Linear.Quaternion:$csin from linear-1.19.1.3

Percentage Accurate: 100.0% → 99.2%

Time: 11.7s

Alternatives: 19

Speedup: 1.0×

• 49.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y}{y}\\ t_1 := \cos x \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot -0.5, 1\right) \cdot \frac{\mathsf{fma}\left(\left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot 0.0001984126984126984\right), y \cdot \left(y \cdot y\right), y\right)}{y}\\ \mathbf{elif}\;t\_1 \leq 1.0000005:\\ \;\;\;\;\cos x \cdot \mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sinh y) y)) (t_1 (* (cos x) t_0)))
   (if (<= t_1 (- INFINITY))
     (*
      (fma x (* x -0.5) 1.0)
      (/
       (fma (* (* y y) (* (* y y) 0.0001984126984126984)) (* y (* y y)) y)
       y))
     (if (<= t_1 1.0000005)
       (* (cos x) (fma 0.16666666666666666 (* y y) 1.0))
       t_0))))

C

double code(double x, double y) {
	double t_0 = sinh(y) / y;
	double t_1 = cos(x) * t_0;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma(x, (x * -0.5), 1.0) * (fma(((y * y) * ((y * y) * 0.0001984126984126984)), (y * (y * y)), y) / y);
	} else if (t_1 <= 1.0000005) {
		tmp = cos(x) * fma(0.16666666666666666, (y * y), 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sinh(y) / y)
	t_1 = Float64(cos(x) * t_0)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(fma(x, Float64(x * -0.5), 1.0) * Float64(fma(Float64(Float64(y * y) * Float64(Float64(y * y) * 0.0001984126984126984)), Float64(y * Float64(y * y)), y) / y));
	elseif (t_1 <= 1.0000005)
		tmp = Float64(cos(x) * fma(0.16666666666666666, Float64(y * y), 1.0));
	else
		tmp = 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[Cos[x], $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(x * N[(x * -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984), $MachinePrecision]), $MachinePrecision] * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1.0000005], N[(N[Cos[x], $MachinePrecision] * N[(0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 100.0

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot -0.5}, 1\right) \cdot \frac{\sinh y}{y} \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot -0.5, 1\right)} \cdot \frac{\sinh y}{y} \]

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-rgt-identity N/A

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

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

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

   8. Simplified 100.0%

      \[\leadsto \mathsf{fma}\left(x, x \cdot -0.5, 1\right) \cdot \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)}}{y} \]

   9. Taylor expanded in y around inf

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

       1. metadata-eval N/A

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

       2. pow-sqr N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

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

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

       6. unpow2 N/A

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

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

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

       8. *-commutative N/A

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

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

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

       10. unpow2 N/A

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

       11. *-lowering-*.f64 100.0

           \[\leadsto \mathsf{fma}\left(x, x \cdot -0.5, 1\right) \cdot \frac{\mathsf{fma}\left(\left(y \cdot y\right) \cdot \left(\color{blue}{\left(y \cdot y\right)} \cdot 0.0001984126984126984\right), y \cdot \left(y \cdot y\right), y\right)}{y} \]

   11. Simplified 100.0%

       \[\leadsto \mathsf{fma}\left(x, x \cdot -0.5, 1\right) \cdot \frac{\mathsf{fma}\left(\color{blue}{\left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot 0.0001984126984126984\right)}, y \cdot \left(y \cdot y\right), y\right)}{y} \]

   if -inf.0 < (*.f64 (cos.f64 x) (/.f64 (sinh.f64 y) y)) < 1.0000005000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-lft-identity N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-in N/A

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

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

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

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

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

      6. +-commutative N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 100.0

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

   5. Simplified 100.0%

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

   if 1.0000005000000001 < (*.f64 (cos.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \cdot \frac{\sinh y}{y} \]
   4. Step-by-step derivation

   5. Simplified 100.0%

      \[\leadsto \color{blue}{1} \cdot \frac{\sinh y}{y} \]
   6. Step-by-step derivation

      1. *-lft-identity N/A

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

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

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

      3. sinh-lowering-sinh.f64 100.0

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

   7. Applied egg-rr 100.0%

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

Alternative 2: 99.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y}{y}\\ t_1 := \cos x \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot -0.5, 1\right) \cdot \frac{\mathsf{fma}\left(\left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot 0.0001984126984126984\right), y \cdot \left(y \cdot y\right), y\right)}{y}\\ \mathbf{elif}\;t\_1 \leq 0.9999999999992064:\\ \;\;\;\;\cos x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sinh y) y)) (t_1 (* (cos x) t_0)))
   (if (<= t_1 (- INFINITY))
     (*
      (fma x (* x -0.5) 1.0)
      (/
       (fma (* (* y y) (* (* y y) 0.0001984126984126984)) (* y (* y y)) y)
       y))
     (if (<= t_1 0.9999999999992064) (cos x) t_0))))

C

double code(double x, double y) {
	double t_0 = sinh(y) / y;
	double t_1 = cos(x) * t_0;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma(x, (x * -0.5), 1.0) * (fma(((y * y) * ((y * y) * 0.0001984126984126984)), (y * (y * y)), y) / y);
	} else if (t_1 <= 0.9999999999992064) {
		tmp = cos(x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sinh(y) / y)
	t_1 = Float64(cos(x) * t_0)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(fma(x, Float64(x * -0.5), 1.0) * Float64(fma(Float64(Float64(y * y) * Float64(Float64(y * y) * 0.0001984126984126984)), Float64(y * Float64(y * y)), y) / y));
	elseif (t_1 <= 0.9999999999992064)
		tmp = cos(x);
	else
		tmp = 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[Cos[x], $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(x * N[(x * -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984), $MachinePrecision]), $MachinePrecision] * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.9999999999992064], N[Cos[x], $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 100.0

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot -0.5}, 1\right) \cdot \frac{\sinh y}{y} \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot -0.5, 1\right)} \cdot \frac{\sinh y}{y} \]

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-rgt-identity N/A

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

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

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

   8. Simplified 100.0%

      \[\leadsto \mathsf{fma}\left(x, x \cdot -0.5, 1\right) \cdot \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)}}{y} \]

   9. Taylor expanded in y around inf

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

       1. metadata-eval N/A

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

       2. pow-sqr N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

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

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

       6. unpow2 N/A

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

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

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

       8. *-commutative N/A

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

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

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

       10. unpow2 N/A

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

       11. *-lowering-*.f64 100.0

           \[\leadsto \mathsf{fma}\left(x, x \cdot -0.5, 1\right) \cdot \frac{\mathsf{fma}\left(\left(y \cdot y\right) \cdot \left(\color{blue}{\left(y \cdot y\right)} \cdot 0.0001984126984126984\right), y \cdot \left(y \cdot y\right), y\right)}{y} \]

   11. Simplified 100.0%

       \[\leadsto \mathsf{fma}\left(x, x \cdot -0.5, 1\right) \cdot \frac{\mathsf{fma}\left(\color{blue}{\left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot 0.0001984126984126984\right)}, y \cdot \left(y \cdot y\right), y\right)}{y} \]

   if -inf.0 < (*.f64 (cos.f64 x) (/.f64 (sinh.f64 y) y)) < 0.99999999999920641

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. cos-lowering-cos.f64 100.0

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

   5. Simplified 100.0%

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

   if 0.99999999999920641 < (*.f64 (cos.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \cdot \frac{\sinh y}{y} \]
   4. Step-by-step derivation

   5. Simplified 100.0%

      \[\leadsto \color{blue}{1} \cdot \frac{\sinh y}{y} \]
   6. Step-by-step derivation

      1. *-lft-identity N/A

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

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

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

      3. sinh-lowering-sinh.f64 100.0

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

   7. Applied egg-rr 100.0%

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

Alternative 3: 93.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x \cdot \frac{\sinh y}{y}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot -0.5, 1\right) \cdot \frac{\mathsf{fma}\left(\left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot 0.0001984126984126984\right), y \cdot \left(y \cdot y\right), y\right)}{y}\\ \mathbf{elif}\;t\_0 \leq 0.9999999999992064:\\ \;\;\;\;\cos x\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y\right)}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (cos x) (/ (sinh y) y))))
   (if (<= t_0 (- INFINITY))
     (*
      (fma x (* x -0.5) 1.0)
      (/
       (fma (* (* y y) (* (* y y) 0.0001984126984126984)) (* y (* y y)) y)
       y))
     (if (<= t_0 0.9999999999992064)
       (cos x)
       (/
        (fma
         (* y y)
         (*
          y
          (fma
           (* y y)
           (fma (* y y) 0.0001984126984126984 0.008333333333333333)
           0.16666666666666666))
         y)
        y)))))

C

double code(double x, double y) {
	double t_0 = cos(x) * (sinh(y) / y);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma(x, (x * -0.5), 1.0) * (fma(((y * y) * ((y * y) * 0.0001984126984126984)), (y * (y * y)), y) / y);
	} else if (t_0 <= 0.9999999999992064) {
		tmp = cos(x);
	} else {
		tmp = fma((y * y), (y * fma((y * y), fma((y * y), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666)), y) / y;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) * Float64(sinh(y) / y))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(fma(x, Float64(x * -0.5), 1.0) * Float64(fma(Float64(Float64(y * y) * Float64(Float64(y * y) * 0.0001984126984126984)), Float64(y * Float64(y * y)), y) / y));
	elseif (t_0 <= 0.9999999999992064)
		tmp = cos(x);
	else
		tmp = Float64(fma(Float64(y * y), Float64(y * fma(Float64(y * y), fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666)), y) / y);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(x * N[(x * -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984), $MachinePrecision]), $MachinePrecision] * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.9999999999992064], N[Cos[x], $MachinePrecision], N[(N[(N[(y * y), $MachinePrecision] * N[(y * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] / y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 100.0

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot -0.5}, 1\right) \cdot \frac{\sinh y}{y} \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot -0.5, 1\right)} \cdot \frac{\sinh y}{y} \]

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-rgt-identity N/A

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

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

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

   8. Simplified 100.0%

      \[\leadsto \mathsf{fma}\left(x, x \cdot -0.5, 1\right) \cdot \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)}}{y} \]

   9. Taylor expanded in y around inf

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

       1. metadata-eval N/A

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

       2. pow-sqr N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

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

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

       6. unpow2 N/A

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

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

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

       8. *-commutative N/A

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

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

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

       10. unpow2 N/A

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

       11. *-lowering-*.f64 100.0

           \[\leadsto \mathsf{fma}\left(x, x \cdot -0.5, 1\right) \cdot \frac{\mathsf{fma}\left(\left(y \cdot y\right) \cdot \left(\color{blue}{\left(y \cdot y\right)} \cdot 0.0001984126984126984\right), y \cdot \left(y \cdot y\right), y\right)}{y} \]

   11. Simplified 100.0%

       \[\leadsto \mathsf{fma}\left(x, x \cdot -0.5, 1\right) \cdot \frac{\mathsf{fma}\left(\color{blue}{\left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot 0.0001984126984126984\right)}, y \cdot \left(y \cdot y\right), y\right)}{y} \]

   if -inf.0 < (*.f64 (cos.f64 x) (/.f64 (sinh.f64 y) y)) < 0.99999999999920641

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. cos-lowering-cos.f64 100.0

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

   5. Simplified 100.0%

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

   if 0.99999999999920641 < (*.f64 (cos.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \cdot \frac{\sinh y}{y} \]
   4. Step-by-step derivation

   5. Simplified 100.0%

      \[\leadsto \color{blue}{1} \cdot \frac{\sinh y}{y} \]
   6. Step-by-step derivation

      1. *-lft-identity N/A

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

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

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

      3. sinh-lowering-sinh.f64 100.0

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

   7. Applied egg-rr 100.0%

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

   8. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-*l* N/A

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

      4. *-lft-identity N/A

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

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

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

   10. Simplified 93.1%

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

Alternative 4: 62.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x \cdot \frac{\sinh y}{y}\\ \mathbf{if}\;t\_0 \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(-0.5, x \cdot x, 1\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(y \cdot \left(y \cdot y\right)\right) \cdot 0.008333333333333333\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (cos x) (/ (sinh y) y))))
   (if (<= t_0 -0.05)
     (fma -0.5 (* x x) 1.0)
     (if (<= t_0 2.0)
       (fma 0.16666666666666666 (* y y) 1.0)
       (* y (* (* y (* y y)) 0.008333333333333333))))))

C

double code(double x, double y) {
	double t_0 = cos(x) * (sinh(y) / y);
	double tmp;
	if (t_0 <= -0.05) {
		tmp = fma(-0.5, (x * x), 1.0);
	} else if (t_0 <= 2.0) {
		tmp = fma(0.16666666666666666, (y * y), 1.0);
	} else {
		tmp = y * ((y * (y * y)) * 0.008333333333333333);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) * Float64(sinh(y) / y))
	tmp = 0.0
	if (t_0 <= -0.05)
		tmp = fma(-0.5, Float64(x * x), 1.0);
	elseif (t_0 <= 2.0)
		tmp = fma(0.16666666666666666, Float64(y * y), 1.0);
	else
		tmp = Float64(y * Float64(Float64(y * Float64(y * y)) * 0.008333333333333333));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.05], N[(-0.5 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision], N[(y * N[(N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision] * 0.008333333333333333), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (cos.f64 x) (/.f64 (sinh.f64 y) y)) < -0.050000000000000003

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. cos-lowering-cos.f64 49.3

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

   5. Simplified 49.3%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{-1}{2} \cdot {x}^{2} + 1} \]

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 25.9

         \[\leadsto \mathsf{fma}\left(-0.5, \color{blue}{x \cdot x}, 1\right) \]

   8. Simplified 25.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, x \cdot x, 1\right)} \]

   if -0.050000000000000003 < (*.f64 (cos.f64 x) (/.f64 (sinh.f64 y) y)) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \cdot \frac{\sinh y}{y} \]
   4. Step-by-step derivation

   5. Simplified 68.2%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{6} \cdot {y}^{2} + 1} \]

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 67.9

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

   8. Simplified 67.9%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \cdot \frac{\sinh y}{y} \]
   4. Step-by-step derivation

   5. Simplified 100.0%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

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

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

      3. unpow2 N/A

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

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

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 74.4

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

   8. Simplified 74.4%

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

   9. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{1}{120} \cdot {y}^{4}} \]
   10. Step-by-step derivation

       1. metadata-eval N/A

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

       2. pow-sqr N/A

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

       3. associate-*l* N/A

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

       4. unpow2 N/A

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

       5. associate-*r* N/A

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

       6. *-commutative N/A

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

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

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

       8. *-rgt-identity N/A

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

       9. rgt-mult-inverse N/A

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

       10. associate-*r/ N/A

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

       11. *-rgt-identity N/A

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

       12. associate-/l* N/A

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

       13. associate-*r* N/A

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

       14. unpow2 N/A

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

       15. associate-*l* N/A

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

       16. pow-sqr N/A

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

       17. metadata-eval N/A

           \[\leadsto y \cdot \frac{\frac{1}{120} \cdot {y}^{\color{blue}{4}}}{y} \]

       18. associate-/l* N/A

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

       19. metadata-eval N/A

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

       20. pow-sqr N/A

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

       21. associate-/l* N/A

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

       22. unpow2 N/A

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

   11. Simplified 74.4%

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

4. Final simplification 59.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos x \cdot \frac{\sinh y}{y} \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(-0.5, x \cdot x, 1\right)\\ \mathbf{elif}\;\cos x \cdot \frac{\sinh y}{y} \leq 2:\\ \;\;\;\;\mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(y \cdot \left(y \cdot y\right)\right) \cdot 0.008333333333333333\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 53.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x \cdot \frac{\sinh y}{y}\\ \mathbf{if}\;t\_0 \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(-0.5, x \cdot x, 1\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot y\right) \cdot 0.16666666666666666\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (cos x) (/ (sinh y) y))))
   (if (<= t_0 -0.05)
     (fma -0.5 (* x x) 1.0)
     (if (<= t_0 2.0) 1.0 (* (* y y) 0.16666666666666666)))))

C

double code(double x, double y) {
	double t_0 = cos(x) * (sinh(y) / y);
	double tmp;
	if (t_0 <= -0.05) {
		tmp = fma(-0.5, (x * x), 1.0);
	} else if (t_0 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = (y * y) * 0.16666666666666666;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) * Float64(sinh(y) / y))
	tmp = 0.0
	if (t_0 <= -0.05)
		tmp = fma(-0.5, Float64(x * x), 1.0);
	elseif (t_0 <= 2.0)
		tmp = 1.0;
	else
		tmp = Float64(Float64(y * y) * 0.16666666666666666);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.05], N[(-0.5 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[t$95$0, 2.0], 1.0, N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (cos.f64 x) (/.f64 (sinh.f64 y) y)) < -0.050000000000000003

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. cos-lowering-cos.f64 49.3

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

   5. Simplified 49.3%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{-1}{2} \cdot {x}^{2} + 1} \]

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 25.9

         \[\leadsto \mathsf{fma}\left(-0.5, \color{blue}{x \cdot x}, 1\right) \]

   8. Simplified 25.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, x \cdot x, 1\right)} \]

   if -0.050000000000000003 < (*.f64 (cos.f64 x) (/.f64 (sinh.f64 y) y)) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. cos-lowering-cos.f64 98.8

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

   5. Simplified 98.8%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \]
   7. Step-by-step derivation

   8. Simplified 67.0%

      \[\leadsto \color{blue}{1} \]

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \cdot \frac{\sinh y}{y} \]
   4. Step-by-step derivation

   5. Simplified 100.0%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{6} \cdot {y}^{2} + 1} \]

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 59.8

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

   8. Simplified 59.8%

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

   9. Taylor expanded in y around inf

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

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

          \[\leadsto \color{blue}{\frac{1}{6} \cdot {y}^{2}} \]

       2. unpow2 N/A

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

       3. *-lowering-*.f64 59.8

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

   11. Simplified 59.8%

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

4. Final simplification 54.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos x \cdot \frac{\sinh y}{y} \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(-0.5, x \cdot x, 1\right)\\ \mathbf{elif}\;\cos x \cdot \frac{\sinh y}{y} \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot y\right) \cdot 0.16666666666666666\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos x \cdot \frac{\sinh y}{y} \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot -0.5, 1\right) \cdot \mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cos x) (/ (sinh y) y)) -0.05)
   (* (fma x (* x -0.5) 1.0) (fma 0.16666666666666666 (* y y) 1.0))
   (fma (* y y) (fma y (* y 0.008333333333333333) 0.16666666666666666) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((cos(x) * (sinh(y) / y)) <= -0.05) {
		tmp = fma(x, (x * -0.5), 1.0) * fma(0.16666666666666666, (y * y), 1.0);
	} else {
		tmp = fma((y * y), fma(y, (y * 0.008333333333333333), 0.16666666666666666), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(cos(x) * Float64(sinh(y) / y)) <= -0.05)
		tmp = Float64(fma(x, Float64(x * -0.5), 1.0) * fma(0.16666666666666666, Float64(y * y), 1.0));
	else
		tmp = fma(Float64(y * y), fma(y, Float64(y * 0.008333333333333333), 0.16666666666666666), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Cos[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], -0.05], N[(N[(x * N[(x * -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * N[(0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cos.f64 x) (/.f64 (sinh.f64 y) y)) < -0.050000000000000003

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 54.7

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot -0.5}, 1\right) \cdot \frac{\sinh y}{y} \]

   5. Simplified 54.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot -0.5, 1\right)} \cdot \frac{\sinh y}{y} \]

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 44.3

         \[\leadsto \mathsf{fma}\left(x, x \cdot -0.5, 1\right) \cdot \mathsf{fma}\left(0.16666666666666666, \color{blue}{y \cdot y}, 1\right) \]

   8. Simplified 44.3%

      \[\leadsto \mathsf{fma}\left(x, x \cdot -0.5, 1\right) \cdot \color{blue}{\mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right)} \]

   if -0.050000000000000003 < (*.f64 (cos.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \cdot \frac{\sinh y}{y} \]
   4. Step-by-step derivation

   5. Simplified 83.3%

      \[\leadsto \color{blue}{1} \cdot \frac{\sinh y}{y} \]
   6. Step-by-step derivation

      1. *-lft-identity N/A

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

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

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

      3. sinh-lowering-sinh.f64 83.3

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

   7. Applied egg-rr 83.3%

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

   8. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

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

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

      3. unpow2 N/A

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

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

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

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

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

      10. *-commutative N/A

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

      11. *-lowering-*.f64 71.1

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

   10. Simplified 71.1%

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

Alternative 7: 46.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos x \cdot \frac{\sinh y}{y} \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot y\right) \cdot 0.16666666666666666\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cos x) (/ (sinh y) y)) 2.0) 1.0 (* (* y y) 0.16666666666666666)))

C

double code(double x, double y) {
	double tmp;
	if ((cos(x) * (sinh(y) / y)) <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = (y * y) * 0.16666666666666666;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((cos(x) * (sinh(y) / y)) <= 2.0d0) then
        tmp = 1.0d0
    else
        tmp = (y * y) * 0.16666666666666666d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((Math.cos(x) * (Math.sinh(y) / y)) <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = (y * y) * 0.16666666666666666;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (math.cos(x) * (math.sinh(y) / y)) <= 2.0:
		tmp = 1.0
	else:
		tmp = (y * y) * 0.16666666666666666
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(cos(x) * Float64(sinh(y) / y)) <= 2.0)
		tmp = 1.0;
	else
		tmp = Float64(Float64(y * y) * 0.16666666666666666);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((cos(x) * (sinh(y) / y)) <= 2.0)
		tmp = 1.0;
	else
		tmp = (y * y) * 0.16666666666666666;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Cos[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], 2.0], 1.0, N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. cos-lowering-cos.f64 79.8

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

   5. Simplified 79.8%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \]
   7. Step-by-step derivation

   8. Simplified 41.7%

      \[\leadsto \color{blue}{1} \]

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \cdot \frac{\sinh y}{y} \]
   4. Step-by-step derivation

   5. Simplified 100.0%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{6} \cdot {y}^{2} + 1} \]

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 59.8

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

   8. Simplified 59.8%

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

   9. Taylor expanded in y around inf

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

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

          \[\leadsto \color{blue}{\frac{1}{6} \cdot {y}^{2}} \]

       2. unpow2 N/A

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

       3. *-lowering-*.f64 59.8

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

   11. Simplified 59.8%

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

4. Final simplification 48.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos x \cdot \frac{\sinh y}{y} \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot y\right) \cdot 0.16666666666666666\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

Alternative 9: 71.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos x \leq -0.04:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot -0.5, 1\right) \cdot \frac{\mathsf{fma}\left(\left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot 0.0001984126984126984\right), y \cdot \left(y \cdot y\right), y\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y\right)}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (cos x) -0.04)
   (*
    (fma x (* x -0.5) 1.0)
    (/ (fma (* (* y y) (* (* y y) 0.0001984126984126984)) (* y (* y y)) y) y))
   (/
    (fma
     (* y y)
     (*
      y
      (fma
       (* y y)
       (fma (* y y) 0.0001984126984126984 0.008333333333333333)
       0.16666666666666666))
     y)
    y)))

C

double code(double x, double y) {
	double tmp;
	if (cos(x) <= -0.04) {
		tmp = fma(x, (x * -0.5), 1.0) * (fma(((y * y) * ((y * y) * 0.0001984126984126984)), (y * (y * y)), y) / y);
	} else {
		tmp = fma((y * y), (y * fma((y * y), fma((y * y), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666)), y) / y;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (cos(x) <= -0.04)
		tmp = Float64(fma(x, Float64(x * -0.5), 1.0) * Float64(fma(Float64(Float64(y * y) * Float64(Float64(y * y) * 0.0001984126984126984)), Float64(y * Float64(y * y)), y) / y));
	else
		tmp = Float64(fma(Float64(y * y), Float64(y * fma(Float64(y * y), fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666)), y) / y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[Cos[x], $MachinePrecision], -0.04], N[(N[(x * N[(x * -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984), $MachinePrecision]), $MachinePrecision] * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y * y), $MachinePrecision] * N[(y * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 x) < -0.0400000000000000008

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 54.7

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot -0.5}, 1\right) \cdot \frac{\sinh y}{y} \]

   5. Simplified 54.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot -0.5, 1\right)} \cdot \frac{\sinh y}{y} \]

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-rgt-identity N/A

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

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

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

   8. Simplified 54.7%

      \[\leadsto \mathsf{fma}\left(x, x \cdot -0.5, 1\right) \cdot \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)}}{y} \]

   9. Taylor expanded in y around inf

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

       1. metadata-eval N/A

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

       2. pow-sqr N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

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

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

       6. unpow2 N/A

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

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

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

       8. *-commutative N/A

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

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

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

       10. unpow2 N/A

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

       11. *-lowering-*.f64 54.7

           \[\leadsto \mathsf{fma}\left(x, x \cdot -0.5, 1\right) \cdot \frac{\mathsf{fma}\left(\left(y \cdot y\right) \cdot \left(\color{blue}{\left(y \cdot y\right)} \cdot 0.0001984126984126984\right), y \cdot \left(y \cdot y\right), y\right)}{y} \]

   11. Simplified 54.7%

       \[\leadsto \mathsf{fma}\left(x, x \cdot -0.5, 1\right) \cdot \frac{\mathsf{fma}\left(\color{blue}{\left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot 0.0001984126984126984\right)}, y \cdot \left(y \cdot y\right), y\right)}{y} \]

   if -0.0400000000000000008 < (cos.f64 x)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \cdot \frac{\sinh y}{y} \]
   4. Step-by-step derivation

   5. Simplified 83.3%

      \[\leadsto \color{blue}{1} \cdot \frac{\sinh y}{y} \]
   6. Step-by-step derivation

      1. *-lft-identity N/A

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

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

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

      3. sinh-lowering-sinh.f64 83.3

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

   7. Applied egg-rr 83.3%

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

   8. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-*l* N/A

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

      4. *-lft-identity N/A

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

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

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

   10. Simplified 77.9%

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

Alternative 10: 71.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos x \leq -0.04:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot -0.5, 1\right) \cdot \mathsf{fma}\left(y \cdot y, y \cdot \left(y \cdot \left(\left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y\right)}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (cos x) -0.04)
   (*
    (fma x (* x -0.5) 1.0)
    (fma (* y y) (* y (* y (* (* y y) 0.0001984126984126984))) 1.0))
   (/
    (fma
     (* y y)
     (*
      y
      (fma
       (* y y)
       (fma (* y y) 0.0001984126984126984 0.008333333333333333)
       0.16666666666666666))
     y)
    y)))

C

double code(double x, double y) {
	double tmp;
	if (cos(x) <= -0.04) {
		tmp = fma(x, (x * -0.5), 1.0) * fma((y * y), (y * (y * ((y * y) * 0.0001984126984126984))), 1.0);
	} else {
		tmp = fma((y * y), (y * fma((y * y), fma((y * y), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666)), y) / y;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (cos(x) <= -0.04)
		tmp = Float64(fma(x, Float64(x * -0.5), 1.0) * fma(Float64(y * y), Float64(y * Float64(y * Float64(Float64(y * y) * 0.0001984126984126984))), 1.0));
	else
		tmp = Float64(fma(Float64(y * y), Float64(y * fma(Float64(y * y), fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666)), y) / y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[Cos[x], $MachinePrecision], -0.04], N[(N[(x * N[(x * -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y * y), $MachinePrecision] * N[(y * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 x) < -0.0400000000000000008

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 54.7

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot -0.5}, 1\right) \cdot \frac{\sinh y}{y} \]

   5. Simplified 54.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot -0.5, 1\right)} \cdot \frac{\sinh y}{y} \]

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

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

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

      3. unpow2 N/A

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

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

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

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

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

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

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

      10. +-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

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

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 53.3

          \[\leadsto \mathsf{fma}\left(x, x \cdot -0.5, 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, \color{blue}{y \cdot 0.0001984126984126984}, 0.008333333333333333\right), 0.16666666666666666\right), 1\right) \]

   8. Simplified 53.3%

      \[\leadsto \mathsf{fma}\left(x, x \cdot -0.5, 1\right) \cdot \color{blue}{\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)} \]

   9. Taylor expanded in y around inf

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

       1. metadata-eval N/A

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

       2. pow-sqr N/A

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

       3. associate-*l* N/A

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

       4. unpow2 N/A

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

       5. associate-*r* N/A

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

       6. associate-*r* N/A

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

       7. unpow2 N/A

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

       8. unpow3 N/A

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

       9. *-commutative N/A

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

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

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

       11. unpow3 N/A

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

       12. unpow2 N/A

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

       13. associate-*r* N/A

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

       14. *-commutative N/A

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

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

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

       16. *-commutative N/A

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

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

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

       18. unpow2 N/A

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

       19. *-lowering-*.f64 53.3

           \[\leadsto \mathsf{fma}\left(x, x \cdot -0.5, 1\right) \cdot \mathsf{fma}\left(y \cdot y, y \cdot \left(y \cdot \left(\color{blue}{\left(y \cdot y\right)} \cdot 0.0001984126984126984\right)\right), 1\right) \]

   11. Simplified 53.3%

       \[\leadsto \mathsf{fma}\left(x, x \cdot -0.5, 1\right) \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{y \cdot \left(y \cdot \left(\left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right)}, 1\right) \]

   if -0.0400000000000000008 < (cos.f64 x)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \cdot \frac{\sinh y}{y} \]
   4. Step-by-step derivation

   5. Simplified 83.3%

      \[\leadsto \color{blue}{1} \cdot \frac{\sinh y}{y} \]
   6. Step-by-step derivation

      1. *-lft-identity N/A

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

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

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

      3. sinh-lowering-sinh.f64 83.3

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

   7. Applied egg-rr 83.3%

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

   8. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-*l* N/A

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

      4. *-lft-identity N/A

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

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

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

   10. Simplified 77.9%

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

Alternative 11: 70.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos x \leq -0.04:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot -0.5, 1\right) \cdot \mathsf{fma}\left(y \cdot y, y \cdot \left(y \cdot \left(\left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (cos x) -0.04)
   (*
    (fma x (* x -0.5) 1.0)
    (fma (* y y) (* y (* y (* (* y y) 0.0001984126984126984))) 1.0))
   (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 (cos(x) <= -0.04) {
		tmp = fma(x, (x * -0.5), 1.0) * fma((y * y), (y * (y * ((y * y) * 0.0001984126984126984))), 1.0);
	} else {
		tmp = 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 (cos(x) <= -0.04)
		tmp = Float64(fma(x, Float64(x * -0.5), 1.0) * fma(Float64(y * y), Float64(y * Float64(y * Float64(Float64(y * y) * 0.0001984126984126984))), 1.0));
	else
		tmp = 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[Cos[x], $MachinePrecision], -0.04], N[(N[(x * N[(x * -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 x) < -0.0400000000000000008

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 54.7

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot -0.5}, 1\right) \cdot \frac{\sinh y}{y} \]

   5. Simplified 54.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot -0.5, 1\right)} \cdot \frac{\sinh y}{y} \]

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

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

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

      3. unpow2 N/A

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

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

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

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

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

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

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

      10. +-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

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

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 53.3

          \[\leadsto \mathsf{fma}\left(x, x \cdot -0.5, 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, \color{blue}{y \cdot 0.0001984126984126984}, 0.008333333333333333\right), 0.16666666666666666\right), 1\right) \]

   8. Simplified 53.3%

      \[\leadsto \mathsf{fma}\left(x, x \cdot -0.5, 1\right) \cdot \color{blue}{\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)} \]

   9. Taylor expanded in y around inf

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

       1. metadata-eval N/A

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

       2. pow-sqr N/A

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

       3. associate-*l* N/A

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

       4. unpow2 N/A

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

       5. associate-*r* N/A

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

       6. associate-*r* N/A

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

       7. unpow2 N/A

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

       8. unpow3 N/A

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

       9. *-commutative N/A

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

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

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

       11. unpow3 N/A

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

       12. unpow2 N/A

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

       13. associate-*r* N/A

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

       14. *-commutative N/A

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

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

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

       16. *-commutative N/A

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

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

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

       18. unpow2 N/A

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

       19. *-lowering-*.f64 53.3

           \[\leadsto \mathsf{fma}\left(x, x \cdot -0.5, 1\right) \cdot \mathsf{fma}\left(y \cdot y, y \cdot \left(y \cdot \left(\color{blue}{\left(y \cdot y\right)} \cdot 0.0001984126984126984\right)\right), 1\right) \]

   11. Simplified 53.3%

       \[\leadsto \mathsf{fma}\left(x, x \cdot -0.5, 1\right) \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{y \cdot \left(y \cdot \left(\left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right)}, 1\right) \]

   if -0.0400000000000000008 < (cos.f64 x)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \cdot \frac{\sinh y}{y} \]
   4. Step-by-step derivation

   5. Simplified 83.3%

      \[\leadsto \color{blue}{1} \cdot \frac{\sinh y}{y} \]
   6. Step-by-step derivation

      1. *-lft-identity N/A

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

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

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

      3. sinh-lowering-sinh.f64 83.3

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

   7. Applied egg-rr 83.3%

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

   8. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

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

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

      8. +-commutative N/A

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

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

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

      10. unpow2 N/A

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

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

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

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

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

      15. unpow2 N/A

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

      16. *-lowering-*.f64 76.5

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

   10. Simplified 76.5%

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

Alternative 12: 70.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos x \leq -0.04:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot -0.5, 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (cos x) -0.04)
   (*
    (fma x (* x -0.5) 1.0)
    (fma (* y y) (fma (* y y) 0.008333333333333333 0.16666666666666666) 1.0))
   (fma
    y
    (*
     y
     (fma
      (* y y)
      (fma (* y y) 0.0001984126984126984 0.008333333333333333)
      0.16666666666666666))
    1.0)))

C

double code(double x, double y) {
	double tmp;
	if (cos(x) <= -0.04) {
		tmp = fma(x, (x * -0.5), 1.0) * fma((y * y), fma((y * y), 0.008333333333333333, 0.16666666666666666), 1.0);
	} else {
		tmp = fma(y, (y * fma((y * y), fma((y * y), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666)), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (cos(x) <= -0.04)
		tmp = Float64(fma(x, Float64(x * -0.5), 1.0) * fma(Float64(y * y), fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666), 1.0));
	else
		tmp = 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[Cos[x], $MachinePrecision], -0.04], N[(N[(x * N[(x * -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 x) < -0.0400000000000000008

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 54.7

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot -0.5}, 1\right) \cdot \frac{\sinh y}{y} \]

   5. Simplified 54.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot -0.5, 1\right)} \cdot \frac{\sinh y}{y} \]

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

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

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

      3. unpow2 N/A

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

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

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 53.2

         \[\leadsto \mathsf{fma}\left(x, x \cdot -0.5, 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, 0.008333333333333333, 0.16666666666666666\right), 1\right) \]

   8. Simplified 53.2%

      \[\leadsto \mathsf{fma}\left(x, x \cdot -0.5, 1\right) \cdot \color{blue}{\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), 1\right)} \]

   if -0.0400000000000000008 < (cos.f64 x)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \cdot \frac{\sinh y}{y} \]
   4. Step-by-step derivation

   5. Simplified 83.3%

      \[\leadsto \color{blue}{1} \cdot \frac{\sinh y}{y} \]
   6. Step-by-step derivation

      1. *-lft-identity N/A

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

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

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

      3. sinh-lowering-sinh.f64 83.3

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

   7. Applied egg-rr 83.3%

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

   8. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

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

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

      8. +-commutative N/A

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

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

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

      10. unpow2 N/A

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

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

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

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

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

      15. unpow2 N/A

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

      16. *-lowering-*.f64 76.5

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

   10. Simplified 76.5%

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

Alternative 13: 69.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos x \leq -0.04:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot -0.5, 1\right) \cdot \mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (cos x) -0.04)
   (* (fma x (* x -0.5) 1.0) (fma 0.16666666666666666 (* y y) 1.0))
   (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 (cos(x) <= -0.04) {
		tmp = fma(x, (x * -0.5), 1.0) * fma(0.16666666666666666, (y * y), 1.0);
	} else {
		tmp = fma(y, (y * fma((y * y), fma((y * y), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666)), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (cos(x) <= -0.04)
		tmp = Float64(fma(x, Float64(x * -0.5), 1.0) * fma(0.16666666666666666, Float64(y * y), 1.0));
	else
		tmp = 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[Cos[x], $MachinePrecision], -0.04], N[(N[(x * N[(x * -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * N[(0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 x) < -0.0400000000000000008

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 54.7

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot -0.5}, 1\right) \cdot \frac{\sinh y}{y} \]

   5. Simplified 54.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot -0.5, 1\right)} \cdot \frac{\sinh y}{y} \]

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 44.3

         \[\leadsto \mathsf{fma}\left(x, x \cdot -0.5, 1\right) \cdot \mathsf{fma}\left(0.16666666666666666, \color{blue}{y \cdot y}, 1\right) \]

   8. Simplified 44.3%

      \[\leadsto \mathsf{fma}\left(x, x \cdot -0.5, 1\right) \cdot \color{blue}{\mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right)} \]

   if -0.0400000000000000008 < (cos.f64 x)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \cdot \frac{\sinh y}{y} \]
   4. Step-by-step derivation

   5. Simplified 83.3%

      \[\leadsto \color{blue}{1} \cdot \frac{\sinh y}{y} \]
   6. Step-by-step derivation

      1. *-lft-identity N/A

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

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

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

      3. sinh-lowering-sinh.f64 83.3

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

   7. Applied egg-rr 83.3%

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

   8. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

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

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

      8. +-commutative N/A

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

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

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

      10. unpow2 N/A

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

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

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

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

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

      15. unpow2 N/A

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

      16. *-lowering-*.f64 76.5

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

   10. Simplified 76.5%

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

Alternative 14: 69.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos x \leq -0.04:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot -0.5, 1\right) \cdot \mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \left(\left(y \cdot y\right) \cdot 0.0001984126984126984\right), 0.16666666666666666\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (cos x) -0.04)
   (* (fma x (* x -0.5) 1.0) (fma 0.16666666666666666 (* y y) 1.0))
   (fma
    (* y y)
    (fma y (* y (* (* y y) 0.0001984126984126984)) 0.16666666666666666)
    1.0)))

C

double code(double x, double y) {
	double tmp;
	if (cos(x) <= -0.04) {
		tmp = fma(x, (x * -0.5), 1.0) * fma(0.16666666666666666, (y * y), 1.0);
	} else {
		tmp = fma((y * y), fma(y, (y * ((y * y) * 0.0001984126984126984)), 0.16666666666666666), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (cos(x) <= -0.04)
		tmp = Float64(fma(x, Float64(x * -0.5), 1.0) * fma(0.16666666666666666, Float64(y * y), 1.0));
	else
		tmp = fma(Float64(y * y), fma(y, Float64(y * Float64(Float64(y * y) * 0.0001984126984126984)), 0.16666666666666666), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[Cos[x], $MachinePrecision], -0.04], N[(N[(x * N[(x * -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * N[(0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 x) < -0.0400000000000000008

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 54.7

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot -0.5}, 1\right) \cdot \frac{\sinh y}{y} \]

   5. Simplified 54.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot -0.5, 1\right)} \cdot \frac{\sinh y}{y} \]

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 44.3

         \[\leadsto \mathsf{fma}\left(x, x \cdot -0.5, 1\right) \cdot \mathsf{fma}\left(0.16666666666666666, \color{blue}{y \cdot y}, 1\right) \]

   8. Simplified 44.3%

      \[\leadsto \mathsf{fma}\left(x, x \cdot -0.5, 1\right) \cdot \color{blue}{\mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right)} \]

   if -0.0400000000000000008 < (cos.f64 x)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \cdot \frac{\sinh y}{y} \]
   4. Step-by-step derivation

   5. Simplified 83.3%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

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

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

      3. unpow2 N/A

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

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

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

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

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

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

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

      10. +-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

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

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 76.5

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

   8. Simplified 76.5%

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

   9. Taylor expanded in y around inf

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

       1. unpow3 N/A

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

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

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

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

       6. *-commutative N/A

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

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

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

       8. unpow2 N/A

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

       9. *-lowering-*.f64 76.4

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

   11. Simplified 76.4%

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

Alternative 15: 69.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos x \leq -0.04:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot -0.5, 1\right) \cdot \mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, \left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot 0.0001984126984126984\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (cos x) -0.04)
   (* (fma x (* x -0.5) 1.0) (fma 0.16666666666666666 (* y y) 1.0))
   (fma (* y y) (* (* y y) (* (* y y) 0.0001984126984126984)) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (cos(x) <= -0.04) {
		tmp = fma(x, (x * -0.5), 1.0) * fma(0.16666666666666666, (y * y), 1.0);
	} else {
		tmp = fma((y * y), ((y * y) * ((y * y) * 0.0001984126984126984)), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (cos(x) <= -0.04)
		tmp = Float64(fma(x, Float64(x * -0.5), 1.0) * fma(0.16666666666666666, Float64(y * y), 1.0));
	else
		tmp = fma(Float64(y * y), Float64(Float64(y * y) * Float64(Float64(y * y) * 0.0001984126984126984)), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[Cos[x], $MachinePrecision], -0.04], N[(N[(x * N[(x * -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * N[(0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 x) < -0.0400000000000000008

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 54.7

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot -0.5}, 1\right) \cdot \frac{\sinh y}{y} \]

   5. Simplified 54.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot -0.5, 1\right)} \cdot \frac{\sinh y}{y} \]

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 44.3

         \[\leadsto \mathsf{fma}\left(x, x \cdot -0.5, 1\right) \cdot \mathsf{fma}\left(0.16666666666666666, \color{blue}{y \cdot y}, 1\right) \]

   8. Simplified 44.3%

      \[\leadsto \mathsf{fma}\left(x, x \cdot -0.5, 1\right) \cdot \color{blue}{\mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right)} \]

   if -0.0400000000000000008 < (cos.f64 x)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \cdot \frac{\sinh y}{y} \]
   4. Step-by-step derivation

   5. Simplified 83.3%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

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

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

      3. unpow2 N/A

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

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

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

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

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

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

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

      10. +-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

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

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 76.5

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

   8. Simplified 76.5%

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

   9. Taylor expanded in y around inf

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

       1. metadata-eval N/A

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

       2. pow-sqr N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

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

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

       6. unpow2 N/A

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

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

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

       8. *-commutative N/A

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

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

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

       10. unpow2 N/A

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

       11. *-lowering-*.f64 75.9

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

   11. Simplified 75.9%

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

Alternative 16: 62.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos x \leq -0.04:\\ \;\;\;\;\mathsf{fma}\left(-0.5, x \cdot x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (cos x) -0.04)
   (fma -0.5 (* x x) 1.0)
   (fma (* y y) (fma y (* y 0.008333333333333333) 0.16666666666666666) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (cos(x) <= -0.04) {
		tmp = fma(-0.5, (x * x), 1.0);
	} else {
		tmp = fma((y * y), fma(y, (y * 0.008333333333333333), 0.16666666666666666), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (cos(x) <= -0.04)
		tmp = fma(-0.5, Float64(x * x), 1.0);
	else
		tmp = fma(Float64(y * y), fma(y, Float64(y * 0.008333333333333333), 0.16666666666666666), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[Cos[x], $MachinePrecision], -0.04], N[(-0.5 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 x) < -0.0400000000000000008

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. cos-lowering-cos.f64 49.3

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

   5. Simplified 49.3%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{-1}{2} \cdot {x}^{2} + 1} \]

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 25.9

         \[\leadsto \mathsf{fma}\left(-0.5, \color{blue}{x \cdot x}, 1\right) \]

   8. Simplified 25.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, x \cdot x, 1\right)} \]

   if -0.0400000000000000008 < (cos.f64 x)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \cdot \frac{\sinh y}{y} \]
   4. Step-by-step derivation

   5. Simplified 83.3%

      \[\leadsto \color{blue}{1} \cdot \frac{\sinh y}{y} \]
   6. Step-by-step derivation

      1. *-lft-identity N/A

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

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

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

      3. sinh-lowering-sinh.f64 83.3

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

   7. Applied egg-rr 83.3%

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

   8. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

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

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

      3. unpow2 N/A

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

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

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

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

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

      10. *-commutative N/A

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

      11. *-lowering-*.f64 71.1

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

   10. Simplified 71.1%

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

Alternative 17: 62.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos x \leq -0.04:\\ \;\;\;\;\mathsf{fma}\left(-0.5, x \cdot x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, y \cdot \left(y \cdot 0.008333333333333333\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (cos x) -0.04)
   (fma -0.5 (* x x) 1.0)
   (fma (* y y) (* y (* y 0.008333333333333333)) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (cos(x) <= -0.04) {
		tmp = fma(-0.5, (x * x), 1.0);
	} else {
		tmp = fma((y * y), (y * (y * 0.008333333333333333)), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (cos(x) <= -0.04)
		tmp = fma(-0.5, Float64(x * x), 1.0);
	else
		tmp = fma(Float64(y * y), Float64(y * Float64(y * 0.008333333333333333)), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[Cos[x], $MachinePrecision], -0.04], N[(-0.5 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 x) < -0.0400000000000000008

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. cos-lowering-cos.f64 49.3

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

   5. Simplified 49.3%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{-1}{2} \cdot {x}^{2} + 1} \]

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 25.9

         \[\leadsto \mathsf{fma}\left(-0.5, \color{blue}{x \cdot x}, 1\right) \]

   8. Simplified 25.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, x \cdot x, 1\right)} \]

   if -0.0400000000000000008 < (cos.f64 x)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \cdot \frac{\sinh y}{y} \]
   4. Step-by-step derivation

   5. Simplified 83.3%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

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

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

      3. unpow2 N/A

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

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

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 71.1

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

   8. Simplified 71.1%

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

   9. Taylor expanded in y around inf

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

       1. unpow2 N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

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

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

       5. *-commutative N/A

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

       6. *-lowering-*.f64 70.5

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

   11. Simplified 70.5%

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

Alternative 18: 53.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos x \leq -0.04:\\ \;\;\;\;\mathsf{fma}\left(-0.5, x \cdot x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (cos x) -0.04)
   (fma -0.5 (* x x) 1.0)
   (fma 0.16666666666666666 (* y y) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (cos(x) <= -0.04) {
		tmp = fma(-0.5, (x * x), 1.0);
	} else {
		tmp = fma(0.16666666666666666, (y * y), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (cos(x) <= -0.04)
		tmp = fma(-0.5, Float64(x * x), 1.0);
	else
		tmp = fma(0.16666666666666666, Float64(y * y), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[Cos[x], $MachinePrecision], -0.04], N[(-0.5 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision], N[(0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 x) < -0.0400000000000000008

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. cos-lowering-cos.f64 49.3

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

   5. Simplified 49.3%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{-1}{2} \cdot {x}^{2} + 1} \]

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 25.9

         \[\leadsto \mathsf{fma}\left(-0.5, \color{blue}{x \cdot x}, 1\right) \]

   8. Simplified 25.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, x \cdot x, 1\right)} \]

   if -0.0400000000000000008 < (cos.f64 x)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \cdot \frac{\sinh y}{y} \]
   4. Step-by-step derivation

   5. Simplified 83.3%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{6} \cdot {y}^{2} + 1} \]

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 64.0

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

   8. Simplified 64.0%

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

Alternative 19: 28.1% accurate, 217.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 1.0)

C

double code(double x, double y) {
	return 1.0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0
end function

Java

public static double code(double x, double y) {
	return 1.0;
}

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0;
end

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in y around 0

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

   1. cos-lowering-cos.f64 52.2

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

5. Simplified 52.2%

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

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1} \]
7. Step-by-step derivation

8. Simplified 27.8%

   \[\leadsto \color{blue}{1} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024205 
(FPCore (x y)
  :name "Linear.Quaternion:$csin from linear-1.19.1.3"
  :precision binary64
  (* (cos x) (/ (sinh y) y)))