right)\\ \mathbf{elif}\;\sin x \leq 5 \cdot 10^{-8}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, x \cdot \mathsf{fma}\left(x, x \cdot 0.008333333333333333, -0.16666666666666666\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sin x) -0.005)
   (*
    (* -0.16666666666666666 (* x (* x x)))
    (fma y (* y 0.16666666666666666) 1.0))
   (if (<= (sin x) 5e-8)
     (*
      x
      (fma (* y y) (fma (* y y) 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) <= -0.005) {
		tmp = (-0.16666666666666666 * (x * (x * x))) * fma(y, (y * 0.16666666666666666), 1.0);
	} else if (sin(x) <= 5e-8) {
		tmp = x * fma((y * y), fma((y * y), 0.008333333333333333, 0.16666666666666666), 1.0);
	} else {
		tmp = 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) <= -0.005)
		tmp = Float64(Float64(-0.16666666666666666 * Float64(x * Float64(x * x))) * fma(y, Float64(y * 0.16666666666666666), 1.0));
	elseif (sin(x) <= 5e-8)
		tmp = Float64(x * fma(Float64(y * y), fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666), 1.0));
	else
		tmp = 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], -0.005], N[(N[(-0.16666666666666666 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y * N[(y * 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[x], $MachinePrecision], 5e-8], N[(x * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(x * 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 x) < -0.0050000000000000001

   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. accelerator-lowering-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. *-lowering-*.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. *-lowering-*.f64 26.0

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

   5. Simplified 26.0%

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

   6. Taylor expanded in y around 0

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

      1. associate-+r+ N/A

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

      2. distribute-lft-in N/A

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

      3. associate-+r+ N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt1-in N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

   8. Simplified 23.6%

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

   9. Taylor expanded in y around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

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

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

       7. *-commutative N/A

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

       8. *-lowering-*.f64 23.6

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

   11. Simplified 23.6%

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

   12. Taylor expanded in x around inf

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

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

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

       2. cube-mult N/A

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

       3. unpow2 N/A

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

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

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

       5. unpow2 N/A

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

       6. *-lowering-*.f64 22.6

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

   14. Simplified 22.6%

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

   if -0.0050000000000000001 < (sin.f64 x) < 4.9999999999999998e-8

   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. accelerator-lowering-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. *-lowering-*.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. *-lowering-*.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around 0

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

      1. associate-+r+ N/A

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

      2. distribute-lft-in N/A

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

      3. associate-+r+ N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt1-in N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

   8. Simplified 82.6%

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

   9. Taylor expanded in x around 0

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

   11. Simplified 82.6%

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

   if 4.9999999999999998e-8 < (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

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

      1. sin-lowering-sin.f64 47.5

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

   5. Simplified 47.5%

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

   6. Taylor expanded in x around 0

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. +-commutative N/A

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

      4. associate-*l* N/A

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

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

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

      6. unpow2 N/A

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

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

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

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

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

      9. sub-neg N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. metadata-eval N/A

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

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

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

      16. *-commutative N/A

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

      17. *-lowering-*.f64 27.2

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

   8. Simplified 27.2%

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

4. Final simplification 50.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin x \leq -0.005:\\ \;\;\;\;\left(-0.16666666666666666 \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \mathsf{fma}\left(y, y \cdot 0.16666666666666666, 1\right)\\ \mathbf{elif}\;\sin x \leq 5 \cdot 10^{-8}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, x \cdot \mathsf{fma}\left(x, x \cdot 0.008333333333333333, -0.16666666666666666\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 54.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin x \leq -0.005:\\ \;\;\;\;y \cdot \left(x \cdot \left(y \cdot \mathsf{fma}\left(x \cdot x, -0.027777777777777776, 0.16666666666666666\right)\right)\right)\\ \mathbf{elif}\;\sin x \leq 5 \cdot 10^{-8}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, x \cdot \mathsf{fma}\left(x, x \cdot 0.008333333333333333, -0.16666666666666666\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sin x) -0.005)
   (* y (* x (* y (fma (* x x) -0.027777777777777776 0.16666666666666666))))
   (if (<= (sin x) 5e-8)
     (*
      x
      (fma (* y y) (fma (* y y) 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) <= -0.005) {
		tmp = y * (x * (y * fma((x * x), -0.027777777777777776, 0.16666666666666666)));
	} else if (sin(x) <= 5e-8) {
		tmp = x * fma((y * y), fma((y * y), 0.008333333333333333, 0.16666666666666666), 1.0);
	} else {
		tmp = 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) <= -0.005)
		tmp = Float64(y * Float64(x * Float64(y * fma(Float64(x * x), -0.027777777777777776, 0.16666666666666666))));
	elseif (sin(x) <= 5e-8)
		tmp = Float64(x * fma(Float64(y * y), fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666), 1.0));
	else
		tmp = 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], -0.005], N[(y * N[(x * N[(y * N[(N[(x * x), $MachinePrecision] * -0.027777777777777776 + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[x], $MachinePrecision], 5e-8], N[(x * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(x * 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 x) < -0.0050000000000000001

   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. *-lowering-*.f64 N/A

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

      5. sin-lowering-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. accelerator-lowering-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. *-lowering-*.f64 72.5

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

   5. Simplified 72.5%

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

   6. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

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

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

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

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

      8. *-commutative N/A

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

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

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

      10. sin-lowering-sin.f64 32.6

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

   8. Simplified 32.6%

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

   9. Taylor expanded in x around 0

      \[\leadsto y \cdot \color{blue}{\left(x \cdot \left(\frac{-1}{36} \cdot \left({x}^{2} \cdot y\right) + \frac{1}{6} \cdot y\right)\right)} \]
   10. Step-by-step derivation

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

       4. +-commutative N/A

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

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

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

       6. *-commutative N/A

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

       7. associate-*r* N/A

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

       8. *-commutative N/A

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

       9. associate-*r* N/A

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

       10. distribute-rgt-out N/A

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

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

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

       12. +-commutative N/A

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

       13. *-commutative N/A

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

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

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

       15. unpow2 N/A

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

       16. *-lowering-*.f64 22.7

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

   11. Simplified 22.7%

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

   if -0.0050000000000000001 < (sin.f64 x) < 4.9999999999999998e-8

   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. accelerator-lowering-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. *-lowering-*.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. *-lowering-*.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around 0

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

      1. associate-+r+ N/A

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

      2. distribute-lft-in N/A

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

      3. associate-+r+ N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt1-in N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

   8. Simplified 82.6%

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

   9. Taylor expanded in x around 0

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

   11. Simplified 82.6%

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

   if 4.9999999999999998e-8 < (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

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

      1. sin-lowering-sin.f64 47.5

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

   5. Simplified 47.5%

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

   6. Taylor expanded in x around 0

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. +-commutative N/A

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

      4. associate-*l* N/A

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

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

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

      6. unpow2 N/A

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

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

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

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

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

      9. sub-neg N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. metadata-eval N/A

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

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

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

      16. *-commutative N/A

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

      17. *-lowering-*.f64 27.2

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

   8. Simplified 27.2%

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

4. Final simplification 50.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin x \leq -0.005:\\ \;\;\;\;y \cdot \left(x \cdot \left(y \cdot \mathsf{fma}\left(x \cdot x, -0.027777777777777776, 0.16666666666666666\right)\right)\right)\\ \mathbf{elif}\;\sin x \leq 5 \cdot 10^{-8}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, x \cdot \mathsf{fma}\left(x, x \cdot 0.008333333333333333, -0.16666666666666666\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 48.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin x \leq -0.005:\\ \;\;\;\;y \cdot \left(x \cdot \left(y \cdot \mathsf{fma}\left(x \cdot x, -0.027777777777777776, 0.16666666666666666\right)\right)\right)\\ \mathbf{elif}\;\sin x \leq 0.13:\\ \;\;\;\;\mathsf{fma}\left(x, 0.16666666666666666 \cdot \left(y \cdot y\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, 0.008333333333333333 \cdot \left(x \cdot \left(x \cdot x\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sin x) -0.005)
   (* y (* x (* y (fma (* x x) -0.027777777777777776 0.16666666666666666))))
   (if (<= (sin x) 0.13)
     (fma x (* 0.16666666666666666 (* y y)) x)
     (fma (* x x) (* 0.008333333333333333 (* x (* x x))) x))))

C

double code(double x, double y) {
	double tmp;
	if (sin(x) <= -0.005) {
		tmp = y * (x * (y * fma((x * x), -0.027777777777777776, 0.16666666666666666)));
	} else if (sin(x) <= 0.13) {
		tmp = fma(x, (0.16666666666666666 * (y * y)), x);
	} else {
		tmp = fma((x * x), (0.008333333333333333 * (x * (x * x))), x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (sin(x) <= -0.005)
		tmp = Float64(y * Float64(x * Float64(y * fma(Float64(x * x), -0.027777777777777776, 0.16666666666666666))));
	elseif (sin(x) <= 0.13)
		tmp = fma(x, Float64(0.16666666666666666 * Float64(y * y)), x);
	else
		tmp = fma(Float64(x * x), Float64(0.008333333333333333 * Float64(x * Float64(x * x))), x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[Sin[x], $MachinePrecision], -0.005], N[(y * N[(x * N[(y * N[(N[(x * x), $MachinePrecision] * -0.027777777777777776 + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[x], $MachinePrecision], 0.13], N[(x * N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(N[(x * x), $MachinePrecision] * N[(0.008333333333333333 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 x) < -0.0050000000000000001

   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. *-lowering-*.f64 N/A

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

      5. sin-lowering-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. accelerator-lowering-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. *-lowering-*.f64 72.5

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

   5. Simplified 72.5%

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

   6. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

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

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

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

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

      8. *-commutative N/A

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

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

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

      10. sin-lowering-sin.f64 32.6

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

   8. Simplified 32.6%

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

   9. Taylor expanded in x around 0

      \[\leadsto y \cdot \color{blue}{\left(x \cdot \left(\frac{-1}{36} \cdot \left({x}^{2} \cdot y\right) + \frac{1}{6} \cdot y\right)\right)} \]
   10. Step-by-step derivation

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

       4. +-commutative N/A

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

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

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

       6. *-commutative N/A

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

       7. associate-*r* N/A

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

       8. *-commutative N/A

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

       9. associate-*r* N/A

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

       10. distribute-rgt-out N/A

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

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

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

       12. +-commutative N/A

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

       13. *-commutative N/A

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

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

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

       15. unpow2 N/A

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

       16. *-lowering-*.f64 22.7

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

   11. Simplified 22.7%

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

   if -0.0050000000000000001 < (sin.f64 x) < 0.13

   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. *-lowering-*.f64 N/A

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

      5. sin-lowering-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. accelerator-lowering-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. *-lowering-*.f64 70.6

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

   5. Simplified 70.6%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

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

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

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

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 67.3

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

   8. Simplified 67.3%

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

   if 0.13 < (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

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

      1. sin-lowering-sin.f64 44.9

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

   5. Simplified 44.9%

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

   6. Taylor expanded in x around 0

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. +-commutative N/A

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

      4. associate-*l* N/A

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

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

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

      6. unpow2 N/A

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

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

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

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

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

      9. sub-neg N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. metadata-eval N/A

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

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

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

      16. *-commutative N/A

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

      17. *-lowering-*.f64 22.6

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

   8. Simplified 22.6%

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

   9. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{120} \cdot {x}^{3}}, x\right) \]
   10. Step-by-step derivation

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

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

       2. cube-mult N/A

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

       3. unpow2 N/A

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

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

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

       5. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{1}{120} \cdot \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right), x\right) \]

       6. *-lowering-*.f64 22.6

          \[\leadsto \mathsf{fma}\left(x \cdot x, 0.008333333333333333 \cdot \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right), x\right) \]

   11. Simplified 22.6%

       \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{0.008333333333333333 \cdot \left(x \cdot \left(x \cdot x\right)\right)}, x\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 18: 41.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin x \cdot \frac{\sinh y}{y} \leq 0.01:\\ \;\;\;\;\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot -0.16666666666666666, x\right)\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot \left(x \cdot \left(y \cdot y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (sin x) (/ (sinh y) y)) 0.01)
   (fma x (* (* x x) -0.16666666666666666) x)
   (* 0.16666666666666666 (* x (* y y)))))

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);
	} else {
		tmp = 0.16666666666666666 * (x * (y * y));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(sin(x) * Float64(sinh(y) / y)) <= 0.01)
		tmp = fma(x, Float64(Float64(x * x) * -0.16666666666666666), x);
	else
		tmp = Float64(0.16666666666666666 * Float64(x * Float64(y * y)));
	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[(x * x), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + x), $MachinePrecision], N[(0.16666666666666666 * N[(x * N[(y * y), $MachinePrecision]), $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. sin-lowering-sin.f64 57.3

         \[\leadsto \color{blue}{\sin x} \]

   5. Simplified 57.3%

      \[\leadsto \color{blue}{\sin x} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right)} \]
   7. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \color{blue}{1 \cdot x + \left({x}^{2} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right) \cdot x} \]

      2. *-lft-identity N/A

         \[\leadsto \color{blue}{x} + \left({x}^{2} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right) \cdot x \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right) \cdot x + x} \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{{x}^{2} \cdot \left(\left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) \cdot x\right)} + x \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{2}, \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) \cdot x, x\right)} \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) \cdot x, x\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) \cdot x, x\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{\left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) \cdot x}, x\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{\left(\frac{1}{120} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)} \cdot x, x\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \left(\color{blue}{{x}^{2} \cdot \frac{1}{120}} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right) \cdot x, x\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{120} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right) \cdot x, x\right) \]

      12. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \left(\color{blue}{x \cdot \left(x \cdot \frac{1}{120}\right)} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right) \cdot x, x\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \left(x \cdot \color{blue}{\left(\frac{1}{120} \cdot x\right)} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right) \cdot x, x\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \left(x \cdot \left(\frac{1}{120} \cdot x\right) + \color{blue}{\frac{-1}{6}}\right) \cdot x, x\right) \]

      15. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, \frac{1}{120} \cdot x, \frac{-1}{6}\right)} \cdot x, x\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{120}}, \frac{-1}{6}\right) \cdot x, x\right) \]

      17. *-lowering-*.f64 50.7

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot 0.008333333333333333}, -0.16666666666666666\right) \cdot x, x\right) \]

   8. Simplified 50.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.008333333333333333, -0.16666666666666666\right) \cdot x, x\right)} \]

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto x \cdot \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2} + 1\right)} \]

       2. distribute-lft-in N/A

          \[\leadsto \color{blue}{x \cdot \left(\frac{-1}{6} \cdot {x}^{2}\right) + x \cdot 1} \]

       3. *-rgt-identity N/A

          \[\leadsto x \cdot \left(\frac{-1}{6} \cdot {x}^{2}\right) + \color{blue}{x} \]

       4. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{-1}{6} \cdot {x}^{2}, x\right)} \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{-1}{6} \cdot {x}^{2}}, x\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(x, \frac{-1}{6} \cdot \color{blue}{\left(x \cdot x\right)}, x\right) \]

       7. *-lowering-*.f64 43.6

          \[\leadsto \mathsf{fma}\left(x, -0.16666666666666666 \cdot \color{blue}{\left(x \cdot x\right)}, x\right) \]

   11. Simplified 43.6%

       \[\leadsto \color{blue}{\mathsf{fma}\left(x, -0.16666666666666666 \cdot \left(x \cdot x\right), x\right)} \]

   if 0.0100000000000000002 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

      \[\sin x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\sin x + \frac{1}{6} \cdot \left({y}^{2} \cdot \sin x\right)} \]
   4. Step-by-step derivation

      1. *-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. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\sin x \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)} \]

      5. sin-lowering-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. accelerator-lowering-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. *-lowering-*.f64 62.2

         \[\leadsto \sin x \cdot \mathsf{fma}\left(0.16666666666666666, \color{blue}{y \cdot y}, 1\right) \]

   5. Simplified 62.2%

      \[\leadsto \color{blue}{\sin x \cdot \mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right)} \]

   6. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{1}{6} \cdot \left({y}^{2} \cdot \sin x\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({y}^{2} \cdot \sin x\right) \cdot \frac{1}{6}} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{{y}^{2} \cdot \left(\sin x \cdot \frac{1}{6}\right)} \]

      3. *-commutative N/A

         \[\leadsto {y}^{2} \cdot \color{blue}{\left(\frac{1}{6} \cdot \sin x\right)} \]

      4. unpow2 N/A

         \[\leadsto \color{blue}{\left(y \cdot y\right)} \cdot \left(\frac{1}{6} \cdot \sin x\right) \]

      5. associate-*l* N/A

         \[\leadsto \color{blue}{y \cdot \left(y \cdot \left(\frac{1}{6} \cdot \sin x\right)\right)} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{y \cdot \left(y \cdot \left(\frac{1}{6} \cdot \sin x\right)\right)} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto y \cdot \color{blue}{\left(y \cdot \left(\frac{1}{6} \cdot \sin x\right)\right)} \]

      8. *-commutative N/A

         \[\leadsto y \cdot \left(y \cdot \color{blue}{\left(\sin x \cdot \frac{1}{6}\right)}\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto y \cdot \left(y \cdot \color{blue}{\left(\sin x \cdot \frac{1}{6}\right)}\right) \]

      10. sin-lowering-sin.f64 27.0

          \[\leadsto y \cdot \left(y \cdot \left(\color{blue}{\sin x} \cdot 0.16666666666666666\right)\right) \]

   8. Simplified 27.0%

      \[\leadsto \color{blue}{y \cdot \left(y \cdot \left(\sin x \cdot 0.16666666666666666\right)\right)} \]

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{6} \cdot \left(x \cdot {y}^{2}\right)} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{6} \cdot \left(x \cdot {y}^{2}\right)} \]

       2. *-commutative N/A

          \[\leadsto \frac{1}{6} \cdot \color{blue}{\left({y}^{2} \cdot x\right)} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \frac{1}{6} \cdot \color{blue}{\left({y}^{2} \cdot x\right)} \]

       4. unpow2 N/A

          \[\leadsto \frac{1}{6} \cdot \left(\color{blue}{\left(y \cdot y\right)} \cdot x\right) \]

       5. *-lowering-*.f64 31.5

          \[\leadsto 0.16666666666666666 \cdot \left(\color{blue}{\left(y \cdot y\right)} \cdot x\right) \]

   11. Simplified 31.5%

       \[\leadsto \color{blue}{0.16666666666666666 \cdot \left(\left(y \cdot y\right) \cdot x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 38.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin x \cdot \frac{\sinh y}{y} \leq 0.01:\\ \;\;\;\;\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot -0.16666666666666666, x\right)\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot \left(x \cdot \left(y \cdot y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 29.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin x \cdot \frac{\sinh y}{y} \leq -0.005:\\ \;\;\;\;-0.16666666666666666 \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (sin x) (/ (sinh y) y)) -0.005)
   (* -0.16666666666666666 (* x (* x x)))
   x))

C

double code(double x, double y) {
	double tmp;
	if ((sin(x) * (sinh(y) / y)) <= -0.005) {
		tmp = -0.16666666666666666 * (x * (x * x));
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((sin(x) * (sinh(y) / y)) <= (-0.005d0)) then
        tmp = (-0.16666666666666666d0) * (x * (x * x))
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((Math.sin(x) * (Math.sinh(y) / y)) <= -0.005) {
		tmp = -0.16666666666666666 * (x * (x * x));
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (math.sin(x) * (math.sinh(y) / y)) <= -0.005:
		tmp = -0.16666666666666666 * (x * (x * x))
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(sin(x) * Float64(sinh(y) / y)) <= -0.005)
		tmp = Float64(-0.16666666666666666 * Float64(x * Float64(x * x)));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((sin(x) * (sinh(y) / y)) <= -0.005)
		tmp = -0.16666666666666666 * (x * (x * x));
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Sin[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], -0.005], N[(-0.16666666666666666 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < -0.0050000000000000001

   1. Initial program 100.0%

      \[\sin x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\sin x} \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 34.2

         \[\leadsto \color{blue}{\sin x} \]

   5. Simplified 34.2%

      \[\leadsto \color{blue}{\sin x} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right)} \]
   7. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \color{blue}{1 \cdot x + \left({x}^{2} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right) \cdot x} \]

      2. *-lft-identity N/A

         \[\leadsto \color{blue}{x} + \left({x}^{2} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right) \cdot x \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right) \cdot x + x} \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{{x}^{2} \cdot \left(\left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) \cdot x\right)} + x \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{2}, \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) \cdot x, x\right)} \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) \cdot x, x\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) \cdot x, x\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{\left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) \cdot x}, x\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{\left(\frac{1}{120} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)} \cdot x, x\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \left(\color{blue}{{x}^{2} \cdot \frac{1}{120}} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right) \cdot x, x\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{120} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right) \cdot x, x\right) \]

      12. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \left(\color{blue}{x \cdot \left(x \cdot \frac{1}{120}\right)} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right) \cdot x, x\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \left(x \cdot \color{blue}{\left(\frac{1}{120} \cdot x\right)} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right) \cdot x, x\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \left(x \cdot \left(\frac{1}{120} \cdot x\right) + \color{blue}{\frac{-1}{6}}\right) \cdot x, x\right) \]

      15. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, \frac{1}{120} \cdot x, \frac{-1}{6}\right)} \cdot x, x\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{120}}, \frac{-1}{6}\right) \cdot x, x\right) \]

      17. *-lowering-*.f64 25.0

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot 0.008333333333333333}, -0.16666666666666666\right) \cdot x, x\right) \]

   8. Simplified 25.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.008333333333333333, -0.16666666666666666\right) \cdot x, x\right)} \]

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto x \cdot \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2} + 1\right)} \]

       2. distribute-lft-in N/A

          \[\leadsto \color{blue}{x \cdot \left(\frac{-1}{6} \cdot {x}^{2}\right) + x \cdot 1} \]

       3. *-rgt-identity N/A

          \[\leadsto x \cdot \left(\frac{-1}{6} \cdot {x}^{2}\right) + \color{blue}{x} \]

       4. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{-1}{6} \cdot {x}^{2}, x\right)} \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{-1}{6} \cdot {x}^{2}}, x\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(x, \frac{-1}{6} \cdot \color{blue}{\left(x \cdot x\right)}, x\right) \]

       7. *-lowering-*.f64 14.2

          \[\leadsto \mathsf{fma}\left(x, -0.16666666666666666 \cdot \color{blue}{\left(x \cdot x\right)}, x\right) \]

   11. Simplified 14.2%

       \[\leadsto \color{blue}{\mathsf{fma}\left(x, -0.16666666666666666 \cdot \left(x \cdot x\right), x\right)} \]

   12. Taylor expanded in x around inf

       \[\leadsto \color{blue}{\frac{-1}{6} \cdot {x}^{3}} \]
   13. Step-by-step derivation

       1. *-lowering-*.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. *-lowering-*.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. *-lowering-*.f64 13.3

          \[\leadsto -0.16666666666666666 \cdot \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \]

   14. Simplified 13.3%

       \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(x \cdot \left(x \cdot x\right)\right)} \]

   if -0.0050000000000000001 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

      \[\sin x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\sin x} \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 52.5

         \[\leadsto \color{blue}{\sin x} \]

   5. Simplified 52.5%

      \[\leadsto \color{blue}{\sin x} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x} \]
   7. Step-by-step derivation

   8. Simplified 36.1%

      \[\leadsto \color{blue}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 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 21: 58.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin x \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, x \cdot \left(x \cdot x\right), x\right) \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot 0.0001984126984126984, y, 0.008333333333333333\right), 0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right) \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.008333333333333333, -0.16666666666666666\right), 1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sin x) 5e-8)
   (*
    (fma -0.16666666666666666 (* x (* x x)) x)
    (/
     (fma
      (fma
       y
       (* y (fma (* y 0.0001984126984126984) y 0.008333333333333333))
       0.16666666666666666)
      (* y (* y y))
      y)
     y))
   (*
    x
    (*
     (fma 0.16666666666666666 (* y y) 1.0)
     (fma
      x
      (* x (fma x (* x 0.008333333333333333) -0.16666666666666666))
      1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (sin(x) <= 5e-8) {
		tmp = fma(-0.16666666666666666, (x * (x * x)), x) * (fma(fma(y, (y * fma((y * 0.0001984126984126984), y, 0.008333333333333333)), 0.16666666666666666), (y * (y * y)), y) / y);
	} else {
		tmp = x * (fma(0.16666666666666666, (y * y), 1.0) * fma(x, (x * fma(x, (x * 0.008333333333333333), -0.16666666666666666)), 1.0));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (sin(x) <= 5e-8)
		tmp = Float64(fma(-0.16666666666666666, Float64(x * Float64(x * x)), x) * Float64(fma(fma(y, Float64(y * fma(Float64(y * 0.0001984126984126984), y, 0.008333333333333333)), 0.16666666666666666), Float64(y * Float64(y * y)), y) / y));
	else
		tmp = Float64(x * Float64(fma(0.16666666666666666, Float64(y * y), 1.0) * fma(x, Float64(x * fma(x, Float64(x * 0.008333333333333333), -0.16666666666666666)), 1.0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[Sin[x], $MachinePrecision], 5e-8], N[(N[(-0.16666666666666666 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] * N[(N[(N[(y * N[(y * N[(N[(y * 0.0001984126984126984), $MachinePrecision] * y + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x * N[(x * N[(x * N[(x * 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 x) < 4.9999999999999998e-8

   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. accelerator-lowering-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. *-lowering-*.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. *-lowering-*.f64 70.0

          \[\leadsto \mathsf{fma}\left(-0.16666666666666666, x \cdot \color{blue}{\left(x \cdot x\right)}, x\right) \cdot \frac{\sinh y}{y} \]

   5. Simplified 70.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.16666666666666666, x \cdot \left(x \cdot x\right), x\right)} \cdot \frac{\sinh y}{y} \]

   6. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma}\left(\frac{-1}{6}, x \cdot \left(x \cdot x\right), x\right) \cdot \frac{\color{blue}{y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)}}{y} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{6}, x \cdot \left(x \cdot x\right), x\right) \cdot \frac{\color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right) \cdot y}}{y} \]

      2. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{6}, x \cdot \left(x \cdot x\right), x\right) \cdot \frac{\color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) + 1\right)} \cdot y}{y} \]

      3. distribute-lft1-in N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{6}, x \cdot \left(x \cdot x\right), x\right) \cdot \frac{\color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right) \cdot y + y}}{y} \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{6}, x \cdot \left(x \cdot x\right), x\right) \cdot \frac{\color{blue}{\left(\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot {y}^{2}\right)} \cdot y + y}{y} \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{6}, x \cdot \left(x \cdot x\right), x\right) \cdot \frac{\color{blue}{\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot \left({y}^{2} \cdot y\right)} + y}{y} \]

      6. pow-plus N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{6}, x \cdot \left(x \cdot x\right), x\right) \cdot \frac{\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot \color{blue}{{y}^{\left(2 + 1\right)}} + y}{y} \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{6}, x \cdot \left(x \cdot x\right), x\right) \cdot \frac{\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot {y}^{\color{blue}{3}} + y}{y} \]

      8. cube-unmult N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{6}, x \cdot \left(x \cdot x\right), x\right) \cdot \frac{\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot \color{blue}{\left(y \cdot \left(y \cdot y\right)\right)} + y}{y} \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{6}, x \cdot \left(x \cdot x\right), x\right) \cdot \frac{\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot \left(y \cdot \color{blue}{{y}^{2}}\right) + y}{y} \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{6}, x \cdot \left(x \cdot x\right), x\right) \cdot \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right), y \cdot {y}^{2}, y\right)}}{y} \]

   8. Simplified 64.1%

      \[\leadsto \mathsf{fma}\left(-0.16666666666666666, x \cdot \left(x \cdot x\right), x\right) \cdot \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)}}{y} \]
   9. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{6}, x \cdot \left(x \cdot x\right), x\right) \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \left(\color{blue}{y \cdot \left(y \cdot \frac{1}{5040}\right)} + \frac{1}{120}\right), \frac{1}{6}\right), y \cdot \left(y \cdot y\right), y\right)}{y} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{6}, x \cdot \left(x \cdot x\right), x\right) \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \left(\color{blue}{\left(y \cdot \frac{1}{5040}\right) \cdot y} + \frac{1}{120}\right), \frac{1}{6}\right), y \cdot \left(y \cdot y\right), y\right)}{y} \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{6}, x \cdot \left(x \cdot x\right), x\right) \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \color{blue}{\mathsf{fma}\left(y \cdot \frac{1}{5040}, y, \frac{1}{120}\right)}, \frac{1}{6}\right), y \cdot \left(y \cdot y\right), y\right)}{y} \]

      4. *-lowering-*.f64 64.1

         \[\leadsto \mathsf{fma}\left(-0.16666666666666666, x \cdot \left(x \cdot x\right), x\right) \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(\color{blue}{y \cdot 0.0001984126984126984}, y, 0.008333333333333333\right), 0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)}{y} \]

   10. Applied egg-rr 64.1%

       \[\leadsto \mathsf{fma}\left(-0.16666666666666666, x \cdot \left(x \cdot x\right), x\right) \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \color{blue}{\mathsf{fma}\left(y \cdot 0.0001984126984126984, y, 0.008333333333333333\right)}, 0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)}{y} \]

   if 4.9999999999999998e-8 < (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

      \[\leadsto \color{blue}{\sin x + \frac{1}{6} \cdot \left({y}^{2} \cdot \sin x\right)} \]
   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. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\sin x \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)} \]

      5. sin-lowering-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. accelerator-lowering-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. *-lowering-*.f64 75.0

         \[\leadsto \sin x \cdot \mathsf{fma}\left(0.16666666666666666, \color{blue}{y \cdot y}, 1\right) \]

   5. Simplified 75.0%

      \[\leadsto \color{blue}{\sin x \cdot \mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + \left(\frac{1}{6} \cdot {y}^{2} + {x}^{2} \cdot \left(\frac{-1}{6} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right) + \frac{1}{120} \cdot \left({x}^{2} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)\right)\right)\right)} \]

   8. Simplified 30.4%

      \[\leadsto \color{blue}{x \cdot \left(\mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right) \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.008333333333333333, -0.16666666666666666\right), 1\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 22: 58.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin x \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, x \cdot \left(x \cdot x\right), x\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right) \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.008333333333333333, -0.16666666666666666\right), 1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sin x) 5e-8)
   (*
    (fma -0.16666666666666666 (* x (* x x)) x)
    (fma
     (* y y)
     (fma
      y
      (* y (fma (* y y) 0.0001984126984126984 0.008333333333333333))
      0.16666666666666666)
     1.0))
   (*
    x
    (*
     (fma 0.16666666666666666 (* y y) 1.0)
     (fma
      x
      (* x (fma x (* x 0.008333333333333333) -0.16666666666666666))
      1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (sin(x) <= 5e-8) {
		tmp = fma(-0.16666666666666666, (x * (x * x)), x) * fma((y * y), fma(y, (y * fma((y * y), 0.0001984126984126984, 0.008333333333333333)), 0.16666666666666666), 1.0);
	} else {
		tmp = x * (fma(0.16666666666666666, (y * y), 1.0) * fma(x, (x * fma(x, (x * 0.008333333333333333), -0.16666666666666666)), 1.0));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (sin(x) <= 5e-8)
		tmp = Float64(fma(-0.16666666666666666, Float64(x * Float64(x * x)), x) * fma(Float64(y * y), fma(y, Float64(y * fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333)), 0.16666666666666666), 1.0));
	else
		tmp = Float64(x * Float64(fma(0.16666666666666666, Float64(y * y), 1.0) * fma(x, Float64(x * fma(x, Float64(x * 0.008333333333333333), -0.16666666666666666)), 1.0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[Sin[x], $MachinePrecision], 5e-8], N[(N[(-0.16666666666666666 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x * N[(x * N[(x * N[(x * 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 x) < 4.9999999999999998e-8

   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. accelerator-lowering-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. *-lowering-*.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. *-lowering-*.f64 70.0

          \[\leadsto \mathsf{fma}\left(-0.16666666666666666, x \cdot \color{blue}{\left(x \cdot x\right)}, x\right) \cdot \frac{\sinh y}{y} \]

   5. Simplified 70.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.16666666666666666, x \cdot \left(x \cdot x\right), x\right)} \cdot \frac{\sinh y}{y} \]

   6. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma}\left(\frac{-1}{6}, x \cdot \left(x \cdot x\right), x\right) \cdot \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{6}, x \cdot \left(x \cdot x\right), x\right) \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) + 1\right)} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{6}, x \cdot \left(x \cdot x\right), x\right) \cdot \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right), 1\right)} \]

      3. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{6}, x \cdot \left(x \cdot x\right), x\right) \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right), 1\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{6}, x \cdot \left(x \cdot x\right), x\right) \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right), 1\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{6}, x \cdot \left(x \cdot x\right), x\right) \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{{y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right) + \frac{1}{6}}, 1\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{6}, x \cdot \left(x \cdot x\right), x\right) \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{\left(y \cdot y\right)} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right) + \frac{1}{6}, 1\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{6}, x \cdot \left(x \cdot x\right), x\right) \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{y \cdot \left(y \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)} + \frac{1}{6}, 1\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{6}, x \cdot \left(x \cdot x\right), x\right) \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{\mathsf{fma}\left(y, y \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right), \frac{1}{6}\right)}, 1\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{6}, x \cdot \left(x \cdot x\right), x\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, \color{blue}{y \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)}, \frac{1}{6}\right), 1\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{6}, x \cdot \left(x \cdot x\right), x\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \color{blue}{\left(\frac{1}{5040} \cdot {y}^{2} + \frac{1}{120}\right)}, \frac{1}{6}\right), 1\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{6}, x \cdot \left(x \cdot x\right), x\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \left(\color{blue}{{y}^{2} \cdot \frac{1}{5040}} + \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{6}, x \cdot \left(x \cdot x\right), x\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{1}{5040}, \frac{1}{120}\right)}, \frac{1}{6}\right), 1\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{6}, x \cdot \left(x \cdot x\right), x\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      14. *-lowering-*.f64 62.6

          \[\leadsto \mathsf{fma}\left(-0.16666666666666666, x \cdot \left(x \cdot x\right), x\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right) \]

   8. Simplified 62.6%

      \[\leadsto \mathsf{fma}\left(-0.16666666666666666, x \cdot \left(x \cdot x\right), x\right) \cdot \color{blue}{\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)} \]

   if 4.9999999999999998e-8 < (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

      \[\leadsto \color{blue}{\sin x + \frac{1}{6} \cdot \left({y}^{2} \cdot \sin x\right)} \]
   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. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\sin x \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)} \]

      5. sin-lowering-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. accelerator-lowering-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. *-lowering-*.f64 75.0

         \[\leadsto \sin x \cdot \mathsf{fma}\left(0.16666666666666666, \color{blue}{y \cdot y}, 1\right) \]

   5. Simplified 75.0%

      \[\leadsto \color{blue}{\sin x \cdot \mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + \left(\frac{1}{6} \cdot {y}^{2} + {x}^{2} \cdot \left(\frac{-1}{6} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right) + \frac{1}{120} \cdot \left({x}^{2} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)\right)\right)\right)} \]

   8. Simplified 30.4%

      \[\leadsto \color{blue}{x \cdot \left(\mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right) \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.008333333333333333, -0.16666666666666666\right), 1\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 23: 55.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin x \leq -0.005:\\ \;\;\;\;y \cdot \left(x \cdot \left(y \cdot \mathsf{fma}\left(x \cdot x, -0.027777777777777776, 0.16666666666666666\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sin x) -0.005)
   (* y (* x (* y (fma (* x x) -0.027777777777777776 0.16666666666666666))))
   (*
    x
    (fma (* y y) (fma (* y y) 0.008333333333333333 0.16666666666666666) 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (sin(x) <= -0.005) {
		tmp = y * (x * (y * fma((x * x), -0.027777777777777776, 0.16666666666666666)));
	} else {
		tmp = x * fma((y * y), fma((y * y), 0.008333333333333333, 0.16666666666666666), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (sin(x) <= -0.005)
		tmp = Float64(y * Float64(x * Float64(y * fma(Float64(x * x), -0.027777777777777776, 0.16666666666666666))));
	else
		tmp = Float64(x * fma(Float64(y * y), fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[Sin[x], $MachinePrecision], -0.005], N[(y * N[(x * N[(y * N[(N[(x * x), $MachinePrecision] * -0.027777777777777776 + 0.16666666666666666), $MachinePrecision]), $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) < -0.0050000000000000001

   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. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\sin x \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)} \]

      5. sin-lowering-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. accelerator-lowering-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. *-lowering-*.f64 72.5

         \[\leadsto \sin x \cdot \mathsf{fma}\left(0.16666666666666666, \color{blue}{y \cdot y}, 1\right) \]

   5. Simplified 72.5%

      \[\leadsto \color{blue}{\sin x \cdot \mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right)} \]

   6. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{1}{6} \cdot \left({y}^{2} \cdot \sin x\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({y}^{2} \cdot \sin x\right) \cdot \frac{1}{6}} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{{y}^{2} \cdot \left(\sin x \cdot \frac{1}{6}\right)} \]

      3. *-commutative N/A

         \[\leadsto {y}^{2} \cdot \color{blue}{\left(\frac{1}{6} \cdot \sin x\right)} \]

      4. unpow2 N/A

         \[\leadsto \color{blue}{\left(y \cdot y\right)} \cdot \left(\frac{1}{6} \cdot \sin x\right) \]

      5. associate-*l* N/A

         \[\leadsto \color{blue}{y \cdot \left(y \cdot \left(\frac{1}{6} \cdot \sin x\right)\right)} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{y \cdot \left(y \cdot \left(\frac{1}{6} \cdot \sin x\right)\right)} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto y \cdot \color{blue}{\left(y \cdot \left(\frac{1}{6} \cdot \sin x\right)\right)} \]

      8. *-commutative N/A

         \[\leadsto y \cdot \left(y \cdot \color{blue}{\left(\sin x \cdot \frac{1}{6}\right)}\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto y \cdot \left(y \cdot \color{blue}{\left(\sin x \cdot \frac{1}{6}\right)}\right) \]

      10. sin-lowering-sin.f64 32.6

          \[\leadsto y \cdot \left(y \cdot \left(\color{blue}{\sin x} \cdot 0.16666666666666666\right)\right) \]

   8. Simplified 32.6%

      \[\leadsto \color{blue}{y \cdot \left(y \cdot \left(\sin x \cdot 0.16666666666666666\right)\right)} \]

   9. Taylor expanded in x around 0

      \[\leadsto y \cdot \color{blue}{\left(x \cdot \left(\frac{-1}{36} \cdot \left({x}^{2} \cdot y\right) + \frac{1}{6} \cdot y\right)\right)} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto y \cdot \left(x \cdot \left(\color{blue}{\left({x}^{2} \cdot y\right) \cdot \frac{-1}{36}} + \frac{1}{6} \cdot y\right)\right) \]

       2. associate-*r* N/A

          \[\leadsto y \cdot \left(x \cdot \left(\color{blue}{{x}^{2} \cdot \left(y \cdot \frac{-1}{36}\right)} + \frac{1}{6} \cdot y\right)\right) \]

       3. *-commutative N/A

          \[\leadsto y \cdot \left(x \cdot \left({x}^{2} \cdot \color{blue}{\left(\frac{-1}{36} \cdot y\right)} + \frac{1}{6} \cdot y\right)\right) \]

       4. +-commutative N/A

          \[\leadsto y \cdot \left(x \cdot \color{blue}{\left(\frac{1}{6} \cdot y + {x}^{2} \cdot \left(\frac{-1}{36} \cdot y\right)\right)}\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto y \cdot \color{blue}{\left(x \cdot \left(\frac{1}{6} \cdot y + {x}^{2} \cdot \left(\frac{-1}{36} \cdot y\right)\right)\right)} \]

       6. *-commutative N/A

          \[\leadsto y \cdot \left(x \cdot \left(\frac{1}{6} \cdot y + {x}^{2} \cdot \color{blue}{\left(y \cdot \frac{-1}{36}\right)}\right)\right) \]

       7. associate-*r* N/A

          \[\leadsto y \cdot \left(x \cdot \left(\frac{1}{6} \cdot y + \color{blue}{\left({x}^{2} \cdot y\right) \cdot \frac{-1}{36}}\right)\right) \]

       8. *-commutative N/A

          \[\leadsto y \cdot \left(x \cdot \left(\frac{1}{6} \cdot y + \color{blue}{\frac{-1}{36} \cdot \left({x}^{2} \cdot y\right)}\right)\right) \]

       9. associate-*r* N/A

          \[\leadsto y \cdot \left(x \cdot \left(\frac{1}{6} \cdot y + \color{blue}{\left(\frac{-1}{36} \cdot {x}^{2}\right) \cdot y}\right)\right) \]

       10. distribute-rgt-out N/A

           \[\leadsto y \cdot \left(x \cdot \color{blue}{\left(y \cdot \left(\frac{1}{6} + \frac{-1}{36} \cdot {x}^{2}\right)\right)}\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto y \cdot \left(x \cdot \color{blue}{\left(y \cdot \left(\frac{1}{6} + \frac{-1}{36} \cdot {x}^{2}\right)\right)}\right) \]

       12. +-commutative N/A

           \[\leadsto y \cdot \left(x \cdot \left(y \cdot \color{blue}{\left(\frac{-1}{36} \cdot {x}^{2} + \frac{1}{6}\right)}\right)\right) \]

       13. *-commutative N/A

           \[\leadsto y \cdot \left(x \cdot \left(y \cdot \left(\color{blue}{{x}^{2} \cdot \frac{-1}{36}} + \frac{1}{6}\right)\right)\right) \]

       14. accelerator-lowering-fma.f64 N/A

           \[\leadsto y \cdot \left(x \cdot \left(y \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{-1}{36}, \frac{1}{6}\right)}\right)\right) \]

       15. unpow2 N/A

           \[\leadsto y \cdot \left(x \cdot \left(y \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{36}, \frac{1}{6}\right)\right)\right) \]

       16. *-lowering-*.f64 22.7

           \[\leadsto y \cdot \left(x \cdot \left(y \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, -0.027777777777777776, 0.16666666666666666\right)\right)\right) \]

   11. Simplified 22.7%

       \[\leadsto y \cdot \color{blue}{\left(x \cdot \left(y \cdot \mathsf{fma}\left(x \cdot x, -0.027777777777777776, 0.16666666666666666\right)\right)\right)} \]

   if -0.0050000000000000001 < (sin.f64 x)

   1. Initial program 100.0%

      \[\sin x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)} \cdot \frac{\sinh y}{y} \]
   4. Step-by-step derivation

      1. +-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. accelerator-lowering-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. *-lowering-*.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. *-lowering-*.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%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.16666666666666666, x \cdot \left(x \cdot x\right), x\right)} \cdot \frac{\sinh y}{y} \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + \left(\frac{-1}{6} \cdot {x}^{3} + {y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \left(x + \frac{-1}{6} \cdot {x}^{3}\right)\right) + \frac{1}{6} \cdot \left(x + \frac{-1}{6} \cdot {x}^{3}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(x + \frac{-1}{6} \cdot {x}^{3}\right) + {y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \left(x + \frac{-1}{6} \cdot {x}^{3}\right)\right) + \frac{1}{6} \cdot \left(x + \frac{-1}{6} \cdot {x}^{3}\right)\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \left(x + \frac{-1}{6} \cdot {x}^{3}\right) + \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \left(x + \frac{-1}{6} \cdot {x}^{3}\right)\right)\right) + {y}^{2} \cdot \left(\frac{1}{6} \cdot \left(x + \frac{-1}{6} \cdot {x}^{3}\right)\right)\right)} \]

      3. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\left(x + \frac{-1}{6} \cdot {x}^{3}\right) + {y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \left(x + \frac{-1}{6} \cdot {x}^{3}\right)\right)\right)\right) + {y}^{2} \cdot \left(\frac{1}{6} \cdot \left(x + \frac{-1}{6} \cdot {x}^{3}\right)\right)} \]

      4. associate-*r* N/A

         \[\leadsto \left(\left(x + \frac{-1}{6} \cdot {x}^{3}\right) + {y}^{2} \cdot \color{blue}{\left(\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot \left(x + \frac{-1}{6} \cdot {x}^{3}\right)\right)}\right) + {y}^{2} \cdot \left(\frac{1}{6} \cdot \left(x + \frac{-1}{6} \cdot {x}^{3}\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(\left(x + \frac{-1}{6} \cdot {x}^{3}\right) + \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2}\right)\right) \cdot \left(x + \frac{-1}{6} \cdot {x}^{3}\right)}\right) + {y}^{2} \cdot \left(\frac{1}{6} \cdot \left(x + \frac{-1}{6} \cdot {x}^{3}\right)\right) \]

      6. distribute-rgt1-in N/A

         \[\leadsto \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2}\right) + 1\right) \cdot \left(x + \frac{-1}{6} \cdot {x}^{3}\right)} + {y}^{2} \cdot \left(\frac{1}{6} \cdot \left(x + \frac{-1}{6} \cdot {x}^{3}\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \left({y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2}\right) + 1\right) \cdot \left(x + \frac{-1}{6} \cdot {x}^{3}\right) + \color{blue}{\left({y}^{2} \cdot \frac{1}{6}\right) \cdot \left(x + \frac{-1}{6} \cdot {x}^{3}\right)} \]

      8. *-commutative N/A

         \[\leadsto \left({y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2}\right) + 1\right) \cdot \left(x + \frac{-1}{6} \cdot {x}^{3}\right) + \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right)} \cdot \left(x + \frac{-1}{6} \cdot {x}^{3}\right) \]

   8. Simplified 66.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot \left(x \cdot -0.16666666666666666\right), x\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), 1\right)} \]

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x} \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \frac{1}{120}, \frac{1}{6}\right), 1\right) \]
   10. Step-by-step derivation

   11. Simplified 62.0%

       \[\leadsto \color{blue}{x} \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), 1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 24: 48.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin x \leq -0.005:\\ \;\;\;\;y \cdot \left(x \cdot \left(y \cdot \mathsf{fma}\left(x \cdot x, -0.027777777777777776, 0.16666666666666666\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 0.16666666666666666 \cdot \left(y \cdot y\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sin x) -0.005)
   (* y (* x (* y (fma (* x x) -0.027777777777777776 0.16666666666666666))))
   (fma x (* 0.16666666666666666 (* y y)) x)))

C

double code(double x, double y) {
	double tmp;
	if (sin(x) <= -0.005) {
		tmp = y * (x * (y * fma((x * x), -0.027777777777777776, 0.16666666666666666)));
	} else {
		tmp = fma(x, (0.16666666666666666 * (y * y)), x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (sin(x) <= -0.005)
		tmp = Float64(y * Float64(x * Float64(y * fma(Float64(x * x), -0.027777777777777776, 0.16666666666666666))));
	else
		tmp = fma(x, Float64(0.16666666666666666 * Float64(y * y)), x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[Sin[x], $MachinePrecision], -0.005], N[(y * N[(x * N[(y * N[(N[(x * x), $MachinePrecision] * -0.027777777777777776 + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 x) < -0.0050000000000000001

   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. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\sin x \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)} \]

      5. sin-lowering-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. accelerator-lowering-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. *-lowering-*.f64 72.5

         \[\leadsto \sin x \cdot \mathsf{fma}\left(0.16666666666666666, \color{blue}{y \cdot y}, 1\right) \]

   5. Simplified 72.5%

      \[\leadsto \color{blue}{\sin x \cdot \mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right)} \]

   6. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{1}{6} \cdot \left({y}^{2} \cdot \sin x\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({y}^{2} \cdot \sin x\right) \cdot \frac{1}{6}} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{{y}^{2} \cdot \left(\sin x \cdot \frac{1}{6}\right)} \]

      3. *-commutative N/A

         \[\leadsto {y}^{2} \cdot \color{blue}{\left(\frac{1}{6} \cdot \sin x\right)} \]

      4. unpow2 N/A

         \[\leadsto \color{blue}{\left(y \cdot y\right)} \cdot \left(\frac{1}{6} \cdot \sin x\right) \]

      5. associate-*l* N/A

         \[\leadsto \color{blue}{y \cdot \left(y \cdot \left(\frac{1}{6} \cdot \sin x\right)\right)} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{y \cdot \left(y \cdot \left(\frac{1}{6} \cdot \sin x\right)\right)} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto y \cdot \color{blue}{\left(y \cdot \left(\frac{1}{6} \cdot \sin x\right)\right)} \]

      8. *-commutative N/A

         \[\leadsto y \cdot \left(y \cdot \color{blue}{\left(\sin x \cdot \frac{1}{6}\right)}\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto y \cdot \left(y \cdot \color{blue}{\left(\sin x \cdot \frac{1}{6}\right)}\right) \]

      10. sin-lowering-sin.f64 32.6

          \[\leadsto y \cdot \left(y \cdot \left(\color{blue}{\sin x} \cdot 0.16666666666666666\right)\right) \]

   8. Simplified 32.6%

      \[\leadsto \color{blue}{y \cdot \left(y \cdot \left(\sin x \cdot 0.16666666666666666\right)\right)} \]

   9. Taylor expanded in x around 0

      \[\leadsto y \cdot \color{blue}{\left(x \cdot \left(\frac{-1}{36} \cdot \left({x}^{2} \cdot y\right) + \frac{1}{6} \cdot y\right)\right)} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto y \cdot \left(x \cdot \left(\color{blue}{\left({x}^{2} \cdot y\right) \cdot \frac{-1}{36}} + \frac{1}{6} \cdot y\right)\right) \]

       2. associate-*r* N/A

          \[\leadsto y \cdot \left(x \cdot \left(\color{blue}{{x}^{2} \cdot \left(y \cdot \frac{-1}{36}\right)} + \frac{1}{6} \cdot y\right)\right) \]

       3. *-commutative N/A

          \[\leadsto y \cdot \left(x \cdot \left({x}^{2} \cdot \color{blue}{\left(\frac{-1}{36} \cdot y\right)} + \frac{1}{6} \cdot y\right)\right) \]

       4. +-commutative N/A

          \[\leadsto y \cdot \left(x \cdot \color{blue}{\left(\frac{1}{6} \cdot y + {x}^{2} \cdot \left(\frac{-1}{36} \cdot y\right)\right)}\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto y \cdot \color{blue}{\left(x \cdot \left(\frac{1}{6} \cdot y + {x}^{2} \cdot \left(\frac{-1}{36} \cdot y\right)\right)\right)} \]

       6. *-commutative N/A

          \[\leadsto y \cdot \left(x \cdot \left(\frac{1}{6} \cdot y + {x}^{2} \cdot \color{blue}{\left(y \cdot \frac{-1}{36}\right)}\right)\right) \]

       7. associate-*r* N/A

          \[\leadsto y \cdot \left(x \cdot \left(\frac{1}{6} \cdot y + \color{blue}{\left({x}^{2} \cdot y\right) \cdot \frac{-1}{36}}\right)\right) \]

       8. *-commutative N/A

          \[\leadsto y \cdot \left(x \cdot \left(\frac{1}{6} \cdot y + \color{blue}{\frac{-1}{36} \cdot \left({x}^{2} \cdot y\right)}\right)\right) \]

       9. associate-*r* N/A

          \[\leadsto y \cdot \left(x \cdot \left(\frac{1}{6} \cdot y + \color{blue}{\left(\frac{-1}{36} \cdot {x}^{2}\right) \cdot y}\right)\right) \]

       10. distribute-rgt-out N/A

           \[\leadsto y \cdot \left(x \cdot \color{blue}{\left(y \cdot \left(\frac{1}{6} + \frac{-1}{36} \cdot {x}^{2}\right)\right)}\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto y \cdot \left(x \cdot \color{blue}{\left(y \cdot \left(\frac{1}{6} + \frac{-1}{36} \cdot {x}^{2}\right)\right)}\right) \]

       12. +-commutative N/A

           \[\leadsto y \cdot \left(x \cdot \left(y \cdot \color{blue}{\left(\frac{-1}{36} \cdot {x}^{2} + \frac{1}{6}\right)}\right)\right) \]

       13. *-commutative N/A

           \[\leadsto y \cdot \left(x \cdot \left(y \cdot \left(\color{blue}{{x}^{2} \cdot \frac{-1}{36}} + \frac{1}{6}\right)\right)\right) \]

       14. accelerator-lowering-fma.f64 N/A

           \[\leadsto y \cdot \left(x \cdot \left(y \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{-1}{36}, \frac{1}{6}\right)}\right)\right) \]

       15. unpow2 N/A

           \[\leadsto y \cdot \left(x \cdot \left(y \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{36}, \frac{1}{6}\right)\right)\right) \]

       16. *-lowering-*.f64 22.7

           \[\leadsto y \cdot \left(x \cdot \left(y \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, -0.027777777777777776, 0.16666666666666666\right)\right)\right) \]

   11. Simplified 22.7%

       \[\leadsto y \cdot \color{blue}{\left(x \cdot \left(y \cdot \mathsf{fma}\left(x \cdot x, -0.027777777777777776, 0.16666666666666666\right)\right)\right)} \]

   if -0.0050000000000000001 < (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

      \[\leadsto \color{blue}{\sin x + \frac{1}{6} \cdot \left({y}^{2} \cdot \sin x\right)} \]
   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. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\sin x \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)} \]

      5. sin-lowering-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. accelerator-lowering-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. *-lowering-*.f64 71.6

         \[\leadsto \sin x \cdot \mathsf{fma}\left(0.16666666666666666, \color{blue}{y \cdot y}, 1\right) \]

   5. Simplified 71.6%

      \[\leadsto \color{blue}{\sin x \cdot \mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(\frac{1}{6} \cdot {y}^{2} + 1\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{x \cdot \left(\frac{1}{6} \cdot {y}^{2}\right) + x \cdot 1} \]

      3. *-rgt-identity N/A

         \[\leadsto x \cdot \left(\frac{1}{6} \cdot {y}^{2}\right) + \color{blue}{x} \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{1}{6} \cdot {y}^{2}, x\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{1}{6} \cdot {y}^{2}}, x\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(x, \frac{1}{6} \cdot \color{blue}{\left(y \cdot y\right)}, x\right) \]

      7. *-lowering-*.f64 53.0

         \[\leadsto \mathsf{fma}\left(x, 0.16666666666666666 \cdot \color{blue}{\left(y \cdot y\right)}, x\right) \]

   8. Simplified 53.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, 0.16666666666666666 \cdot \left(y \cdot y\right), x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 25: 47.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin x \leq -0.005:\\ \;\;\;\;-0.16666666666666666 \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(y, y \cdot 0.16666666666666666, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sin x) -0.005)
   (* -0.16666666666666666 (* x (* x x)))
   (* x (fma y (* y 0.16666666666666666) 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (sin(x) <= -0.005) {
		tmp = -0.16666666666666666 * (x * (x * x));
	} else {
		tmp = x * fma(y, (y * 0.16666666666666666), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (sin(x) <= -0.005)
		tmp = Float64(-0.16666666666666666 * Float64(x * Float64(x * x)));
	else
		tmp = Float64(x * fma(y, Float64(y * 0.16666666666666666), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[Sin[x], $MachinePrecision], -0.005], N[(-0.16666666666666666 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(y * N[(y * 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 x) < -0.0050000000000000001

   1. Initial program 100.0%

      \[\sin x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\sin x} \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 43.2

         \[\leadsto \color{blue}{\sin x} \]

   5. Simplified 43.2%

      \[\leadsto \color{blue}{\sin x} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right)} \]
   7. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \color{blue}{1 \cdot x + \left({x}^{2} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right) \cdot x} \]

      2. *-lft-identity N/A

         \[\leadsto \color{blue}{x} + \left({x}^{2} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right) \cdot x \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right) \cdot x + x} \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{{x}^{2} \cdot \left(\left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) \cdot x\right)} + x \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{2}, \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) \cdot x, x\right)} \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) \cdot x, x\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) \cdot x, x\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{\left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) \cdot x}, x\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{\left(\frac{1}{120} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)} \cdot x, x\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \left(\color{blue}{{x}^{2} \cdot \frac{1}{120}} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right) \cdot x, x\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{120} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right) \cdot x, x\right) \]

      12. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \left(\color{blue}{x \cdot \left(x \cdot \frac{1}{120}\right)} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right) \cdot x, x\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \left(x \cdot \color{blue}{\left(\frac{1}{120} \cdot x\right)} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right) \cdot x, x\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \left(x \cdot \left(\frac{1}{120} \cdot x\right) + \color{blue}{\frac{-1}{6}}\right) \cdot x, x\right) \]

      15. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, \frac{1}{120} \cdot x, \frac{-1}{6}\right)} \cdot x, x\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{120}}, \frac{-1}{6}\right) \cdot x, x\right) \]

      17. *-lowering-*.f64 31.3

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot 0.008333333333333333}, -0.16666666666666666\right) \cdot x, x\right) \]

   8. Simplified 31.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.008333333333333333, -0.16666666666666666\right) \cdot x, x\right)} \]

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto x \cdot \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2} + 1\right)} \]

       2. distribute-lft-in N/A

          \[\leadsto \color{blue}{x \cdot \left(\frac{-1}{6} \cdot {x}^{2}\right) + x \cdot 1} \]

       3. *-rgt-identity N/A

          \[\leadsto x \cdot \left(\frac{-1}{6} \cdot {x}^{2}\right) + \color{blue}{x} \]

       4. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{-1}{6} \cdot {x}^{2}, x\right)} \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{-1}{6} \cdot {x}^{2}}, x\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(x, \frac{-1}{6} \cdot \color{blue}{\left(x \cdot x\right)}, x\right) \]

       7. *-lowering-*.f64 17.6

          \[\leadsto \mathsf{fma}\left(x, -0.16666666666666666 \cdot \color{blue}{\left(x \cdot x\right)}, x\right) \]

   11. Simplified 17.6%

       \[\leadsto \color{blue}{\mathsf{fma}\left(x, -0.16666666666666666 \cdot \left(x \cdot x\right), x\right)} \]

   12. Taylor expanded in x around inf

       \[\leadsto \color{blue}{\frac{-1}{6} \cdot {x}^{3}} \]
   13. Step-by-step derivation

       1. *-lowering-*.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. *-lowering-*.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. *-lowering-*.f64 16.6

          \[\leadsto -0.16666666666666666 \cdot \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \]

   14. Simplified 16.6%

       \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(x \cdot \left(x \cdot x\right)\right)} \]

   if -0.0050000000000000001 < (sin.f64 x)

   1. Initial program 100.0%

      \[\sin x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)} \cdot \frac{\sinh y}{y} \]
   4. Step-by-step derivation

      1. +-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. accelerator-lowering-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. *-lowering-*.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. *-lowering-*.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%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.16666666666666666, x \cdot \left(x \cdot x\right), x\right)} \cdot \frac{\sinh y}{y} \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + \left(\frac{-1}{6} \cdot {x}^{3} + {y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \left(x + \frac{-1}{6} \cdot {x}^{3}\right)\right) + \frac{1}{6} \cdot \left(x + \frac{-1}{6} \cdot {x}^{3}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(x + \frac{-1}{6} \cdot {x}^{3}\right) + {y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \left(x + \frac{-1}{6} \cdot {x}^{3}\right)\right) + \frac{1}{6} \cdot \left(x + \frac{-1}{6} \cdot {x}^{3}\right)\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \left(x + \frac{-1}{6} \cdot {x}^{3}\right) + \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \left(x + \frac{-1}{6} \cdot {x}^{3}\right)\right)\right) + {y}^{2} \cdot \left(\frac{1}{6} \cdot \left(x + \frac{-1}{6} \cdot {x}^{3}\right)\right)\right)} \]

      3. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\left(x + \frac{-1}{6} \cdot {x}^{3}\right) + {y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \left(x + \frac{-1}{6} \cdot {x}^{3}\right)\right)\right)\right) + {y}^{2} \cdot \left(\frac{1}{6} \cdot \left(x + \frac{-1}{6} \cdot {x}^{3}\right)\right)} \]

      4. associate-*r* N/A

         \[\leadsto \left(\left(x + \frac{-1}{6} \cdot {x}^{3}\right) + {y}^{2} \cdot \color{blue}{\left(\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot \left(x + \frac{-1}{6} \cdot {x}^{3}\right)\right)}\right) + {y}^{2} \cdot \left(\frac{1}{6} \cdot \left(x + \frac{-1}{6} \cdot {x}^{3}\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(\left(x + \frac{-1}{6} \cdot {x}^{3}\right) + \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2}\right)\right) \cdot \left(x + \frac{-1}{6} \cdot {x}^{3}\right)}\right) + {y}^{2} \cdot \left(\frac{1}{6} \cdot \left(x + \frac{-1}{6} \cdot {x}^{3}\right)\right) \]

      6. distribute-rgt1-in N/A

         \[\leadsto \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2}\right) + 1\right) \cdot \left(x + \frac{-1}{6} \cdot {x}^{3}\right)} + {y}^{2} \cdot \left(\frac{1}{6} \cdot \left(x + \frac{-1}{6} \cdot {x}^{3}\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \left({y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2}\right) + 1\right) \cdot \left(x + \frac{-1}{6} \cdot {x}^{3}\right) + \color{blue}{\left({y}^{2} \cdot \frac{1}{6}\right) \cdot \left(x + \frac{-1}{6} \cdot {x}^{3}\right)} \]

      8. *-commutative N/A

         \[\leadsto \left({y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2}\right) + 1\right) \cdot \left(x + \frac{-1}{6} \cdot {x}^{3}\right) + \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right)} \cdot \left(x + \frac{-1}{6} \cdot {x}^{3}\right) \]

   8. Simplified 66.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot \left(x \cdot -0.16666666666666666\right), x\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), 1\right)} \]

   9. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma}\left(x, x \cdot \left(x \cdot \frac{-1}{6}\right), x\right) \cdot \color{blue}{\left(1 + \frac{1}{6} \cdot {y}^{2}\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(x, x \cdot \left(x \cdot \frac{-1}{6}\right), x\right) \cdot \color{blue}{\left(\frac{1}{6} \cdot {y}^{2} + 1\right)} \]

       2. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x, x \cdot \left(x \cdot \frac{-1}{6}\right), x\right) \cdot \left(\color{blue}{{y}^{2} \cdot \frac{1}{6}} + 1\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(x, x \cdot \left(x \cdot \frac{-1}{6}\right), x\right) \cdot \left(\color{blue}{\left(y \cdot y\right)} \cdot \frac{1}{6} + 1\right) \]

       4. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(x, x \cdot \left(x \cdot \frac{-1}{6}\right), x\right) \cdot \left(\color{blue}{y \cdot \left(y \cdot \frac{1}{6}\right)} + 1\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x, x \cdot \left(x \cdot \frac{-1}{6}\right), x\right) \cdot \left(y \cdot \color{blue}{\left(\frac{1}{6} \cdot y\right)} + 1\right) \]

       6. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(x, x \cdot \left(x \cdot \frac{-1}{6}\right), x\right) \cdot \color{blue}{\mathsf{fma}\left(y, \frac{1}{6} \cdot y, 1\right)} \]

       7. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x, x \cdot \left(x \cdot \frac{-1}{6}\right), x\right) \cdot \mathsf{fma}\left(y, \color{blue}{y \cdot \frac{1}{6}}, 1\right) \]

       8. *-lowering-*.f64 57.8

          \[\leadsto \mathsf{fma}\left(x, x \cdot \left(x \cdot -0.16666666666666666\right), x\right) \cdot \mathsf{fma}\left(y, \color{blue}{y \cdot 0.16666666666666666}, 1\right) \]

   11. Simplified 57.8%

       \[\leadsto \mathsf{fma}\left(x, x \cdot \left(x \cdot -0.16666666666666666\right), x\right) \cdot \color{blue}{\mathsf{fma}\left(y, y \cdot 0.16666666666666666, 1\right)} \]

   12. Taylor expanded in x around 0

       \[\leadsto \color{blue}{x} \cdot \mathsf{fma}\left(y, y \cdot \frac{1}{6}, 1\right) \]
   13. Step-by-step derivation

   14. Simplified 53.0%

       \[\leadsto \color{blue}{x} \cdot \mathsf{fma}\left(y, y \cdot 0.16666666666666666, 1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 26: 26.3% accurate, 217.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

(FPCore (x y) :precision binary64 x)

C

double code(double x, double y) {
	return x;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x
end function

Java

public static double code(double x, double y) {
	return x;
}

Python

def code(x, y):
	return x

Julia

function code(x, y)
	return x
end

MATLAB

function tmp = code(x, y)
	tmp = x;
end

Wolfram

code[x_, y_] := x

Derivation

1. Initial program 100.0%

   \[\sin x \cdot \frac{\sinh y}{y} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\sin x} \]
4. Step-by-step derivation

   1. sin-lowering-sin.f64 45.3

      \[\leadsto \color{blue}{\sin x} \]

5. Simplified 45.3%

   \[\leadsto \color{blue}{\sin x} \]

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x} \]
7. Step-by-step derivation

8. Simplified 23.1%

   \[\leadsto \color{blue}{x} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024199 
(FPCore (x y)
  :name "Linear.Quaternion:$ccos from linear-1.19.1.3"
  :precision binary64
  (* (sin x) (/ (sinh y) y)))