Linear.Quaternion:$csin from linear-1.19.1.3

Percentage Accurate: 100.0% → 100.0%

Time: 14.4s

Alternatives: 26

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: 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 2: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y}{y}\\ t_1 := \cos x \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(x, x \cdot -0.5, 1\right)\\ \mathbf{elif}\;t\_1 \leq 0.9999999999999495:\\ \;\;\;\;\cos x \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sinh y) y)) (t_1 (* (cos x) t_0)))
   (if (<= t_1 (- INFINITY))
     (* t_0 (fma x (* x -0.5) 1.0))
     (if (<= t_1 0.9999999999999495)
       (*
        (cos x)
        (fma
         y
         (*
          y
          (fma
           y
           (* y (fma (* y y) 0.0001984126984126984 0.008333333333333333))
           0.16666666666666666))
         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 = t_0 * fma(x, (x * -0.5), 1.0);
	} else if (t_1 <= 0.9999999999999495) {
		tmp = cos(x) * fma(y, (y * fma(y, (y * fma((y * y), 0.0001984126984126984, 0.008333333333333333)), 0.16666666666666666)), 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(t_0 * fma(x, Float64(x * -0.5), 1.0));
	elseif (t_1 <= 0.9999999999999495)
		tmp = Float64(cos(x) * fma(y, Float64(y * fma(y, Float64(y * fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333)), 0.16666666666666666)), 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[(t$95$0 * N[(x * N[(x * -0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.9999999999999495], N[(N[Cos[x], $MachinePrecision] * N[(y * N[(y * N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $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. lower-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. lower-*.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%

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

   if -inf.0 < (*.f64 (cos.f64 x) (/.f64 (sinh.f64 y) y)) < 0.999999999999949485

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-rgt-identity N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt-out N/A

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

      9. +-commutative N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

   5. Simplified 99.0%

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

   if 0.999999999999949485 < (*.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. lift-sinh.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-lft-identity 100.0

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

   7. Applied egg-rr 100.0%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos x \cdot \frac{\sinh y}{y} \leq -\infty:\\ \;\;\;\;\frac{\sinh y}{y} \cdot \mathsf{fma}\left(x, x \cdot -0.5, 1\right)\\ \mathbf{elif}\;\cos x \cdot \frac{\sinh y}{y} \leq 0.9999999999999495:\\ \;\;\;\;\cos x \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sinh y}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y}{y}\\ t_1 := \cos x \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(x, x \cdot -0.5, 1\right)\\ \mathbf{elif}\;t\_1 \leq 0.9999999999999495:\\ \;\;\;\;\cos x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sinh y) y)) (t_1 (* (cos x) t_0)))
   (if (<= t_1 (- INFINITY))
     (* t_0 (fma x (* x -0.5) 1.0))
     (if (<= t_1 0.9999999999999495)
       (*
        (cos x)
        (fma
         (* y y)
         (fma y (* y 0.008333333333333333) 0.16666666666666666)
         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 = t_0 * fma(x, (x * -0.5), 1.0);
	} else if (t_1 <= 0.9999999999999495) {
		tmp = cos(x) * fma((y * y), fma(y, (y * 0.008333333333333333), 0.16666666666666666), 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(t_0 * fma(x, Float64(x * -0.5), 1.0));
	elseif (t_1 <= 0.9999999999999495)
		tmp = Float64(cos(x) * fma(Float64(y * y), fma(y, Float64(y * 0.008333333333333333), 0.16666666666666666), 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[(t$95$0 * N[(x * N[(x * -0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.9999999999999495], N[(N[Cos[x], $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $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. lower-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. lower-*.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%

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

   if -inf.0 < (*.f64 (cos.f64 x) (/.f64 (sinh.f64 y) y)) < 0.999999999999949485

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. +-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

   5. Simplified 98.9%

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

   if 0.999999999999949485 < (*.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. lift-sinh.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-lft-identity 100.0

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

   7. Applied egg-rr 100.0%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos x \cdot \frac{\sinh y}{y} \leq -\infty:\\ \;\;\;\;\frac{\sinh y}{y} \cdot \mathsf{fma}\left(x, x \cdot -0.5, 1\right)\\ \mathbf{elif}\;\cos x \cdot \frac{\sinh y}{y} \leq 0.9999999999999495:\\ \;\;\;\;\cos x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sinh y}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y}{y}\\ t_1 := \cos x \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right) \cdot \mathsf{fma}\left(x \cdot x, -0.5, 1\right)\\ \mathbf{elif}\;t\_1 \leq 0.9999999999999495:\\ \;\;\;\;\cos x \cdot \mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sinh y) y)) (t_1 (* (cos x) t_0)))
   (if (<= t_1 (- INFINITY))
     (*
      (fma
       y
       (*
        y
        (fma
         (* y y)
         (fma (* y y) 0.0001984126984126984 0.008333333333333333)
         0.16666666666666666))
       1.0)
      (fma (* x x) -0.5 1.0))
     (if (<= t_1 0.9999999999999495)
       (* (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(y, (y * fma((y * y), fma((y * y), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666)), 1.0) * fma((x * x), -0.5, 1.0);
	} else if (t_1 <= 0.9999999999999495) {
		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(y, Float64(y * fma(Float64(y * y), fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666)), 1.0) * fma(Float64(x * x), -0.5, 1.0));
	elseif (t_1 <= 0.9999999999999495)
		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[(y * N[(y * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.9999999999999495], 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 y around 0

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

      1. *-rgt-identity N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt-out N/A

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

      9. +-commutative N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

   5. Simplified 88.1%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + \left({x}^{2} \cdot \left(\frac{-1}{2} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right) + \frac{1}{24} \cdot \left({x}^{2} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right) + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)} \]

   8. Simplified 2.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.041666666666666664, -0.5\right), 1\right)} \]

   9. Taylor expanded in x around 0

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

   11. Simplified 97.6%

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

   if -inf.0 < (*.f64 (cos.f64 x) (/.f64 (sinh.f64 y) y)) < 0.999999999999949485

   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. lower-*.f64 N/A

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

      5. lower-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. lower-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. lower-*.f64 98.9

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

   5. Simplified 98.9%

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

   if 0.999999999999949485 < (*.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. lift-sinh.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-lft-identity 100.0

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

   7. Applied egg-rr 100.0%

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

Alternative 5: 99.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y}{y}\\ t_1 := \cos x \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right) \cdot \mathsf{fma}\left(x \cdot x, -0.5, 1\right)\\ \mathbf{elif}\;t\_1 \leq 0.9999999999999495:\\ \;\;\;\;\cos x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sinh y) y)) (t_1 (* (cos x) t_0)))
   (if (<= t_1 (- INFINITY))
     (*
      (fma
       y
       (*
        y
        (fma
         (* y y)
         (fma (* y y) 0.0001984126984126984 0.008333333333333333)
         0.16666666666666666))
       1.0)
      (fma (* x x) -0.5 1.0))
     (if (<= t_1 0.9999999999999495) (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(y, (y * fma((y * y), fma((y * y), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666)), 1.0) * fma((x * x), -0.5, 1.0);
	} else if (t_1 <= 0.9999999999999495) {
		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(y, Float64(y * fma(Float64(y * y), fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666)), 1.0) * fma(Float64(x * x), -0.5, 1.0));
	elseif (t_1 <= 0.9999999999999495)
		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[(y * N[(y * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.9999999999999495], 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 y around 0

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

      1. *-rgt-identity N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt-out N/A

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

      9. +-commutative N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

   5. Simplified 88.1%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + \left({x}^{2} \cdot \left(\frac{-1}{2} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right) + \frac{1}{24} \cdot \left({x}^{2} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right) + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)} \]

   8. Simplified 2.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.041666666666666664, -0.5\right), 1\right)} \]

   9. Taylor expanded in x around 0

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

   11. Simplified 97.6%

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

   if -inf.0 < (*.f64 (cos.f64 x) (/.f64 (sinh.f64 y) y)) < 0.999999999999949485

   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. lower-cos.f64 98.7

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

   5. Simplified 98.7%

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

   if 0.999999999999949485 < (*.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. lift-sinh.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-lft-identity 100.0

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

   7. Applied egg-rr 100.0%

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

Alternative 6: 94.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x \cdot \frac{\sinh y}{y}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right) \cdot \mathsf{fma}\left(x \cdot x, -0.5, 1\right)\\ \mathbf{elif}\;t\_0 \leq 0.99995:\\ \;\;\;\;\cos x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right) \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.041666666666666664, -0.5\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (cos x) (/ (sinh y) y))))
   (if (<= t_0 (- INFINITY))
     (*
      (fma
       y
       (*
        y
        (fma
         (* y y)
         (fma (* y y) 0.0001984126984126984 0.008333333333333333)
         0.16666666666666666))
       1.0)
      (fma (* x x) -0.5 1.0))
     (if (<= t_0 0.99995)
       (cos x)
       (*
        (fma
         (* y y)
         (fma
          y
          (* y (fma y (* y 0.0001984126984126984) 0.008333333333333333))
          0.16666666666666666)
         1.0)
        (fma x (* x (fma x (* x 0.041666666666666664) -0.5)) 1.0))))))

C

double code(double x, double y) {
	double t_0 = cos(x) * (sinh(y) / y);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma(y, (y * fma((y * y), fma((y * y), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666)), 1.0) * fma((x * x), -0.5, 1.0);
	} else if (t_0 <= 0.99995) {
		tmp = cos(x);
	} else {
		tmp = fma((y * y), fma(y, (y * fma(y, (y * 0.0001984126984126984), 0.008333333333333333)), 0.16666666666666666), 1.0) * fma(x, (x * fma(x, (x * 0.041666666666666664), -0.5)), 1.0);
	}
	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(y, Float64(y * fma(Float64(y * y), fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666)), 1.0) * fma(Float64(x * x), -0.5, 1.0));
	elseif (t_0 <= 0.99995)
		tmp = cos(x);
	else
		tmp = Float64(fma(Float64(y * y), fma(y, Float64(y * fma(y, Float64(y * 0.0001984126984126984), 0.008333333333333333)), 0.16666666666666666), 1.0) * fma(x, Float64(x * fma(x, Float64(x * 0.041666666666666664), -0.5)), 1.0));
	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[(y * N[(y * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.99995], N[Cos[x], $MachinePrecision], N[(N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * N[(y * N[(y * 0.0001984126984126984), $MachinePrecision] + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x * N[(x * N[(x * N[(x * 0.041666666666666664), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $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 y around 0

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

      1. *-rgt-identity N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt-out N/A

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

      9. +-commutative N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

   5. Simplified 88.1%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + \left({x}^{2} \cdot \left(\frac{-1}{2} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right) + \frac{1}{24} \cdot \left({x}^{2} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right) + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)} \]

   8. Simplified 2.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.041666666666666664, -0.5\right), 1\right)} \]

   9. Taylor expanded in x around 0

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

   11. Simplified 97.6%

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

   if -inf.0 < (*.f64 (cos.f64 x) (/.f64 (sinh.f64 y) y)) < 0.999950000000000006

   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. lower-cos.f64 98.6

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

   5. Simplified 98.6%

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

   if 0.999950000000000006 < (*.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 y around 0

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

      1. *-rgt-identity N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt-out N/A

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

      9. +-commutative N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

   5. Simplified 86.2%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + \left({x}^{2} \cdot \left(\frac{-1}{2} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right) + \frac{1}{24} \cdot \left({x}^{2} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right) + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)} \]

   8. Simplified 92.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.041666666666666664, -0.5\right), 1\right)} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, y \cdot \left(\color{blue}{\left(y \cdot y\right)} \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{5040} + \frac{1}{120}\right) + \frac{1}{6}\right), 1\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{24}, \frac{-1}{2}\right), 1\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, y \cdot \left(\left(y \cdot y\right) \cdot \left(\color{blue}{\left(y \cdot y\right)} \cdot \frac{1}{5040} + \frac{1}{120}\right) + \frac{1}{6}\right), 1\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{24}, \frac{-1}{2}\right), 1\right) \]

      3. lift-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, y \cdot \left(\left(y \cdot y\right) \cdot \color{blue}{\mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right)} + \frac{1}{6}\right), 1\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{24}, \frac{-1}{2}\right), 1\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, y \cdot \color{blue}{\left(\frac{1}{6} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right)\right)}, 1\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{24}, \frac{-1}{2}\right), 1\right) \]

      5. lift-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, y \cdot \left(\frac{1}{6} + \left(y \cdot y\right) \cdot \color{blue}{\left(\left(y \cdot y\right) \cdot \frac{1}{5040} + \frac{1}{120}\right)}\right), 1\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{24}, \frac{-1}{2}\right), 1\right) \]

      6. distribute-lft-in N/A

         \[\leadsto \mathsf{fma}\left(y, y \cdot \left(\frac{1}{6} + \color{blue}{\left(\left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{5040}\right) + \left(y \cdot y\right) \cdot \frac{1}{120}\right)}\right), 1\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{24}, \frac{-1}{2}\right), 1\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, y \cdot \left(\frac{1}{6} + \left(\left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{5040}\right) + \color{blue}{\left(y \cdot y\right)} \cdot \frac{1}{120}\right)\right), 1\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{24}, \frac{-1}{2}\right), 1\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(y, y \cdot \left(\frac{1}{6} + \left(\left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{5040}\right) + \color{blue}{y \cdot \left(y \cdot \frac{1}{120}\right)}\right)\right), 1\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{24}, \frac{-1}{2}\right), 1\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, y \cdot \left(\frac{1}{6} + \left(\left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{5040}\right) + y \cdot \color{blue}{\left(y \cdot \frac{1}{120}\right)}\right)\right), 1\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{24}, \frac{-1}{2}\right), 1\right) \]

      10. associate-+r+ N/A

          \[\leadsto \mathsf{fma}\left(y, y \cdot \color{blue}{\left(\left(\frac{1}{6} + \left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{5040}\right)\right) + y \cdot \left(y \cdot \frac{1}{120}\right)\right)}, 1\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{24}, \frac{-1}{2}\right), 1\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(y, y \cdot \left(\left(\frac{1}{6} + \color{blue}{\left(\left(y \cdot y\right) \cdot \frac{1}{5040}\right) \cdot \left(y \cdot y\right)}\right) + y \cdot \left(y \cdot \frac{1}{120}\right)\right), 1\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{24}, \frac{-1}{2}\right), 1\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, y \cdot \color{blue}{\left(\left(\frac{1}{6} + \left(\left(y \cdot y\right) \cdot \frac{1}{5040}\right) \cdot \left(y \cdot y\right)\right) + y \cdot \left(y \cdot \frac{1}{120}\right)\right)}, 1\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{24}, \frac{-1}{2}\right), 1\right) \]

   10. Applied egg-rr 92.0%

       \[\leadsto \mathsf{fma}\left(y, y \cdot \color{blue}{\left(\left(0.16666666666666666 + y \cdot \left(y \cdot \left(y \cdot \left(y \cdot 0.0001984126984126984\right)\right)\right)\right) + y \cdot \left(y \cdot 0.008333333333333333\right)\right)}, 1\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.041666666666666664, -0.5\right), 1\right) \]

   12. Applied egg-rr 92.0%

       \[\leadsto \color{blue}{\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right) \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.041666666666666664, -0.5\right), 1\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 72.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x \cdot \frac{\sinh y}{y}\\ \mathbf{if}\;t\_0 \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right) \cdot \mathsf{fma}\left(x \cdot x, -0.5, 1\right)\\ \mathbf{elif}\;t\_0 \leq 0.99995:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right) \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.041666666666666664, -0.5\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (cos x) (/ (sinh y) y))))
   (if (<= t_0 -0.02)
     (*
      (fma
       y
       (*
        y
        (fma
         (* y y)
         (fma (* y y) 0.0001984126984126984 0.008333333333333333)
         0.16666666666666666))
       1.0)
      (fma (* x x) -0.5 1.0))
     (if (<= t_0 0.99995)
       1.0
       (*
        (fma
         (* y y)
         (fma
          y
          (* y (fma y (* y 0.0001984126984126984) 0.008333333333333333))
          0.16666666666666666)
         1.0)
        (fma x (* x (fma x (* x 0.041666666666666664) -0.5)) 1.0))))))

C

double code(double x, double y) {
	double t_0 = cos(x) * (sinh(y) / y);
	double tmp;
	if (t_0 <= -0.02) {
		tmp = fma(y, (y * fma((y * y), fma((y * y), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666)), 1.0) * fma((x * x), -0.5, 1.0);
	} else if (t_0 <= 0.99995) {
		tmp = 1.0;
	} else {
		tmp = fma((y * y), fma(y, (y * fma(y, (y * 0.0001984126984126984), 0.008333333333333333)), 0.16666666666666666), 1.0) * fma(x, (x * fma(x, (x * 0.041666666666666664), -0.5)), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) * Float64(sinh(y) / y))
	tmp = 0.0
	if (t_0 <= -0.02)
		tmp = Float64(fma(y, Float64(y * fma(Float64(y * y), fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666)), 1.0) * fma(Float64(x * x), -0.5, 1.0));
	elseif (t_0 <= 0.99995)
		tmp = 1.0;
	else
		tmp = Float64(fma(Float64(y * y), fma(y, Float64(y * fma(y, Float64(y * 0.0001984126984126984), 0.008333333333333333)), 0.16666666666666666), 1.0) * fma(x, Float64(x * fma(x, Float64(x * 0.041666666666666664), -0.5)), 1.0));
	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, -0.02], N[(N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.99995], 1.0, N[(N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * N[(y * N[(y * 0.0001984126984126984), $MachinePrecision] + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x * N[(x * N[(x * N[(x * 0.041666666666666664), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (cos.f64 x) (/.f64 (sinh.f64 y) y)) < -0.0200000000000000004

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-rgt-identity N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt-out N/A

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

      9. +-commutative N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

   5. Simplified 92.6%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + \left({x}^{2} \cdot \left(\frac{-1}{2} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right) + \frac{1}{24} \cdot \left({x}^{2} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right) + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)} \]

   8. Simplified 1.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.041666666666666664, -0.5\right), 1\right)} \]

   9. Taylor expanded in x around 0

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

   11. Simplified 53.9%

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

   if -0.0200000000000000004 < (*.f64 (cos.f64 x) (/.f64 (sinh.f64 y) y)) < 0.999950000000000006

   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. lower-cos.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Simplified 22.3%

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

   if 0.999950000000000006 < (*.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 y around 0

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

      1. *-rgt-identity N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt-out N/A

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

      9. +-commutative N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

   5. Simplified 86.2%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + \left({x}^{2} \cdot \left(\frac{-1}{2} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right) + \frac{1}{24} \cdot \left({x}^{2} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right) + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)} \]

   8. Simplified 92.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.041666666666666664, -0.5\right), 1\right)} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, y \cdot \left(\color{blue}{\left(y \cdot y\right)} \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{5040} + \frac{1}{120}\right) + \frac{1}{6}\right), 1\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{24}, \frac{-1}{2}\right), 1\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, y \cdot \left(\left(y \cdot y\right) \cdot \left(\color{blue}{\left(y \cdot y\right)} \cdot \frac{1}{5040} + \frac{1}{120}\right) + \frac{1}{6}\right), 1\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{24}, \frac{-1}{2}\right), 1\right) \]

      3. lift-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, y \cdot \left(\left(y \cdot y\right) \cdot \color{blue}{\mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right)} + \frac{1}{6}\right), 1\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{24}, \frac{-1}{2}\right), 1\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, y \cdot \color{blue}{\left(\frac{1}{6} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right)\right)}, 1\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{24}, \frac{-1}{2}\right), 1\right) \]

      5. lift-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, y \cdot \left(\frac{1}{6} + \left(y \cdot y\right) \cdot \color{blue}{\left(\left(y \cdot y\right) \cdot \frac{1}{5040} + \frac{1}{120}\right)}\right), 1\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{24}, \frac{-1}{2}\right), 1\right) \]

      6. distribute-lft-in N/A

         \[\leadsto \mathsf{fma}\left(y, y \cdot \left(\frac{1}{6} + \color{blue}{\left(\left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{5040}\right) + \left(y \cdot y\right) \cdot \frac{1}{120}\right)}\right), 1\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{24}, \frac{-1}{2}\right), 1\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, y \cdot \left(\frac{1}{6} + \left(\left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{5040}\right) + \color{blue}{\left(y \cdot y\right)} \cdot \frac{1}{120}\right)\right), 1\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{24}, \frac{-1}{2}\right), 1\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(y, y \cdot \left(\frac{1}{6} + \left(\left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{5040}\right) + \color{blue}{y \cdot \left(y \cdot \frac{1}{120}\right)}\right)\right), 1\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{24}, \frac{-1}{2}\right), 1\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, y \cdot \left(\frac{1}{6} + \left(\left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{5040}\right) + y \cdot \color{blue}{\left(y \cdot \frac{1}{120}\right)}\right)\right), 1\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{24}, \frac{-1}{2}\right), 1\right) \]

      10. associate-+r+ N/A

          \[\leadsto \mathsf{fma}\left(y, y \cdot \color{blue}{\left(\left(\frac{1}{6} + \left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{5040}\right)\right) + y \cdot \left(y \cdot \frac{1}{120}\right)\right)}, 1\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{24}, \frac{-1}{2}\right), 1\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(y, y \cdot \left(\left(\frac{1}{6} + \color{blue}{\left(\left(y \cdot y\right) \cdot \frac{1}{5040}\right) \cdot \left(y \cdot y\right)}\right) + y \cdot \left(y \cdot \frac{1}{120}\right)\right), 1\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{24}, \frac{-1}{2}\right), 1\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, y \cdot \color{blue}{\left(\left(\frac{1}{6} + \left(\left(y \cdot y\right) \cdot \frac{1}{5040}\right) \cdot \left(y \cdot y\right)\right) + y \cdot \left(y \cdot \frac{1}{120}\right)\right)}, 1\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{24}, \frac{-1}{2}\right), 1\right) \]

   10. Applied egg-rr 92.0%

       \[\leadsto \mathsf{fma}\left(y, y \cdot \color{blue}{\left(\left(0.16666666666666666 + y \cdot \left(y \cdot \left(y \cdot \left(y \cdot 0.0001984126984126984\right)\right)\right)\right) + y \cdot \left(y \cdot 0.008333333333333333\right)\right)}, 1\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.041666666666666664, -0.5\right), 1\right) \]

   12. Applied egg-rr 92.0%

       \[\leadsto \color{blue}{\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right) \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.041666666666666664, -0.5\right), 1\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 72.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x \cdot \frac{\sinh y}{y}\\ t_1 := \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)\\ \mathbf{if}\;t\_0 \leq -0.02:\\ \;\;\;\;t\_1 \cdot \mathsf{fma}\left(x \cdot x, -0.5, 1\right)\\ \mathbf{elif}\;t\_0 \leq 0.99995:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.041666666666666664, -0.5\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (cos x) (/ (sinh y) y)))
        (t_1
         (fma
          y
          (*
           y
           (fma
            (* y y)
            (fma (* y y) 0.0001984126984126984 0.008333333333333333)
            0.16666666666666666))
          1.0)))
   (if (<= t_0 -0.02)
     (* t_1 (fma (* x x) -0.5 1.0))
     (if (<= t_0 0.99995)
       1.0
       (* t_1 (fma (* x x) (fma (* x x) 0.041666666666666664 -0.5) 1.0))))))

C

double code(double x, double y) {
	double t_0 = cos(x) * (sinh(y) / y);
	double t_1 = fma(y, (y * fma((y * y), fma((y * y), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666)), 1.0);
	double tmp;
	if (t_0 <= -0.02) {
		tmp = t_1 * fma((x * x), -0.5, 1.0);
	} else if (t_0 <= 0.99995) {
		tmp = 1.0;
	} else {
		tmp = t_1 * fma((x * x), fma((x * x), 0.041666666666666664, -0.5), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) * Float64(sinh(y) / y))
	t_1 = fma(y, Float64(y * fma(Float64(y * y), fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666)), 1.0)
	tmp = 0.0
	if (t_0 <= -0.02)
		tmp = Float64(t_1 * fma(Float64(x * x), -0.5, 1.0));
	elseif (t_0 <= 0.99995)
		tmp = 1.0;
	else
		tmp = Float64(t_1 * fma(Float64(x * x), fma(Float64(x * x), 0.041666666666666664, -0.5), 1.0));
	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]}, Block[{t$95$1 = 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]}, If[LessEqual[t$95$0, -0.02], N[(t$95$1 * N[(N[(x * x), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.99995], 1.0, N[(t$95$1 * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.041666666666666664 + -0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (cos.f64 x) (/.f64 (sinh.f64 y) y)) < -0.0200000000000000004

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-rgt-identity N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt-out N/A

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

      9. +-commutative N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

   5. Simplified 92.6%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + \left({x}^{2} \cdot \left(\frac{-1}{2} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right) + \frac{1}{24} \cdot \left({x}^{2} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right) + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)} \]

   8. Simplified 1.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.041666666666666664, -0.5\right), 1\right)} \]

   9. Taylor expanded in x around 0

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

   11. Simplified 53.9%

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

   if -0.0200000000000000004 < (*.f64 (cos.f64 x) (/.f64 (sinh.f64 y) y)) < 0.999950000000000006

   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. lower-cos.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Simplified 22.3%

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

   if 0.999950000000000006 < (*.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 y around 0

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

      1. *-rgt-identity N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt-out N/A

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

      9. +-commutative N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

   5. Simplified 86.2%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + \left({x}^{2} \cdot \left(\frac{-1}{2} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right) + \frac{1}{24} \cdot \left({x}^{2} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right) + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)} \]

   8. Simplified 92.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.041666666666666664, -0.5\right), 1\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 72.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right)\\ t_1 := \cos x \cdot \frac{\sinh y}{y}\\ t_2 := \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, t\_0, 0.16666666666666666\right), 1\right)\\ \mathbf{if}\;t\_1 \leq -0.02:\\ \;\;\;\;t\_2 \cdot \mathsf{fma}\left(x \cdot x, -0.5, 1\right)\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot t\_0, 0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2 \cdot \mathsf{fma}\left(x \cdot x, x \cdot \left(x \cdot 0.041666666666666664\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma (* y y) 0.0001984126984126984 0.008333333333333333))
        (t_1 (* (cos x) (/ (sinh y) y)))
        (t_2 (fma y (* y (fma (* y y) t_0 0.16666666666666666)) 1.0)))
   (if (<= t_1 -0.02)
     (* t_2 (fma (* x x) -0.5 1.0))
     (if (<= t_1 2.0)
       (fma (* y y) (fma y (* y t_0) 0.16666666666666666) 1.0)
       (* t_2 (fma (* x x) (* x (* x 0.041666666666666664)) 1.0))))))

C

double code(double x, double y) {
	double t_0 = fma((y * y), 0.0001984126984126984, 0.008333333333333333);
	double t_1 = cos(x) * (sinh(y) / y);
	double t_2 = fma(y, (y * fma((y * y), t_0, 0.16666666666666666)), 1.0);
	double tmp;
	if (t_1 <= -0.02) {
		tmp = t_2 * fma((x * x), -0.5, 1.0);
	} else if (t_1 <= 2.0) {
		tmp = fma((y * y), fma(y, (y * t_0), 0.16666666666666666), 1.0);
	} else {
		tmp = t_2 * fma((x * x), (x * (x * 0.041666666666666664)), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333)
	t_1 = Float64(cos(x) * Float64(sinh(y) / y))
	t_2 = fma(y, Float64(y * fma(Float64(y * y), t_0, 0.16666666666666666)), 1.0)
	tmp = 0.0
	if (t_1 <= -0.02)
		tmp = Float64(t_2 * fma(Float64(x * x), -0.5, 1.0));
	elseif (t_1 <= 2.0)
		tmp = fma(Float64(y * y), fma(y, Float64(y * t_0), 0.16666666666666666), 1.0);
	else
		tmp = Float64(t_2 * fma(Float64(x * x), Float64(x * Float64(x * 0.041666666666666664)), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * t$95$0 + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[t$95$1, -0.02], N[(t$95$2 * N[(N[(x * x), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * t$95$0), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision], N[(t$95$2 * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (cos.f64 x) (/.f64 (sinh.f64 y) y)) < -0.0200000000000000004

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-rgt-identity N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt-out N/A

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

      9. +-commutative N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

   5. Simplified 92.6%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + \left({x}^{2} \cdot \left(\frac{-1}{2} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right) + \frac{1}{24} \cdot \left({x}^{2} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right) + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)} \]

   8. Simplified 1.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.041666666666666664, -0.5\right), 1\right)} \]

   9. Taylor expanded in x around 0

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

   11. Simplified 53.9%

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

   if -0.0200000000000000004 < (*.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

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

      1. *-rgt-identity N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt-out N/A

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

      9. +-commutative N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

   5. Simplified 99.4%

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

      1. lift-cos.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. flip-+ N/A

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

      8. clear-num N/A

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

   7. Applied egg-rr 99.4%

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

   8. Taylor expanded in x 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. lower-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. lower-*.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. lower-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. lower-*.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. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 66.8

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

   10. Simplified 66.8%

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

   if 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 y around 0

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

      1. *-rgt-identity N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt-out N/A

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

      9. +-commutative N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

   5. Simplified 79.3%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + \left({x}^{2} \cdot \left(\frac{-1}{2} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right) + \frac{1}{24} \cdot \left({x}^{2} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right) + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)} \]

   8. Simplified 88.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.041666666666666664, -0.5\right), 1\right)} \]

   9. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{24} \cdot {x}^{2}}, 1\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{1}{24}}, 1\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{24}, 1\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(x \cdot \frac{1}{24}\right)}, 1\right) \]

       4. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(x \cdot \frac{1}{24}\right)}, 1\right) \]

       5. lower-*.f64 88.3

          \[\leadsto \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right) \cdot \mathsf{fma}\left(x \cdot x, x \cdot \color{blue}{\left(x \cdot 0.041666666666666664\right)}, 1\right) \]

   11. Simplified 88.3%

       \[\leadsto \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right) \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(x \cdot 0.041666666666666664\right)}, 1\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 66.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x \cdot \frac{\sinh y}{y}\\ \mathbf{if}\;t\_0 \leq -0.02:\\ \;\;\;\;x \cdot \left(x \cdot \mathsf{fma}\left(y \cdot y, -0.08333333333333333, -0.5\right)\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(y, y \cdot 0.16666666666666666, 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot \left(\left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (cos x) (/ (sinh y) y))))
   (if (<= t_0 -0.02)
     (* x (* x (fma (* y y) -0.08333333333333333 -0.5)))
     (if (<= t_0 2.0)
       (fma y (* y 0.16666666666666666) 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 <= -0.02) {
		tmp = x * (x * fma((y * y), -0.08333333333333333, -0.5));
	} else if (t_0 <= 2.0) {
		tmp = fma(y, (y * 0.16666666666666666), 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 <= -0.02)
		tmp = Float64(x * Float64(x * fma(Float64(y * y), -0.08333333333333333, -0.5)));
	elseif (t_0 <= 2.0)
		tmp = fma(y, Float64(y * 0.16666666666666666), 1.0);
	else
		tmp = Float64(y * Float64(y * Float64(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, -0.02], N[(x * N[(x * N[(N[(y * y), $MachinePrecision] * -0.08333333333333333 + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(y * N[(y * 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision], N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (cos.f64 x) (/.f64 (sinh.f64 y) y)) < -0.0200000000000000004

   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. lower-*.f64 N/A

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

      5. lower-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. lower-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. lower-*.f64 77.2

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

   5. Simplified 77.2%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt1-in N/A

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

      5. +-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 51.2

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

   8. Simplified 51.2%

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

   9. Taylor expanded in x around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. unpow2 N/A

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

       5. associate-*l* N/A

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

       6. lower-*.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. +-commutative N/A

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

       9. distribute-rgt-in N/A

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

       10. *-commutative N/A

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

       11. associate-*l* N/A

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

       12. metadata-eval N/A

           \[\leadsto x \cdot \left(x \cdot \left({y}^{2} \cdot \color{blue}{\frac{-1}{12}} + 1 \cdot \frac{-1}{2}\right)\right) \]

       13. metadata-eval N/A

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

       14. metadata-eval N/A

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

       15. lower-fma.f64 N/A

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

       16. unpow2 N/A

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

       17. lower-*.f64 N/A

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

       18. metadata-eval 51.2

           \[\leadsto x \cdot \left(x \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{-0.08333333333333333}, -0.5\right)\right) \]

   11. Simplified 51.2%

       \[\leadsto \color{blue}{x \cdot \left(x \cdot \mathsf{fma}\left(y \cdot y, -0.08333333333333333, -0.5\right)\right)} \]

   if -0.0200000000000000004 < (*.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

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

      1. *-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. +-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

   5. Simplified 99.3%

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

   6. Taylor expanded in x around 0

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

   8. Simplified 66.7%

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

   9. Taylor expanded in y around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. lower-fma.f64 N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 66.6

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

   11. Simplified 66.6%

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

   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 y around 0

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

      1. *-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. +-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

   5. Simplified 74.1%

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

   6. Taylor expanded in x around 0

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

   8. Simplified 74.1%

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

   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. *-commutative N/A

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

       5. unpow2 N/A

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

       6. associate-*l* N/A

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

       7. lower-*.f64 N/A

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

       8. lower-*.f64 N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 N/A

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

       11. unpow2 N/A

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

       12. lower-*.f64 74.1

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

   11. Simplified 74.1%

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

Alternative 11: 62.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x \cdot \frac{\sinh y}{y}\\ \mathbf{if}\;t\_0 \leq -0.02:\\ \;\;\;\;-0.5 \cdot \left(x \cdot x\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(y, y \cdot 0.16666666666666666, 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot \left(\left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (cos x) (/ (sinh y) y))))
   (if (<= t_0 -0.02)
     (* -0.5 (* x x))
     (if (<= t_0 2.0)
       (fma y (* y 0.16666666666666666) 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 <= -0.02) {
		tmp = -0.5 * (x * x);
	} else if (t_0 <= 2.0) {
		tmp = fma(y, (y * 0.16666666666666666), 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 <= -0.02)
		tmp = Float64(-0.5 * Float64(x * x));
	elseif (t_0 <= 2.0)
		tmp = fma(y, Float64(y * 0.16666666666666666), 1.0);
	else
		tmp = Float64(y * Float64(y * Float64(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, -0.02], N[(-0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(y * N[(y * 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision], N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (cos.f64 x) (/.f64 (sinh.f64 y) y)) < -0.0200000000000000004

   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. lower-cos.f64 46.9

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

   5. Simplified 46.9%

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

   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. lower-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. lower-*.f64 28.0

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

   8. Simplified 28.0%

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

   9. Taylor expanded in x around inf

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

       1. lower-*.f64 N/A

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

       2. unpow2 N/A

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

       3. lower-*.f64 28.0

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

   11. Simplified 28.0%

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

   if -0.0200000000000000004 < (*.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

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

      1. *-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. +-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

   5. Simplified 99.3%

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

   6. Taylor expanded in x around 0

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

   8. Simplified 66.7%

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

   9. Taylor expanded in y around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. lower-fma.f64 N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 66.6

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

   11. Simplified 66.6%

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

   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 y around 0

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

      1. *-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. +-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

   5. Simplified 74.1%

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

   6. Taylor expanded in x around 0

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

   8. Simplified 74.1%

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

   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. *-commutative N/A

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

       5. unpow2 N/A

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

       6. associate-*l* N/A

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

       7. lower-*.f64 N/A

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

       8. lower-*.f64 N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 N/A

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

       11. unpow2 N/A

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

       12. lower-*.f64 74.1

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

   11. Simplified 74.1%

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

Alternative 12: 96.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y}{y}\\ \mathbf{if}\;\cos x \cdot t\_0 \leq 0.9999999999999495:\\ \;\;\;\;\cos x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sinh y) y)))
   (if (<= (* (cos x) t_0) 0.9999999999999495)
     (*
      (cos x)
      (fma (* y y) (fma y (* y 0.008333333333333333) 0.16666666666666666) 1.0))
     t_0)))

C

double code(double x, double y) {
	double t_0 = sinh(y) / y;
	double tmp;
	if ((cos(x) * t_0) <= 0.9999999999999495) {
		tmp = cos(x) * fma((y * y), fma(y, (y * 0.008333333333333333), 0.16666666666666666), 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sinh(y) / y)
	tmp = 0.0
	if (Float64(cos(x) * t_0) <= 0.9999999999999495)
		tmp = Float64(cos(x) * fma(Float64(y * y), fma(y, Float64(y * 0.008333333333333333), 0.16666666666666666), 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]}, If[LessEqual[N[(N[Cos[x], $MachinePrecision] * t$95$0), $MachinePrecision], 0.9999999999999495], N[(N[Cos[x], $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cos.f64 x) (/.f64 (sinh.f64 y) y)) < 0.999999999999949485

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. +-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

   5. Simplified 95.0%

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

   if 0.999999999999949485 < (*.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. lift-sinh.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-lft-identity 100.0

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

   7. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{\sinh y}{y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 70.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right)\\ \mathbf{if}\;\cos x \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, t\_0, 0.16666666666666666\right), 1\right) \cdot \mathsf{fma}\left(x \cdot x, -0.5, 1\right)\\ \mathbf{elif}\;\cos x \leq 0.78:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.041666666666666664, -0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot t\_0, 0.16666666666666666\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma (* y y) 0.0001984126984126984 0.008333333333333333)))
   (if (<= (cos x) -0.02)
     (*
      (fma y (* y (fma (* y y) t_0 0.16666666666666666)) 1.0)
      (fma (* x x) -0.5 1.0))
     (if (<= (cos x) 0.78)
       (*
        (fma
         (* y y)
         (fma y (* y 0.008333333333333333) 0.16666666666666666)
         1.0)
        (fma (* x x) (fma (* x x) 0.041666666666666664 -0.5) 1.0))
       (fma (* y y) (fma y (* y t_0) 0.16666666666666666) 1.0)))))

C

double code(double x, double y) {
	double t_0 = fma((y * y), 0.0001984126984126984, 0.008333333333333333);
	double tmp;
	if (cos(x) <= -0.02) {
		tmp = fma(y, (y * fma((y * y), t_0, 0.16666666666666666)), 1.0) * fma((x * x), -0.5, 1.0);
	} else if (cos(x) <= 0.78) {
		tmp = fma((y * y), fma(y, (y * 0.008333333333333333), 0.16666666666666666), 1.0) * fma((x * x), fma((x * x), 0.041666666666666664, -0.5), 1.0);
	} else {
		tmp = fma((y * y), fma(y, (y * t_0), 0.16666666666666666), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333)
	tmp = 0.0
	if (cos(x) <= -0.02)
		tmp = Float64(fma(y, Float64(y * fma(Float64(y * y), t_0, 0.16666666666666666)), 1.0) * fma(Float64(x * x), -0.5, 1.0));
	elseif (cos(x) <= 0.78)
		tmp = Float64(fma(Float64(y * y), fma(y, Float64(y * 0.008333333333333333), 0.16666666666666666), 1.0) * fma(Float64(x * x), fma(Float64(x * x), 0.041666666666666664, -0.5), 1.0));
	else
		tmp = fma(Float64(y * y), fma(y, Float64(y * t_0), 0.16666666666666666), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision]}, If[LessEqual[N[Cos[x], $MachinePrecision], -0.02], N[(N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * t$95$0 + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Cos[x], $MachinePrecision], 0.78], N[(N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.041666666666666664 + -0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * t$95$0), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (cos.f64 x) < -0.0200000000000000004

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-rgt-identity N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt-out N/A

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

      9. +-commutative N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

   5. Simplified 92.6%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + \left({x}^{2} \cdot \left(\frac{-1}{2} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right) + \frac{1}{24} \cdot \left({x}^{2} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right) + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)} \]

   8. Simplified 1.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.041666666666666664, -0.5\right), 1\right)} \]

   9. Taylor expanded in x around 0

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

   11. Simplified 53.9%

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

   if -0.0200000000000000004 < (cos.f64 x) < 0.78000000000000003

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. +-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

   5. Simplified 76.7%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right)} \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right) + 1\right)} \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{24} \cdot {x}^{2} - \frac{1}{2}, 1\right)} \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24} \cdot {x}^{2} - \frac{1}{2}, 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24} \cdot {x}^{2} - \frac{1}{2}, 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{24} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{1}{24}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, {x}^{2} \cdot \frac{1}{24} + \color{blue}{\frac{-1}{2}}, 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{24}, \frac{-1}{2}\right)}, 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24}, \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      10. lower-*.f64 58.2

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.041666666666666664, -0.5\right), 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right) \]

   8. Simplified 58.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.041666666666666664, -0.5\right), 1\right)} \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right) \]

   if 0.78000000000000003 < (cos.f64 x)

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-rgt-identity N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt-out N/A

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

      9. +-commutative N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

   5. Simplified 92.3%

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

      1. lift-cos.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. flip-+ N/A

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

      8. clear-num N/A

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

   7. Applied egg-rr 92.3%

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

   8. Taylor expanded in x 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. lower-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. lower-*.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. lower-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. lower-*.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. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 81.4

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

   10. Simplified 81.4%

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

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos x \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right) \cdot \mathsf{fma}\left(x \cdot x, -0.5, 1\right)\\ \mathbf{elif}\;\cos x \leq 0.78:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.041666666666666664, -0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 70.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right)\\ \mathbf{if}\;\cos x \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, t\_0, 0.16666666666666666\right), 1\right) \cdot \mathsf{fma}\left(x \cdot x, -0.5, 1\right)\\ \mathbf{elif}\;\cos x \leq 0.78:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), \left(y \cdot y\right) \cdot 0.041666666666666664, 0.041666666666666664\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot t\_0, 0.16666666666666666\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma (* y y) 0.0001984126984126984 0.008333333333333333)))
   (if (<= (cos x) -0.02)
     (*
      (fma y (* y (fma (* y y) t_0 0.16666666666666666)) 1.0)
      (fma (* x x) -0.5 1.0))
     (if (<= (cos x) 0.78)
       (*
        (* (* x x) (* x x))
        (fma
         (fma (* y y) 0.008333333333333333 0.16666666666666666)
         (* (* y y) 0.041666666666666664)
         0.041666666666666664))
       (fma (* y y) (fma y (* y t_0) 0.16666666666666666) 1.0)))))

C

double code(double x, double y) {
	double t_0 = fma((y * y), 0.0001984126984126984, 0.008333333333333333);
	double tmp;
	if (cos(x) <= -0.02) {
		tmp = fma(y, (y * fma((y * y), t_0, 0.16666666666666666)), 1.0) * fma((x * x), -0.5, 1.0);
	} else if (cos(x) <= 0.78) {
		tmp = ((x * x) * (x * x)) * fma(fma((y * y), 0.008333333333333333, 0.16666666666666666), ((y * y) * 0.041666666666666664), 0.041666666666666664);
	} else {
		tmp = fma((y * y), fma(y, (y * t_0), 0.16666666666666666), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333)
	tmp = 0.0
	if (cos(x) <= -0.02)
		tmp = Float64(fma(y, Float64(y * fma(Float64(y * y), t_0, 0.16666666666666666)), 1.0) * fma(Float64(x * x), -0.5, 1.0));
	elseif (cos(x) <= 0.78)
		tmp = Float64(Float64(Float64(x * x) * Float64(x * x)) * fma(fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666), Float64(Float64(y * y) * 0.041666666666666664), 0.041666666666666664));
	else
		tmp = fma(Float64(y * y), fma(y, Float64(y * t_0), 0.16666666666666666), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision]}, If[LessEqual[N[Cos[x], $MachinePrecision], -0.02], N[(N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * t$95$0 + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Cos[x], $MachinePrecision], 0.78], N[(N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.041666666666666664), $MachinePrecision] + 0.041666666666666664), $MachinePrecision]), $MachinePrecision], N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * t$95$0), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (cos.f64 x) < -0.0200000000000000004

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-rgt-identity N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt-out N/A

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

      9. +-commutative N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

   5. Simplified 92.6%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + \left({x}^{2} \cdot \left(\frac{-1}{2} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right) + \frac{1}{24} \cdot \left({x}^{2} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right) + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)} \]

   8. Simplified 1.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.041666666666666664, -0.5\right), 1\right)} \]

   9. Taylor expanded in x around 0

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

   11. Simplified 53.9%

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

   if -0.0200000000000000004 < (cos.f64 x) < 0.78000000000000003

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. +-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

   5. Simplified 76.7%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right)} \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right) + 1\right)} \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{24} \cdot {x}^{2} - \frac{1}{2}, 1\right)} \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24} \cdot {x}^{2} - \frac{1}{2}, 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24} \cdot {x}^{2} - \frac{1}{2}, 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{24} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{1}{24}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, {x}^{2} \cdot \frac{1}{24} + \color{blue}{\frac{-1}{2}}, 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{24}, \frac{-1}{2}\right)}, 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24}, \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      10. lower-*.f64 58.2

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.041666666666666664, -0.5\right), 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right) \]

   8. Simplified 58.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.041666666666666664, -0.5\right), 1\right)} \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right) \]

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{24} \cdot \left({x}^{4} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{1}{24} \cdot {x}^{4}\right) \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} \]

       2. *-commutative N/A

          \[\leadsto \color{blue}{\left({x}^{4} \cdot \frac{1}{24}\right)} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) \]

       3. associate-*r* N/A

          \[\leadsto \color{blue}{{x}^{4} \cdot \left(\frac{1}{24} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto \color{blue}{{x}^{4} \cdot \left(\frac{1}{24} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)} \]

       5. metadata-eval N/A

          \[\leadsto {x}^{\color{blue}{\left(2 \cdot 2\right)}} \cdot \left(\frac{1}{24} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) \]

       6. pow-sqr N/A

          \[\leadsto \color{blue}{\left({x}^{2} \cdot {x}^{2}\right)} \cdot \left(\frac{1}{24} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) \]

       7. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left({x}^{2} \cdot {x}^{2}\right)} \cdot \left(\frac{1}{24} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \left(\color{blue}{\left(x \cdot x\right)} \cdot {x}^{2}\right) \cdot \left(\frac{1}{24} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) \]

       9. lower-*.f64 N/A

          \[\leadsto \left(\color{blue}{\left(x \cdot x\right)} \cdot {x}^{2}\right) \cdot \left(\frac{1}{24} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \left(\left(x \cdot x\right) \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(\frac{1}{24} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) \]

       11. lower-*.f64 N/A

           \[\leadsto \left(\left(x \cdot x\right) \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(\frac{1}{24} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) \]

       12. distribute-rgt-in N/A

           \[\leadsto \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(1 \cdot \frac{1}{24} + \left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) \cdot \frac{1}{24}\right)} \]

       13. metadata-eval N/A

           \[\leadsto \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\color{blue}{\frac{1}{24}} + \left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) \cdot \frac{1}{24}\right) \]

       14. +-commutative N/A

           \[\leadsto \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(\left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) \cdot \frac{1}{24} + \frac{1}{24}\right)} \]

       15. *-commutative N/A

           \[\leadsto \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\color{blue}{\left(\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) \cdot {y}^{2}\right)} \cdot \frac{1}{24} + \frac{1}{24}\right) \]

       16. associate-*l* N/A

           \[\leadsto \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\color{blue}{\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) \cdot \left({y}^{2} \cdot \frac{1}{24}\right)} + \frac{1}{24}\right) \]

       17. lower-fma.f64 N/A

           \[\leadsto \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}, {y}^{2} \cdot \frac{1}{24}, \frac{1}{24}\right)} \]

   11. Simplified 58.2%

       \[\leadsto \color{blue}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), \left(y \cdot y\right) \cdot 0.041666666666666664, 0.041666666666666664\right)} \]

   if 0.78000000000000003 < (cos.f64 x)

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-rgt-identity N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt-out N/A

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

      9. +-commutative N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

   5. Simplified 92.3%

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

      1. lift-cos.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. flip-+ N/A

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

      8. clear-num N/A

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

   7. Applied egg-rr 92.3%

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

   8. Taylor expanded in x 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. lower-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. lower-*.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. lower-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. lower-*.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. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 81.4

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

   10. Simplified 81.4%

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

Alternative 15: 70.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos x \leq 0.01:\\ \;\;\;\;\mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right) \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right)\\ \mathbf{elif}\;\cos x \leq 0.78:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), \left(y \cdot y\right) \cdot 0.041666666666666664, 0.041666666666666664\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (cos x) 0.01)
   (*
    (fma 0.16666666666666666 (* y y) 1.0)
    (fma
     x
     (*
      x
      (fma
       (* x x)
       (fma x (* x -0.001388888888888889) 0.041666666666666664)
       -0.5))
     1.0))
   (if (<= (cos x) 0.78)
     (*
      (* (* x x) (* x x))
      (fma
       (fma (* y y) 0.008333333333333333 0.16666666666666666)
       (* (* y y) 0.041666666666666664)
       0.041666666666666664))
     (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) <= 0.01) {
		tmp = fma(0.16666666666666666, (y * y), 1.0) * fma(x, (x * fma((x * x), fma(x, (x * -0.001388888888888889), 0.041666666666666664), -0.5)), 1.0);
	} else if (cos(x) <= 0.78) {
		tmp = ((x * x) * (x * x)) * fma(fma((y * y), 0.008333333333333333, 0.16666666666666666), ((y * y) * 0.041666666666666664), 0.041666666666666664);
	} 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) <= 0.01)
		tmp = Float64(fma(0.16666666666666666, Float64(y * y), 1.0) * fma(x, Float64(x * fma(Float64(x * x), fma(x, Float64(x * -0.001388888888888889), 0.041666666666666664), -0.5)), 1.0));
	elseif (cos(x) <= 0.78)
		tmp = Float64(Float64(Float64(x * x) * Float64(x * x)) * fma(fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666), Float64(Float64(y * y) * 0.041666666666666664), 0.041666666666666664));
	else
		tmp = fma(Float64(y * y), fma(y, Float64(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], 0.01], N[(N[(0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * -0.001388888888888889), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Cos[x], $MachinePrecision], 0.78], N[(N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.041666666666666664), $MachinePrecision] + 0.041666666666666664), $MachinePrecision]), $MachinePrecision], N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (cos.f64 x) < 0.0100000000000000002

   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. lower-*.f64 N/A

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

      5. lower-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. lower-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. lower-*.f64 77.5

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

   5. Simplified 77.5%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)\right)} \cdot \mathsf{fma}\left(\frac{1}{6}, y \cdot y, 1\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right) + 1\right)} \cdot \mathsf{fma}\left(\frac{1}{6}, y \cdot y, 1\right) \]

      2. unpow2 N/A

         \[\leadsto \left(\color{blue}{\left(x \cdot x\right)} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right) + 1\right) \cdot \mathsf{fma}\left(\frac{1}{6}, y \cdot y, 1\right) \]

      3. associate-*l* N/A

         \[\leadsto \left(\color{blue}{x \cdot \left(x \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)\right)} + 1\right) \cdot \mathsf{fma}\left(\frac{1}{6}, y \cdot y, 1\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right), 1\right)} \cdot \mathsf{fma}\left(\frac{1}{6}, y \cdot y, 1\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)}, 1\right) \cdot \mathsf{fma}\left(\frac{1}{6}, y \cdot y, 1\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(x, x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)}, 1\right) \cdot \mathsf{fma}\left(\frac{1}{6}, y \cdot y, 1\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(x, x \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) + \color{blue}{\frac{-1}{2}}\right), 1\right) \cdot \mathsf{fma}\left(\frac{1}{6}, y \cdot y, 1\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}, \frac{-1}{2}\right)}, 1\right) \cdot \mathsf{fma}\left(\frac{1}{6}, y \cdot y, 1\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}, \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(\frac{1}{6}, y \cdot y, 1\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}, \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(\frac{1}{6}, y \cdot y, 1\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{-1}{720} \cdot {x}^{2} + \frac{1}{24}}, \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(\frac{1}{6}, y \cdot y, 1\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{720} \cdot \color{blue}{\left(x \cdot x\right)} + \frac{1}{24}, \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(\frac{1}{6}, y \cdot y, 1\right) \]

      13. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\left(\frac{-1}{720} \cdot x\right) \cdot x} + \frac{1}{24}, \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(\frac{1}{6}, y \cdot y, 1\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(\frac{-1}{720} \cdot x\right)} + \frac{1}{24}, \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(\frac{1}{6}, y \cdot y, 1\right) \]

      15. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, \frac{-1}{720} \cdot x, \frac{1}{24}\right)}, \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(\frac{1}{6}, y \cdot y, 1\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{-1}{720}}, \frac{1}{24}\right), \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(\frac{1}{6}, y \cdot y, 1\right) \]

      17. lower-*.f64 53.2

          \[\leadsto \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot -0.001388888888888889}, 0.041666666666666664\right), -0.5\right), 1\right) \cdot \mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right) \]

   8. Simplified 53.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right)} \cdot \mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right) \]

   if 0.0100000000000000002 < (cos.f64 x) < 0.78000000000000003

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. +-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

   5. Simplified 76.1%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right)} \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right) + 1\right)} \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{24} \cdot {x}^{2} - \frac{1}{2}, 1\right)} \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24} \cdot {x}^{2} - \frac{1}{2}, 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24} \cdot {x}^{2} - \frac{1}{2}, 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{24} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{1}{24}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, {x}^{2} \cdot \frac{1}{24} + \color{blue}{\frac{-1}{2}}, 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{24}, \frac{-1}{2}\right)}, 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24}, \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      10. lower-*.f64 59.2

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.041666666666666664, -0.5\right), 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right) \]

   8. Simplified 59.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.041666666666666664, -0.5\right), 1\right)} \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right) \]

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{24} \cdot \left({x}^{4} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{1}{24} \cdot {x}^{4}\right) \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} \]

       2. *-commutative N/A

          \[\leadsto \color{blue}{\left({x}^{4} \cdot \frac{1}{24}\right)} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) \]

       3. associate-*r* N/A

          \[\leadsto \color{blue}{{x}^{4} \cdot \left(\frac{1}{24} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto \color{blue}{{x}^{4} \cdot \left(\frac{1}{24} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)} \]

       5. metadata-eval N/A

          \[\leadsto {x}^{\color{blue}{\left(2 \cdot 2\right)}} \cdot \left(\frac{1}{24} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) \]

       6. pow-sqr N/A

          \[\leadsto \color{blue}{\left({x}^{2} \cdot {x}^{2}\right)} \cdot \left(\frac{1}{24} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) \]

       7. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left({x}^{2} \cdot {x}^{2}\right)} \cdot \left(\frac{1}{24} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \left(\color{blue}{\left(x \cdot x\right)} \cdot {x}^{2}\right) \cdot \left(\frac{1}{24} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) \]

       9. lower-*.f64 N/A

          \[\leadsto \left(\color{blue}{\left(x \cdot x\right)} \cdot {x}^{2}\right) \cdot \left(\frac{1}{24} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \left(\left(x \cdot x\right) \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(\frac{1}{24} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) \]

       11. lower-*.f64 N/A

           \[\leadsto \left(\left(x \cdot x\right) \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(\frac{1}{24} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) \]

       12. distribute-rgt-in N/A

           \[\leadsto \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(1 \cdot \frac{1}{24} + \left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) \cdot \frac{1}{24}\right)} \]

       13. metadata-eval N/A

           \[\leadsto \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\color{blue}{\frac{1}{24}} + \left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) \cdot \frac{1}{24}\right) \]

       14. +-commutative N/A

           \[\leadsto \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(\left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) \cdot \frac{1}{24} + \frac{1}{24}\right)} \]

       15. *-commutative N/A

           \[\leadsto \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\color{blue}{\left(\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) \cdot {y}^{2}\right)} \cdot \frac{1}{24} + \frac{1}{24}\right) \]

       16. associate-*l* N/A

           \[\leadsto \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\color{blue}{\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) \cdot \left({y}^{2} \cdot \frac{1}{24}\right)} + \frac{1}{24}\right) \]

       17. lower-fma.f64 N/A

           \[\leadsto \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}, {y}^{2} \cdot \frac{1}{24}, \frac{1}{24}\right)} \]

   11. Simplified 59.2%

       \[\leadsto \color{blue}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), \left(y \cdot y\right) \cdot 0.041666666666666664, 0.041666666666666664\right)} \]

   if 0.78000000000000003 < (cos.f64 x)

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-rgt-identity N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt-out N/A

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

      9. +-commutative N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

   5. Simplified 92.3%

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

      1. lift-cos.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. flip-+ N/A

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

      8. clear-num N/A

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

   7. Applied egg-rr 92.3%

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

   8. Taylor expanded in x 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. lower-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. lower-*.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. lower-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. lower-*.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. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 81.4

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

   10. Simplified 81.4%

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

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos x \leq 0.01:\\ \;\;\;\;\mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right) \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right)\\ \mathbf{elif}\;\cos x \leq 0.78:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), \left(y \cdot y\right) \cdot 0.041666666666666664, 0.041666666666666664\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 70.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos x \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right) \cdot \mathsf{fma}\left(-0.5, x \cdot x, 1\right)\\ \mathbf{elif}\;\cos x \leq 0.78:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), \left(y \cdot y\right) \cdot 0.041666666666666664, 0.041666666666666664\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (cos x) -0.02)
   (*
    (fma (* y y) (fma y (* y 0.008333333333333333) 0.16666666666666666) 1.0)
    (fma -0.5 (* x x) 1.0))
   (if (<= (cos x) 0.78)
     (*
      (* (* x x) (* x x))
      (fma
       (fma (* y y) 0.008333333333333333 0.16666666666666666)
       (* (* y y) 0.041666666666666664)
       0.041666666666666664))
     (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) <= -0.02) {
		tmp = fma((y * y), fma(y, (y * 0.008333333333333333), 0.16666666666666666), 1.0) * fma(-0.5, (x * x), 1.0);
	} else if (cos(x) <= 0.78) {
		tmp = ((x * x) * (x * x)) * fma(fma((y * y), 0.008333333333333333, 0.16666666666666666), ((y * y) * 0.041666666666666664), 0.041666666666666664);
	} 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) <= -0.02)
		tmp = Float64(fma(Float64(y * y), fma(y, Float64(y * 0.008333333333333333), 0.16666666666666666), 1.0) * fma(-0.5, Float64(x * x), 1.0));
	elseif (cos(x) <= 0.78)
		tmp = Float64(Float64(Float64(x * x) * Float64(x * x)) * fma(fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666), Float64(Float64(y * y) * 0.041666666666666664), 0.041666666666666664));
	else
		tmp = fma(Float64(y * y), fma(y, Float64(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], -0.02], N[(N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * N[(-0.5 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Cos[x], $MachinePrecision], 0.78], N[(N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.041666666666666664), $MachinePrecision] + 0.041666666666666664), $MachinePrecision]), $MachinePrecision], N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (cos.f64 x) < -0.0200000000000000004

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. +-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

   5. Simplified 92.4%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 52.6

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

   8. Simplified 52.6%

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

   if -0.0200000000000000004 < (cos.f64 x) < 0.78000000000000003

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. +-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

   5. Simplified 76.7%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right)} \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right) + 1\right)} \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{24} \cdot {x}^{2} - \frac{1}{2}, 1\right)} \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24} \cdot {x}^{2} - \frac{1}{2}, 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24} \cdot {x}^{2} - \frac{1}{2}, 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{24} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{1}{24}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, {x}^{2} \cdot \frac{1}{24} + \color{blue}{\frac{-1}{2}}, 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{24}, \frac{-1}{2}\right)}, 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24}, \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      10. lower-*.f64 58.2

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.041666666666666664, -0.5\right), 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right) \]

   8. Simplified 58.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.041666666666666664, -0.5\right), 1\right)} \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right) \]

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{24} \cdot \left({x}^{4} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{1}{24} \cdot {x}^{4}\right) \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} \]

       2. *-commutative N/A

          \[\leadsto \color{blue}{\left({x}^{4} \cdot \frac{1}{24}\right)} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) \]

       3. associate-*r* N/A

          \[\leadsto \color{blue}{{x}^{4} \cdot \left(\frac{1}{24} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto \color{blue}{{x}^{4} \cdot \left(\frac{1}{24} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)} \]

       5. metadata-eval N/A

          \[\leadsto {x}^{\color{blue}{\left(2 \cdot 2\right)}} \cdot \left(\frac{1}{24} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) \]

       6. pow-sqr N/A

          \[\leadsto \color{blue}{\left({x}^{2} \cdot {x}^{2}\right)} \cdot \left(\frac{1}{24} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) \]

       7. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left({x}^{2} \cdot {x}^{2}\right)} \cdot \left(\frac{1}{24} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \left(\color{blue}{\left(x \cdot x\right)} \cdot {x}^{2}\right) \cdot \left(\frac{1}{24} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) \]

       9. lower-*.f64 N/A

          \[\leadsto \left(\color{blue}{\left(x \cdot x\right)} \cdot {x}^{2}\right) \cdot \left(\frac{1}{24} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \left(\left(x \cdot x\right) \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(\frac{1}{24} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) \]

       11. lower-*.f64 N/A

           \[\leadsto \left(\left(x \cdot x\right) \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(\frac{1}{24} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) \]

       12. distribute-rgt-in N/A

           \[\leadsto \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(1 \cdot \frac{1}{24} + \left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) \cdot \frac{1}{24}\right)} \]

       13. metadata-eval N/A

           \[\leadsto \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\color{blue}{\frac{1}{24}} + \left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) \cdot \frac{1}{24}\right) \]

       14. +-commutative N/A

           \[\leadsto \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(\left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) \cdot \frac{1}{24} + \frac{1}{24}\right)} \]

       15. *-commutative N/A

           \[\leadsto \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\color{blue}{\left(\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) \cdot {y}^{2}\right)} \cdot \frac{1}{24} + \frac{1}{24}\right) \]

       16. associate-*l* N/A

           \[\leadsto \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\color{blue}{\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) \cdot \left({y}^{2} \cdot \frac{1}{24}\right)} + \frac{1}{24}\right) \]

       17. lower-fma.f64 N/A

           \[\leadsto \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}, {y}^{2} \cdot \frac{1}{24}, \frac{1}{24}\right)} \]

   11. Simplified 58.2%

       \[\leadsto \color{blue}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), \left(y \cdot y\right) \cdot 0.041666666666666664, 0.041666666666666664\right)} \]

   if 0.78000000000000003 < (cos.f64 x)

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-rgt-identity N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt-out N/A

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

      9. +-commutative N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

   5. Simplified 92.3%

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

      1. lift-cos.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. flip-+ N/A

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

      8. clear-num N/A

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

   7. Applied egg-rr 92.3%

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

   8. Taylor expanded in x 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. lower-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. lower-*.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. lower-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. lower-*.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. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 81.4

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

   10. Simplified 81.4%

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

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos x \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right) \cdot \mathsf{fma}\left(-0.5, x \cdot x, 1\right)\\ \mathbf{elif}\;\cos x \leq 0.78:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), \left(y \cdot y\right) \cdot 0.041666666666666664, 0.041666666666666664\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 70.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos x \cdot \frac{\sinh y}{y} \leq -0.04:\\ \;\;\;\;\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \mathsf{fma}\left(x \cdot x, -0.004166666666666667, 0.008333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cos x) (/ (sinh y) y)) -0.04)
   (*
    (* (* y y) (* y y))
    (fma (* x x) -0.004166666666666667 0.008333333333333333))
   (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) * (sinh(y) / y)) <= -0.04) {
		tmp = ((y * y) * (y * y)) * fma((x * x), -0.004166666666666667, 0.008333333333333333);
	} 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 (Float64(cos(x) * Float64(sinh(y) / y)) <= -0.04)
		tmp = Float64(Float64(Float64(y * y) * Float64(y * y)) * fma(Float64(x * x), -0.004166666666666667, 0.008333333333333333));
	else
		tmp = fma(Float64(y * y), fma(y, Float64(y * fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333)), 0.16666666666666666), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Cos[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], -0.04], N[(N[(N[(y * y), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.004166666666666667 + 0.008333333333333333), $MachinePrecision]), $MachinePrecision], N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cos.f64 x) (/.f64 (sinh.f64 y) y)) < -0.0400000000000000008

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. +-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

   5. Simplified 92.2%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 54.0

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

   8. Simplified 54.0%

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

   9. Taylor expanded in y around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. lower-*.f64 N/A

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

       5. metadata-eval N/A

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

       6. pow-sqr N/A

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

       7. lower-*.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f64 N/A

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

       12. distribute-rgt-in N/A

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

       13. metadata-eval N/A

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

       14. +-commutative N/A

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

       15. *-commutative N/A

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

       16. associate-*l* N/A

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

       17. metadata-eval N/A

           \[\leadsto \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \left({x}^{2} \cdot \color{blue}{\frac{-1}{240}} + \frac{1}{120}\right) \]

       18. metadata-eval N/A

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

       19. lower-fma.f64 N/A

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

       20. unpow2 N/A

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

       21. lower-*.f64 N/A

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

       22. metadata-eval 53.4

           \[\leadsto \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{-0.004166666666666667}, 0.008333333333333333\right) \]

   11. Simplified 53.4%

       \[\leadsto \color{blue}{\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \mathsf{fma}\left(x \cdot x, -0.004166666666666667, 0.008333333333333333\right)} \]

   if -0.0400000000000000008 < (*.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 y around 0

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

      1. *-rgt-identity N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt-out N/A

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

      9. +-commutative N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

   5. Simplified 89.1%

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

      1. lift-cos.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. flip-+ N/A

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

      8. clear-num N/A

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

   7. Applied egg-rr 89.1%

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

   8. Taylor expanded in x 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. lower-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. lower-*.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. lower-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. lower-*.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. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 72.5

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

   10. Simplified 72.5%

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

Alternative 18: 70.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos x \cdot \frac{\sinh y}{y} \leq -0.04:\\ \;\;\;\;\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \mathsf{fma}\left(x \cdot x, -0.004166666666666667, 0.008333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cos x) (/ (sinh y) y)) -0.04)
   (*
    (* (* y y) (* y y))
    (fma (* x x) -0.004166666666666667 0.008333333333333333))
   (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) * (sinh(y) / y)) <= -0.04) {
		tmp = ((y * y) * (y * y)) * fma((x * x), -0.004166666666666667, 0.008333333333333333);
	} 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 (Float64(cos(x) * Float64(sinh(y) / y)) <= -0.04)
		tmp = Float64(Float64(Float64(y * y) * Float64(y * y)) * fma(Float64(x * x), -0.004166666666666667, 0.008333333333333333));
	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[(N[Cos[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], -0.04], N[(N[(N[(y * y), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.004166666666666667 + 0.008333333333333333), $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 (*.f64 (cos.f64 x) (/.f64 (sinh.f64 y) y)) < -0.0400000000000000008

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. +-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

   5. Simplified 92.2%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 54.0

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

   8. Simplified 54.0%

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

   9. Taylor expanded in y around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. lower-*.f64 N/A

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

       5. metadata-eval N/A

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

       6. pow-sqr N/A

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

       7. lower-*.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f64 N/A

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

       12. distribute-rgt-in N/A

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

       13. metadata-eval N/A

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

       14. +-commutative N/A

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

       15. *-commutative N/A

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

       16. associate-*l* N/A

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

       17. metadata-eval N/A

           \[\leadsto \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \left({x}^{2} \cdot \color{blue}{\frac{-1}{240}} + \frac{1}{120}\right) \]

       18. metadata-eval N/A

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

       19. lower-fma.f64 N/A

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

       20. unpow2 N/A

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

       21. lower-*.f64 N/A

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

       22. metadata-eval 53.4

           \[\leadsto \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{-0.004166666666666667}, 0.008333333333333333\right) \]

   11. Simplified 53.4%

       \[\leadsto \color{blue}{\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \mathsf{fma}\left(x \cdot x, -0.004166666666666667, 0.008333333333333333\right)} \]

   if -0.0400000000000000008 < (*.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 y around 0

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

      1. *-rgt-identity N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt-out N/A

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

      9. +-commutative N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

   5. Simplified 89.1%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)} \]
   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. 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. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 72.5

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

   8. Simplified 72.5%

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

Alternative 19: 67.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos x \cdot \frac{\sinh y}{y} \leq -0.04:\\ \;\;\;\;\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \mathsf{fma}\left(x \cdot x, -0.004166666666666667, 0.008333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cos x) (/ (sinh y) y)) -0.04)
   (*
    (* (* y y) (* y y))
    (fma (* x x) -0.004166666666666667 0.008333333333333333))
   (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)) <= -0.04) {
		tmp = ((y * y) * (y * y)) * fma((x * x), -0.004166666666666667, 0.008333333333333333);
	} 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)) <= -0.04)
		tmp = Float64(Float64(Float64(y * y) * Float64(y * y)) * fma(Float64(x * x), -0.004166666666666667, 0.008333333333333333));
	else
		tmp = fma(y, Float64(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], -0.04], N[(N[(N[(y * y), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.004166666666666667 + 0.008333333333333333), $MachinePrecision]), $MachinePrecision], N[(y * N[(y * N[(y * N[(y * 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cos.f64 x) (/.f64 (sinh.f64 y) y)) < -0.0400000000000000008

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. +-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

   5. Simplified 92.2%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 54.0

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

   8. Simplified 54.0%

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

   9. Taylor expanded in y around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. lower-*.f64 N/A

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

       5. metadata-eval N/A

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

       6. pow-sqr N/A

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

       7. lower-*.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f64 N/A

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

       12. distribute-rgt-in N/A

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

       13. metadata-eval N/A

           \[\leadsto \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \left(\color{blue}{\frac{1}{120}} + \left(\frac{-1}{2} \cdot {x}^{2}\right) \cdot \frac{1}{120}\right) \]

       14. +-commutative N/A

           \[\leadsto \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \color{blue}{\left(\left(\frac{-1}{2} \cdot {x}^{2}\right) \cdot \frac{1}{120} + \frac{1}{120}\right)} \]

       15. *-commutative N/A

           \[\leadsto \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \left(\color{blue}{\left({x}^{2} \cdot \frac{-1}{2}\right)} \cdot \frac{1}{120} + \frac{1}{120}\right) \]

       16. associate-*l* N/A

           \[\leadsto \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \left(\color{blue}{{x}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{1}{120}\right)} + \frac{1}{120}\right) \]

       17. metadata-eval N/A

           \[\leadsto \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \left({x}^{2} \cdot \color{blue}{\frac{-1}{240}} + \frac{1}{120}\right) \]

       18. metadata-eval N/A

           \[\leadsto \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \left({x}^{2} \cdot \color{blue}{\left(\frac{1}{120} \cdot \frac{-1}{2}\right)} + \frac{1}{120}\right) \]

       19. lower-fma.f64 N/A

           \[\leadsto \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{120} \cdot \frac{-1}{2}, \frac{1}{120}\right)} \]

       20. unpow2 N/A

           \[\leadsto \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{120} \cdot \frac{-1}{2}, \frac{1}{120}\right) \]

       21. lower-*.f64 N/A

           \[\leadsto \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{120} \cdot \frac{-1}{2}, \frac{1}{120}\right) \]

       22. metadata-eval 53.4

           \[\leadsto \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{-0.004166666666666667}, 0.008333333333333333\right) \]

   11. Simplified 53.4%

       \[\leadsto \color{blue}{\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \mathsf{fma}\left(x \cdot x, -0.004166666666666667, 0.008333333333333333\right)} \]

   if -0.0400000000000000008 < (*.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 83.5%

      \[\leadsto \color{blue}{1} \cdot \frac{\sinh y}{y} \]

   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. unpow2 N/A

         \[\leadsto \color{blue}{\left(y \cdot y\right)} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) + 1 \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{y \cdot \left(y \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} + 1 \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, y \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right), 1\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{y \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)}, 1\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, y \cdot \color{blue}{\left(\frac{1}{120} \cdot {y}^{2} + \frac{1}{6}\right)}, 1\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(y, y \cdot \left(\frac{1}{120} \cdot \color{blue}{\left(y \cdot y\right)} + \frac{1}{6}\right), 1\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(y, y \cdot \left(\color{blue}{\left(\frac{1}{120} \cdot y\right) \cdot y} + \frac{1}{6}\right), 1\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, y \cdot \left(\color{blue}{y \cdot \left(\frac{1}{120} \cdot y\right)} + \frac{1}{6}\right), 1\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, y \cdot \color{blue}{\mathsf{fma}\left(y, \frac{1}{120} \cdot y, \frac{1}{6}\right)}, 1\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, \color{blue}{y \cdot \frac{1}{120}}, \frac{1}{6}\right), 1\right) \]

      12. lower-*.f64 69.8

          \[\leadsto \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, \color{blue}{y \cdot 0.008333333333333333}, 0.16666666666666666\right), 1\right) \]

   8. Simplified 69.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 70.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos x \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right) \cdot \mathsf{fma}\left(-0.5, x \cdot x, 1\right)\\ \mathbf{elif}\;\cos x \leq 0.78:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.041666666666666664, -0.5\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot 0.16666666666666666, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (cos x) -0.02)
   (*
    (fma (* y y) (fma y (* y 0.008333333333333333) 0.16666666666666666) 1.0)
    (fma -0.5 (* x x) 1.0))
   (if (<= (cos x) 0.78)
     (*
      (fma x (* x (fma x (* x 0.041666666666666664) -0.5)) 1.0)
      (fma y (* y 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) <= -0.02) {
		tmp = fma((y * y), fma(y, (y * 0.008333333333333333), 0.16666666666666666), 1.0) * fma(-0.5, (x * x), 1.0);
	} else if (cos(x) <= 0.78) {
		tmp = fma(x, (x * fma(x, (x * 0.041666666666666664), -0.5)), 1.0) * fma(y, (y * 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) <= -0.02)
		tmp = Float64(fma(Float64(y * y), fma(y, Float64(y * 0.008333333333333333), 0.16666666666666666), 1.0) * fma(-0.5, Float64(x * x), 1.0));
	elseif (cos(x) <= 0.78)
		tmp = Float64(fma(x, Float64(x * fma(x, Float64(x * 0.041666666666666664), -0.5)), 1.0) * fma(y, Float64(y * 0.16666666666666666), 1.0));
	else
		tmp = fma(Float64(y * y), fma(y, Float64(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], -0.02], N[(N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * N[(-0.5 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Cos[x], $MachinePrecision], 0.78], N[(N[(x * N[(x * N[(x * N[(x * 0.041666666666666664), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(y * N[(y * 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (cos.f64 x) < -0.0200000000000000004

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\cos x + {y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right) + \frac{1}{6} \cdot \cos x\right)} \]
   4. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto \color{blue}{\cos x \cdot 1} + {y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right) + \frac{1}{6} \cdot \cos x\right) \]

      2. *-commutative N/A

         \[\leadsto \cos x \cdot 1 + \color{blue}{\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right) + \frac{1}{6} \cdot \cos x\right) \cdot {y}^{2}} \]

      3. associate-*r* N/A

         \[\leadsto \cos x \cdot 1 + \left(\color{blue}{\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot \cos x} + \frac{1}{6} \cdot \cos x\right) \cdot {y}^{2} \]

      4. distribute-rgt-out N/A

         \[\leadsto \cos x \cdot 1 + \color{blue}{\left(\cos x \cdot \left(\frac{1}{120} \cdot {y}^{2} + \frac{1}{6}\right)\right)} \cdot {y}^{2} \]

      5. +-commutative N/A

         \[\leadsto \cos x \cdot 1 + \left(\cos x \cdot \color{blue}{\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)}\right) \cdot {y}^{2} \]

      6. associate-*l* N/A

         \[\leadsto \cos x \cdot 1 + \color{blue}{\cos x \cdot \left(\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) \cdot {y}^{2}\right)} \]

      7. *-commutative N/A

         \[\leadsto \cos x \cdot 1 + \cos x \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} \]

      8. distribute-lft-in N/A

         \[\leadsto \color{blue}{\cos x \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{\cos x \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} \]

      10. lower-cos.f64 N/A

          \[\leadsto \color{blue}{\cos x} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \cos x \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) + 1\right)} \]

      12. lower-fma.f64 N/A

          \[\leadsto \cos x \cdot \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{1}{6} + \frac{1}{120} \cdot {y}^{2}, 1\right)} \]

   5. Simplified 92.4%

      \[\leadsto \color{blue}{\cos x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + \frac{-1}{2} \cdot {x}^{2}\right)} \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{2} \cdot {x}^{2} + 1\right)} \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1}{2}, {x}^{2}, 1\right)} \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{x \cdot x}, 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      4. lower-*.f64 52.6

         \[\leadsto \mathsf{fma}\left(-0.5, \color{blue}{x \cdot x}, 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right) \]

   8. Simplified 52.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, x \cdot x, 1\right)} \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right) \]

   if -0.0200000000000000004 < (cos.f64 x) < 0.78000000000000003

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\cos x + {y}^{2} \cdot \left(\frac{1}{6} \cdot \cos x + {y}^{2} \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \cos x\right) + \frac{1}{120} \cdot \cos x\right)\right)} \]
   4. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto \color{blue}{\cos x \cdot 1} + {y}^{2} \cdot \left(\frac{1}{6} \cdot \cos x + {y}^{2} \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \cos x\right) + \frac{1}{120} \cdot \cos x\right)\right) \]

      2. distribute-rgt-in N/A

         \[\leadsto \cos x \cdot 1 + \color{blue}{\left(\left(\frac{1}{6} \cdot \cos x\right) \cdot {y}^{2} + \left({y}^{2} \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \cos x\right) + \frac{1}{120} \cdot \cos x\right)\right) \cdot {y}^{2}\right)} \]

      3. *-commutative N/A

         \[\leadsto \cos x \cdot 1 + \left(\color{blue}{\left(\cos x \cdot \frac{1}{6}\right)} \cdot {y}^{2} + \left({y}^{2} \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \cos x\right) + \frac{1}{120} \cdot \cos x\right)\right) \cdot {y}^{2}\right) \]

      4. associate-*l* N/A

         \[\leadsto \cos x \cdot 1 + \left(\color{blue}{\cos x \cdot \left(\frac{1}{6} \cdot {y}^{2}\right)} + \left({y}^{2} \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \cos x\right) + \frac{1}{120} \cdot \cos x\right)\right) \cdot {y}^{2}\right) \]

      5. *-commutative N/A

         \[\leadsto \cos x \cdot 1 + \left(\cos x \cdot \left(\frac{1}{6} \cdot {y}^{2}\right) + \color{blue}{\left(\left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \cos x\right) + \frac{1}{120} \cdot \cos x\right) \cdot {y}^{2}\right)} \cdot {y}^{2}\right) \]

      6. associate-*l* N/A

         \[\leadsto \cos x \cdot 1 + \left(\cos x \cdot \left(\frac{1}{6} \cdot {y}^{2}\right) + \color{blue}{\left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \cos x\right) + \frac{1}{120} \cdot \cos x\right) \cdot \left({y}^{2} \cdot {y}^{2}\right)}\right) \]

      7. associate-*r* N/A

         \[\leadsto \cos x \cdot 1 + \left(\cos x \cdot \left(\frac{1}{6} \cdot {y}^{2}\right) + \left(\color{blue}{\left(\frac{1}{5040} \cdot {y}^{2}\right) \cdot \cos x} + \frac{1}{120} \cdot \cos x\right) \cdot \left({y}^{2} \cdot {y}^{2}\right)\right) \]

      8. distribute-rgt-out N/A

         \[\leadsto \cos x \cdot 1 + \left(\cos x \cdot \left(\frac{1}{6} \cdot {y}^{2}\right) + \color{blue}{\left(\cos x \cdot \left(\frac{1}{5040} \cdot {y}^{2} + \frac{1}{120}\right)\right)} \cdot \left({y}^{2} \cdot {y}^{2}\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \cos x \cdot 1 + \left(\cos x \cdot \left(\frac{1}{6} \cdot {y}^{2}\right) + \left(\cos x \cdot \color{blue}{\left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)}\right) \cdot \left({y}^{2} \cdot {y}^{2}\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \cos x \cdot 1 + \left(\cos x \cdot \left(\frac{1}{6} \cdot {y}^{2}\right) + \color{blue}{\cos x \cdot \left(\left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right) \cdot \left({y}^{2} \cdot {y}^{2}\right)\right)}\right) \]

      11. *-commutative N/A

          \[\leadsto \cos x \cdot 1 + \left(\cos x \cdot \left(\frac{1}{6} \cdot {y}^{2}\right) + \cos x \cdot \color{blue}{\left(\left({y}^{2} \cdot {y}^{2}\right) \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)}\right) \]

   5. Simplified 77.2%

      \[\leadsto \color{blue}{\cos x \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + \left({x}^{2} \cdot \left(\frac{-1}{2} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right) + \frac{1}{24} \cdot \left({x}^{2} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right) + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)} \]

   8. Simplified 58.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.041666666666666664, -0.5\right), 1\right)} \]

   9. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1 + \left(\frac{1}{6} \cdot \left({y}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right)\right) + {x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto 1 + \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right) + \frac{1}{6} \cdot \left({y}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right)\right)\right)} \]

       2. associate-+r+ N/A

          \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right) + \frac{1}{6} \cdot \left({y}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right)\right)} \]

       3. associate-*r* N/A

          \[\leadsto \left(1 + {x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right) + \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right)} \]

       4. distribute-rgt1-in N/A

          \[\leadsto \color{blue}{\left(\frac{1}{6} \cdot {y}^{2} + 1\right) \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right)} \]

       5. +-commutative N/A

          \[\leadsto \color{blue}{\left(1 + \frac{1}{6} \cdot {y}^{2}\right)} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right) \]

       6. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(1 + \frac{1}{6} \cdot {y}^{2}\right) \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right)} \]

       7. +-commutative N/A

          \[\leadsto \color{blue}{\left(\frac{1}{6} \cdot {y}^{2} + 1\right)} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right) \]

       8. *-commutative N/A

          \[\leadsto \left(\color{blue}{{y}^{2} \cdot \frac{1}{6}} + 1\right) \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right) \]

       9. unpow2 N/A

          \[\leadsto \left(\color{blue}{\left(y \cdot y\right)} \cdot \frac{1}{6} + 1\right) \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right) \]

       10. associate-*l* N/A

           \[\leadsto \left(\color{blue}{y \cdot \left(y \cdot \frac{1}{6}\right)} + 1\right) \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right) \]

       11. *-commutative N/A

           \[\leadsto \left(y \cdot \color{blue}{\left(\frac{1}{6} \cdot y\right)} + 1\right) \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right) \]

       12. lower-fma.f64 N/A

           \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{1}{6} \cdot y, 1\right)} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right) \]

       13. *-commutative N/A

           \[\leadsto \mathsf{fma}\left(y, \color{blue}{y \cdot \frac{1}{6}}, 1\right) \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right) \]

       14. lower-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(y, \color{blue}{y \cdot \frac{1}{6}}, 1\right) \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right) \]

   11. Simplified 56.0%

       \[\leadsto \color{blue}{\mathsf{fma}\left(y, y \cdot 0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.041666666666666664, -0.5\right), 1\right)} \]

   if 0.78000000000000003 < (cos.f64 x)

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\cos x + {y}^{2} \cdot \left(\frac{1}{6} \cdot \cos x + {y}^{2} \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \cos x\right) + \frac{1}{120} \cdot \cos x\right)\right)} \]
   4. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto \color{blue}{\cos x \cdot 1} + {y}^{2} \cdot \left(\frac{1}{6} \cdot \cos x + {y}^{2} \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \cos x\right) + \frac{1}{120} \cdot \cos x\right)\right) \]

      2. distribute-rgt-in N/A

         \[\leadsto \cos x \cdot 1 + \color{blue}{\left(\left(\frac{1}{6} \cdot \cos x\right) \cdot {y}^{2} + \left({y}^{2} \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \cos x\right) + \frac{1}{120} \cdot \cos x\right)\right) \cdot {y}^{2}\right)} \]

      3. *-commutative N/A

         \[\leadsto \cos x \cdot 1 + \left(\color{blue}{\left(\cos x \cdot \frac{1}{6}\right)} \cdot {y}^{2} + \left({y}^{2} \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \cos x\right) + \frac{1}{120} \cdot \cos x\right)\right) \cdot {y}^{2}\right) \]

      4. associate-*l* N/A

         \[\leadsto \cos x \cdot 1 + \left(\color{blue}{\cos x \cdot \left(\frac{1}{6} \cdot {y}^{2}\right)} + \left({y}^{2} \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \cos x\right) + \frac{1}{120} \cdot \cos x\right)\right) \cdot {y}^{2}\right) \]

      5. *-commutative N/A

         \[\leadsto \cos x \cdot 1 + \left(\cos x \cdot \left(\frac{1}{6} \cdot {y}^{2}\right) + \color{blue}{\left(\left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \cos x\right) + \frac{1}{120} \cdot \cos x\right) \cdot {y}^{2}\right)} \cdot {y}^{2}\right) \]

      6. associate-*l* N/A

         \[\leadsto \cos x \cdot 1 + \left(\cos x \cdot \left(\frac{1}{6} \cdot {y}^{2}\right) + \color{blue}{\left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \cos x\right) + \frac{1}{120} \cdot \cos x\right) \cdot \left({y}^{2} \cdot {y}^{2}\right)}\right) \]

      7. associate-*r* N/A

         \[\leadsto \cos x \cdot 1 + \left(\cos x \cdot \left(\frac{1}{6} \cdot {y}^{2}\right) + \left(\color{blue}{\left(\frac{1}{5040} \cdot {y}^{2}\right) \cdot \cos x} + \frac{1}{120} \cdot \cos x\right) \cdot \left({y}^{2} \cdot {y}^{2}\right)\right) \]

      8. distribute-rgt-out N/A

         \[\leadsto \cos x \cdot 1 + \left(\cos x \cdot \left(\frac{1}{6} \cdot {y}^{2}\right) + \color{blue}{\left(\cos x \cdot \left(\frac{1}{5040} \cdot {y}^{2} + \frac{1}{120}\right)\right)} \cdot \left({y}^{2} \cdot {y}^{2}\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \cos x \cdot 1 + \left(\cos x \cdot \left(\frac{1}{6} \cdot {y}^{2}\right) + \left(\cos x \cdot \color{blue}{\left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)}\right) \cdot \left({y}^{2} \cdot {y}^{2}\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \cos x \cdot 1 + \left(\cos x \cdot \left(\frac{1}{6} \cdot {y}^{2}\right) + \color{blue}{\cos x \cdot \left(\left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right) \cdot \left({y}^{2} \cdot {y}^{2}\right)\right)}\right) \]

      11. *-commutative N/A

          \[\leadsto \cos x \cdot 1 + \left(\cos x \cdot \left(\frac{1}{6} \cdot {y}^{2}\right) + \cos x \cdot \color{blue}{\left(\left({y}^{2} \cdot {y}^{2}\right) \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)}\right) \]

   5. Simplified 92.3%

      \[\leadsto \color{blue}{\cos x \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)} \]
   6. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \color{blue}{\cos x} \cdot \left(y \cdot \left(y \cdot \left(y \cdot \left(y \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{5040} + \frac{1}{120}\right)\right) + \frac{1}{6}\right)\right) + 1\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \cos x \cdot \left(y \cdot \left(y \cdot \left(y \cdot \left(y \cdot \left(\color{blue}{\left(y \cdot y\right)} \cdot \frac{1}{5040} + \frac{1}{120}\right)\right) + \frac{1}{6}\right)\right) + 1\right) \]

      3. lift-fma.f64 N/A

         \[\leadsto \cos x \cdot \left(y \cdot \left(y \cdot \left(y \cdot \left(y \cdot \color{blue}{\mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right)}\right) + \frac{1}{6}\right)\right) + 1\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \cos x \cdot \left(y \cdot \left(y \cdot \left(y \cdot \color{blue}{\left(y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right)\right)} + \frac{1}{6}\right)\right) + 1\right) \]

      5. lift-fma.f64 N/A

         \[\leadsto \cos x \cdot \left(y \cdot \left(y \cdot \color{blue}{\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right)}\right) + 1\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \cos x \cdot \left(y \cdot \color{blue}{\left(y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right)\right)} + 1\right) \]

      7. flip-+ N/A

         \[\leadsto \cos x \cdot \color{blue}{\frac{\left(y \cdot \left(y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right)\right)\right) \cdot \left(y \cdot \left(y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right)\right)\right) - 1 \cdot 1}{y \cdot \left(y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right)\right) - 1}} \]

      8. clear-num N/A

         \[\leadsto \cos x \cdot \color{blue}{\frac{1}{\frac{y \cdot \left(y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right)\right) - 1}{\left(y \cdot \left(y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right)\right)\right) \cdot \left(y \cdot \left(y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right)\right)\right) - 1 \cdot 1}}} \]

   7. Applied egg-rr 92.3%

      \[\leadsto \color{blue}{\frac{\cos x}{\frac{1}{\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)}}} \]

   8. Taylor expanded in x 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. lower-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. lower-*.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. lower-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. lower-*.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. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \left(\color{blue}{{y}^{2} \cdot \frac{1}{5040}} + \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{1}{5040}, \frac{1}{120}\right)}, \frac{1}{6}\right), 1\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      14. lower-*.f64 81.4

          \[\leadsto \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right) \]

   10. Simplified 81.4%

       \[\leadsto \color{blue}{\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos x \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right) \cdot \mathsf{fma}\left(-0.5, x \cdot x, 1\right)\\ \mathbf{elif}\;\cos x \leq 0.78:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.041666666666666664, -0.5\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot 0.16666666666666666, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 66.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos x \cdot \frac{\sinh y}{y} \leq -0.02:\\ \;\;\;\;x \cdot \left(x \cdot \mathsf{fma}\left(y \cdot y, -0.08333333333333333, -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cos x) (/ (sinh y) y)) -0.02)
   (* x (* x (fma (* y y) -0.08333333333333333 -0.5)))
   (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)) <= -0.02) {
		tmp = x * (x * fma((y * y), -0.08333333333333333, -0.5));
	} 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)) <= -0.02)
		tmp = Float64(x * Float64(x * fma(Float64(y * y), -0.08333333333333333, -0.5)));
	else
		tmp = fma(y, Float64(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], -0.02], N[(x * N[(x * N[(N[(y * y), $MachinePrecision] * -0.08333333333333333 + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(y * N[(y * N[(y * 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cos.f64 x) (/.f64 (sinh.f64 y) y)) < -0.0200000000000000004

   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. lower-*.f64 N/A

         \[\leadsto \color{blue}{\cos x \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)} \]

      5. lower-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. lower-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. lower-*.f64 77.2

         \[\leadsto \cos x \cdot \mathsf{fma}\left(0.16666666666666666, \color{blue}{y \cdot y}, 1\right) \]

   5. Simplified 77.2%

      \[\leadsto \color{blue}{\cos x \cdot \mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + \left(\frac{-1}{2} \cdot \left({x}^{2} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right) + \frac{1}{6} \cdot {y}^{2}\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto 1 + \color{blue}{\left(\frac{1}{6} \cdot {y}^{2} + \frac{-1}{2} \cdot \left({x}^{2} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)\right)} \]

      2. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(1 + \frac{1}{6} \cdot {y}^{2}\right) + \frac{-1}{2} \cdot \left({x}^{2} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)} \]

      3. associate-*r* N/A

         \[\leadsto \left(1 + \frac{1}{6} \cdot {y}^{2}\right) + \color{blue}{\left(\frac{-1}{2} \cdot {x}^{2}\right) \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)} \]

      4. distribute-rgt1-in N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{2} \cdot {x}^{2} + 1\right) \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)} \]

      5. +-commutative N/A

         \[\leadsto \color{blue}{\left(1 + \frac{-1}{2} \cdot {x}^{2}\right)} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(1 + \frac{-1}{2} \cdot {x}^{2}\right) \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)} \]

      7. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{2} \cdot {x}^{2} + 1\right)} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1}{2}, {x}^{2}, 1\right)} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{x \cdot x}, 1\right) \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{x \cdot x}, 1\right) \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, x \cdot x, 1\right) \cdot \color{blue}{\left(\frac{1}{6} \cdot {y}^{2} + 1\right)} \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, x \cdot x, 1\right) \cdot \left(\color{blue}{{y}^{2} \cdot \frac{1}{6}} + 1\right) \]

      13. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, x \cdot x, 1\right) \cdot \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{1}{6}, 1\right)} \]

      14. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, x \cdot x, 1\right) \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{6}, 1\right) \]

      15. lower-*.f64 51.2

          \[\leadsto \mathsf{fma}\left(-0.5, x \cdot x, 1\right) \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, 0.16666666666666666, 1\right) \]

   8. Simplified 51.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, x \cdot x, 1\right) \cdot \mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right)} \]

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \left({x}^{2} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{-1}{2} \cdot {x}^{2}\right) \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)} \]

       2. *-commutative N/A

          \[\leadsto \color{blue}{\left({x}^{2} \cdot \frac{-1}{2}\right)} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right) \]

       3. associate-*r* N/A

          \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{-1}{2} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)} \]

       4. unpow2 N/A

          \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{-1}{2} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right) \]

       5. associate-*l* N/A

          \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(\frac{-1}{2} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)\right)} \]

       6. lower-*.f64 N/A

          \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(\frac{-1}{2} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)\right)} \]

       7. lower-*.f64 N/A

          \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(\frac{-1}{2} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)\right)} \]

       8. +-commutative N/A

          \[\leadsto x \cdot \left(x \cdot \left(\frac{-1}{2} \cdot \color{blue}{\left(\frac{1}{6} \cdot {y}^{2} + 1\right)}\right)\right) \]

       9. distribute-rgt-in N/A

          \[\leadsto x \cdot \left(x \cdot \color{blue}{\left(\left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \frac{-1}{2} + 1 \cdot \frac{-1}{2}\right)}\right) \]

       10. *-commutative N/A

           \[\leadsto x \cdot \left(x \cdot \left(\color{blue}{\left({y}^{2} \cdot \frac{1}{6}\right)} \cdot \frac{-1}{2} + 1 \cdot \frac{-1}{2}\right)\right) \]

       11. associate-*l* N/A

           \[\leadsto x \cdot \left(x \cdot \left(\color{blue}{{y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{-1}{2}\right)} + 1 \cdot \frac{-1}{2}\right)\right) \]

       12. metadata-eval N/A

           \[\leadsto x \cdot \left(x \cdot \left({y}^{2} \cdot \color{blue}{\frac{-1}{12}} + 1 \cdot \frac{-1}{2}\right)\right) \]

       13. metadata-eval N/A

           \[\leadsto x \cdot \left(x \cdot \left({y}^{2} \cdot \color{blue}{\left(\frac{-1}{2} \cdot \frac{1}{6}\right)} + 1 \cdot \frac{-1}{2}\right)\right) \]

       14. metadata-eval N/A

           \[\leadsto x \cdot \left(x \cdot \left({y}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{1}{6}\right) + \color{blue}{\frac{-1}{2}}\right)\right) \]

       15. lower-fma.f64 N/A

           \[\leadsto x \cdot \left(x \cdot \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{-1}{2} \cdot \frac{1}{6}, \frac{-1}{2}\right)}\right) \]

       16. unpow2 N/A

           \[\leadsto x \cdot \left(x \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{-1}{2} \cdot \frac{1}{6}, \frac{-1}{2}\right)\right) \]

       17. lower-*.f64 N/A

           \[\leadsto x \cdot \left(x \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{-1}{2} \cdot \frac{1}{6}, \frac{-1}{2}\right)\right) \]

       18. metadata-eval 51.2

           \[\leadsto x \cdot \left(x \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{-0.08333333333333333}, -0.5\right)\right) \]

   11. Simplified 51.2%

       \[\leadsto \color{blue}{x \cdot \left(x \cdot \mathsf{fma}\left(y \cdot y, -0.08333333333333333, -0.5\right)\right)} \]

   if -0.0200000000000000004 < (*.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 84.4%

      \[\leadsto \color{blue}{1} \cdot \frac{\sinh y}{y} \]

   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. unpow2 N/A

         \[\leadsto \color{blue}{\left(y \cdot y\right)} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) + 1 \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{y \cdot \left(y \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} + 1 \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, y \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right), 1\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{y \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)}, 1\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, y \cdot \color{blue}{\left(\frac{1}{120} \cdot {y}^{2} + \frac{1}{6}\right)}, 1\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(y, y \cdot \left(\frac{1}{120} \cdot \color{blue}{\left(y \cdot y\right)} + \frac{1}{6}\right), 1\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(y, y \cdot \left(\color{blue}{\left(\frac{1}{120} \cdot y\right) \cdot y} + \frac{1}{6}\right), 1\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, y \cdot \left(\color{blue}{y \cdot \left(\frac{1}{120} \cdot y\right)} + \frac{1}{6}\right), 1\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, y \cdot \color{blue}{\mathsf{fma}\left(y, \frac{1}{120} \cdot y, \frac{1}{6}\right)}, 1\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, \color{blue}{y \cdot \frac{1}{120}}, \frac{1}{6}\right), 1\right) \]

      12. lower-*.f64 70.6

          \[\leadsto \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, \color{blue}{y \cdot 0.008333333333333333}, 0.16666666666666666\right), 1\right) \]

   8. Simplified 70.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 22: 71.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right)\\ \mathbf{if}\;\cos x \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(y, t\_0, 1\right) \cdot \mathsf{fma}\left(x \cdot x, -0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot y, t\_0, y\right)}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (*
          y
          (fma
           (* y y)
           (fma (* y y) 0.0001984126984126984 0.008333333333333333)
           0.16666666666666666))))
   (if (<= (cos x) -0.02)
     (* (fma y t_0 1.0) (fma (* x x) -0.5 1.0))
     (/ (fma (* y y) t_0 y) y))))

C

double code(double x, double y) {
	double t_0 = y * fma((y * y), fma((y * y), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666);
	double tmp;
	if (cos(x) <= -0.02) {
		tmp = fma(y, t_0, 1.0) * fma((x * x), -0.5, 1.0);
	} else {
		tmp = fma((y * y), t_0, y) / y;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(y * fma(Float64(y * y), fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666))
	tmp = 0.0
	if (cos(x) <= -0.02)
		tmp = Float64(fma(y, t_0, 1.0) * fma(Float64(x * x), -0.5, 1.0));
	else
		tmp = Float64(fma(Float64(y * y), t_0, y) / y);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Cos[x], $MachinePrecision], -0.02], N[(N[(y * t$95$0 + 1.0), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y * y), $MachinePrecision] * t$95$0 + y), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 x) < -0.0200000000000000004

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\cos x + {y}^{2} \cdot \left(\frac{1}{6} \cdot \cos x + {y}^{2} \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \cos x\right) + \frac{1}{120} \cdot \cos x\right)\right)} \]
   4. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto \color{blue}{\cos x \cdot 1} + {y}^{2} \cdot \left(\frac{1}{6} \cdot \cos x + {y}^{2} \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \cos x\right) + \frac{1}{120} \cdot \cos x\right)\right) \]

      2. distribute-rgt-in N/A

         \[\leadsto \cos x \cdot 1 + \color{blue}{\left(\left(\frac{1}{6} \cdot \cos x\right) \cdot {y}^{2} + \left({y}^{2} \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \cos x\right) + \frac{1}{120} \cdot \cos x\right)\right) \cdot {y}^{2}\right)} \]

      3. *-commutative N/A

         \[\leadsto \cos x \cdot 1 + \left(\color{blue}{\left(\cos x \cdot \frac{1}{6}\right)} \cdot {y}^{2} + \left({y}^{2} \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \cos x\right) + \frac{1}{120} \cdot \cos x\right)\right) \cdot {y}^{2}\right) \]

      4. associate-*l* N/A

         \[\leadsto \cos x \cdot 1 + \left(\color{blue}{\cos x \cdot \left(\frac{1}{6} \cdot {y}^{2}\right)} + \left({y}^{2} \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \cos x\right) + \frac{1}{120} \cdot \cos x\right)\right) \cdot {y}^{2}\right) \]

      5. *-commutative N/A

         \[\leadsto \cos x \cdot 1 + \left(\cos x \cdot \left(\frac{1}{6} \cdot {y}^{2}\right) + \color{blue}{\left(\left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \cos x\right) + \frac{1}{120} \cdot \cos x\right) \cdot {y}^{2}\right)} \cdot {y}^{2}\right) \]

      6. associate-*l* N/A

         \[\leadsto \cos x \cdot 1 + \left(\cos x \cdot \left(\frac{1}{6} \cdot {y}^{2}\right) + \color{blue}{\left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \cos x\right) + \frac{1}{120} \cdot \cos x\right) \cdot \left({y}^{2} \cdot {y}^{2}\right)}\right) \]

      7. associate-*r* N/A

         \[\leadsto \cos x \cdot 1 + \left(\cos x \cdot \left(\frac{1}{6} \cdot {y}^{2}\right) + \left(\color{blue}{\left(\frac{1}{5040} \cdot {y}^{2}\right) \cdot \cos x} + \frac{1}{120} \cdot \cos x\right) \cdot \left({y}^{2} \cdot {y}^{2}\right)\right) \]

      8. distribute-rgt-out N/A

         \[\leadsto \cos x \cdot 1 + \left(\cos x \cdot \left(\frac{1}{6} \cdot {y}^{2}\right) + \color{blue}{\left(\cos x \cdot \left(\frac{1}{5040} \cdot {y}^{2} + \frac{1}{120}\right)\right)} \cdot \left({y}^{2} \cdot {y}^{2}\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \cos x \cdot 1 + \left(\cos x \cdot \left(\frac{1}{6} \cdot {y}^{2}\right) + \left(\cos x \cdot \color{blue}{\left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)}\right) \cdot \left({y}^{2} \cdot {y}^{2}\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \cos x \cdot 1 + \left(\cos x \cdot \left(\frac{1}{6} \cdot {y}^{2}\right) + \color{blue}{\cos x \cdot \left(\left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right) \cdot \left({y}^{2} \cdot {y}^{2}\right)\right)}\right) \]

      11. *-commutative N/A

          \[\leadsto \cos x \cdot 1 + \left(\cos x \cdot \left(\frac{1}{6} \cdot {y}^{2}\right) + \cos x \cdot \color{blue}{\left(\left({y}^{2} \cdot {y}^{2}\right) \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)}\right) \]

   5. Simplified 92.6%

      \[\leadsto \color{blue}{\cos x \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + \left({x}^{2} \cdot \left(\frac{-1}{2} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right) + \frac{1}{24} \cdot \left({x}^{2} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right) + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)} \]

   8. Simplified 1.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.041666666666666664, -0.5\right), 1\right)} \]

   9. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{-1}{2}}, 1\right) \]
   10. Step-by-step derivation

   11. Simplified 53.9%

       \[\leadsto \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right) \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{-0.5}, 1\right) \]

   if -0.0200000000000000004 < (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 84.4%

      \[\leadsto \color{blue}{1} \cdot \frac{\sinh y}{y} \]

   6. Taylor expanded in y around 0

      \[\leadsto 1 \cdot \frac{\color{blue}{y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)}}{y} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto 1 \cdot \frac{y \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) + 1\right)}}{y} \]

      2. distribute-rgt-in N/A

         \[\leadsto 1 \cdot \frac{\color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right) \cdot y + 1 \cdot y}}{y} \]

      3. *-lft-identity N/A

         \[\leadsto 1 \cdot \frac{\left({y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right) \cdot y + \color{blue}{y}}{y} \]

      4. associate-*l* N/A

         \[\leadsto 1 \cdot \frac{\color{blue}{{y}^{2} \cdot \left(\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot y\right)} + y}{y} \]

      5. *-commutative N/A

         \[\leadsto 1 \cdot \frac{{y}^{2} \cdot \color{blue}{\left(y \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)} + y}{y} \]

      6. lower-fma.f64 N/A

         \[\leadsto 1 \cdot \frac{\color{blue}{\mathsf{fma}\left({y}^{2}, y \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right), y\right)}}{y} \]

   8. Simplified 75.2%

      \[\leadsto 1 \cdot \frac{\color{blue}{\mathsf{fma}\left(y \cdot y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y\right)}}{y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos x \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right) \cdot \mathsf{fma}\left(x \cdot x, -0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y\right)}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 70.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos x \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right) \cdot \mathsf{fma}\left(-0.5, x \cdot x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (cos x) -0.02)
   (*
    (fma (* y y) (fma y (* y 0.008333333333333333) 0.16666666666666666) 1.0)
    (fma -0.5 (* x x) 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) <= -0.02) {
		tmp = fma((y * y), fma(y, (y * 0.008333333333333333), 0.16666666666666666), 1.0) * fma(-0.5, (x * x), 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) <= -0.02)
		tmp = Float64(fma(Float64(y * y), fma(y, Float64(y * 0.008333333333333333), 0.16666666666666666), 1.0) * fma(-0.5, Float64(x * x), 1.0));
	else
		tmp = fma(Float64(y * y), fma(y, Float64(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], -0.02], N[(N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * N[(-0.5 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 x) < -0.0200000000000000004

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\cos x + {y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right) + \frac{1}{6} \cdot \cos x\right)} \]
   4. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto \color{blue}{\cos x \cdot 1} + {y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right) + \frac{1}{6} \cdot \cos x\right) \]

      2. *-commutative N/A

         \[\leadsto \cos x \cdot 1 + \color{blue}{\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right) + \frac{1}{6} \cdot \cos x\right) \cdot {y}^{2}} \]

      3. associate-*r* N/A

         \[\leadsto \cos x \cdot 1 + \left(\color{blue}{\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot \cos x} + \frac{1}{6} \cdot \cos x\right) \cdot {y}^{2} \]

      4. distribute-rgt-out N/A

         \[\leadsto \cos x \cdot 1 + \color{blue}{\left(\cos x \cdot \left(\frac{1}{120} \cdot {y}^{2} + \frac{1}{6}\right)\right)} \cdot {y}^{2} \]

      5. +-commutative N/A

         \[\leadsto \cos x \cdot 1 + \left(\cos x \cdot \color{blue}{\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)}\right) \cdot {y}^{2} \]

      6. associate-*l* N/A

         \[\leadsto \cos x \cdot 1 + \color{blue}{\cos x \cdot \left(\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) \cdot {y}^{2}\right)} \]

      7. *-commutative N/A

         \[\leadsto \cos x \cdot 1 + \cos x \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} \]

      8. distribute-lft-in N/A

         \[\leadsto \color{blue}{\cos x \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{\cos x \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} \]

      10. lower-cos.f64 N/A

          \[\leadsto \color{blue}{\cos x} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \cos x \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) + 1\right)} \]

      12. lower-fma.f64 N/A

          \[\leadsto \cos x \cdot \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{1}{6} + \frac{1}{120} \cdot {y}^{2}, 1\right)} \]

   5. Simplified 92.4%

      \[\leadsto \color{blue}{\cos x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + \frac{-1}{2} \cdot {x}^{2}\right)} \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{2} \cdot {x}^{2} + 1\right)} \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1}{2}, {x}^{2}, 1\right)} \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{x \cdot x}, 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      4. lower-*.f64 52.6

         \[\leadsto \mathsf{fma}\left(-0.5, \color{blue}{x \cdot x}, 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right) \]

   8. Simplified 52.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, x \cdot x, 1\right)} \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right) \]

   if -0.0200000000000000004 < (cos.f64 x)

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\cos x + {y}^{2} \cdot \left(\frac{1}{6} \cdot \cos x + {y}^{2} \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \cos x\right) + \frac{1}{120} \cdot \cos x\right)\right)} \]
   4. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto \color{blue}{\cos x \cdot 1} + {y}^{2} \cdot \left(\frac{1}{6} \cdot \cos x + {y}^{2} \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \cos x\right) + \frac{1}{120} \cdot \cos x\right)\right) \]

      2. distribute-rgt-in N/A

         \[\leadsto \cos x \cdot 1 + \color{blue}{\left(\left(\frac{1}{6} \cdot \cos x\right) \cdot {y}^{2} + \left({y}^{2} \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \cos x\right) + \frac{1}{120} \cdot \cos x\right)\right) \cdot {y}^{2}\right)} \]

      3. *-commutative N/A

         \[\leadsto \cos x \cdot 1 + \left(\color{blue}{\left(\cos x \cdot \frac{1}{6}\right)} \cdot {y}^{2} + \left({y}^{2} \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \cos x\right) + \frac{1}{120} \cdot \cos x\right)\right) \cdot {y}^{2}\right) \]

      4. associate-*l* N/A

         \[\leadsto \cos x \cdot 1 + \left(\color{blue}{\cos x \cdot \left(\frac{1}{6} \cdot {y}^{2}\right)} + \left({y}^{2} \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \cos x\right) + \frac{1}{120} \cdot \cos x\right)\right) \cdot {y}^{2}\right) \]

      5. *-commutative N/A

         \[\leadsto \cos x \cdot 1 + \left(\cos x \cdot \left(\frac{1}{6} \cdot {y}^{2}\right) + \color{blue}{\left(\left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \cos x\right) + \frac{1}{120} \cdot \cos x\right) \cdot {y}^{2}\right)} \cdot {y}^{2}\right) \]

      6. associate-*l* N/A

         \[\leadsto \cos x \cdot 1 + \left(\cos x \cdot \left(\frac{1}{6} \cdot {y}^{2}\right) + \color{blue}{\left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \cos x\right) + \frac{1}{120} \cdot \cos x\right) \cdot \left({y}^{2} \cdot {y}^{2}\right)}\right) \]

      7. associate-*r* N/A

         \[\leadsto \cos x \cdot 1 + \left(\cos x \cdot \left(\frac{1}{6} \cdot {y}^{2}\right) + \left(\color{blue}{\left(\frac{1}{5040} \cdot {y}^{2}\right) \cdot \cos x} + \frac{1}{120} \cdot \cos x\right) \cdot \left({y}^{2} \cdot {y}^{2}\right)\right) \]

      8. distribute-rgt-out N/A

         \[\leadsto \cos x \cdot 1 + \left(\cos x \cdot \left(\frac{1}{6} \cdot {y}^{2}\right) + \color{blue}{\left(\cos x \cdot \left(\frac{1}{5040} \cdot {y}^{2} + \frac{1}{120}\right)\right)} \cdot \left({y}^{2} \cdot {y}^{2}\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \cos x \cdot 1 + \left(\cos x \cdot \left(\frac{1}{6} \cdot {y}^{2}\right) + \left(\cos x \cdot \color{blue}{\left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)}\right) \cdot \left({y}^{2} \cdot {y}^{2}\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \cos x \cdot 1 + \left(\cos x \cdot \left(\frac{1}{6} \cdot {y}^{2}\right) + \color{blue}{\cos x \cdot \left(\left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right) \cdot \left({y}^{2} \cdot {y}^{2}\right)\right)}\right) \]

      11. *-commutative N/A

          \[\leadsto \cos x \cdot 1 + \left(\cos x \cdot \left(\frac{1}{6} \cdot {y}^{2}\right) + \cos x \cdot \color{blue}{\left(\left({y}^{2} \cdot {y}^{2}\right) \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)}\right) \]

   5. Simplified 88.9%

      \[\leadsto \color{blue}{\cos x \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)} \]
   6. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \color{blue}{\cos x} \cdot \left(y \cdot \left(y \cdot \left(y \cdot \left(y \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{5040} + \frac{1}{120}\right)\right) + \frac{1}{6}\right)\right) + 1\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \cos x \cdot \left(y \cdot \left(y \cdot \left(y \cdot \left(y \cdot \left(\color{blue}{\left(y \cdot y\right)} \cdot \frac{1}{5040} + \frac{1}{120}\right)\right) + \frac{1}{6}\right)\right) + 1\right) \]

      3. lift-fma.f64 N/A

         \[\leadsto \cos x \cdot \left(y \cdot \left(y \cdot \left(y \cdot \left(y \cdot \color{blue}{\mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right)}\right) + \frac{1}{6}\right)\right) + 1\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \cos x \cdot \left(y \cdot \left(y \cdot \left(y \cdot \color{blue}{\left(y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right)\right)} + \frac{1}{6}\right)\right) + 1\right) \]

      5. lift-fma.f64 N/A

         \[\leadsto \cos x \cdot \left(y \cdot \left(y \cdot \color{blue}{\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right)}\right) + 1\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \cos x \cdot \left(y \cdot \color{blue}{\left(y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right)\right)} + 1\right) \]

      7. flip-+ N/A

         \[\leadsto \cos x \cdot \color{blue}{\frac{\left(y \cdot \left(y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right)\right)\right) \cdot \left(y \cdot \left(y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right)\right)\right) - 1 \cdot 1}{y \cdot \left(y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right)\right) - 1}} \]

      8. clear-num N/A

         \[\leadsto \cos x \cdot \color{blue}{\frac{1}{\frac{y \cdot \left(y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right)\right) - 1}{\left(y \cdot \left(y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right)\right)\right) \cdot \left(y \cdot \left(y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right)\right)\right) - 1 \cdot 1}}} \]

   7. Applied egg-rr 88.9%

      \[\leadsto \color{blue}{\frac{\cos x}{\frac{1}{\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)}}} \]

   8. Taylor expanded in x 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. lower-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. lower-*.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. lower-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. lower-*.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. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \left(\color{blue}{{y}^{2} \cdot \frac{1}{5040}} + \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{1}{5040}, \frac{1}{120}\right)}, \frac{1}{6}\right), 1\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      14. lower-*.f64 73.3

          \[\leadsto \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right) \]

   10. Simplified 73.3%

       \[\leadsto \color{blue}{\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos x \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right) \cdot \mathsf{fma}\left(-0.5, x \cdot x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 53.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos x \leq -0.02:\\ \;\;\;\;-0.5 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, y \cdot 0.16666666666666666, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (cos x) -0.02)
   (* -0.5 (* x x))
   (fma y (* y 0.16666666666666666) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (cos(x) <= -0.02) {
		tmp = -0.5 * (x * x);
	} else {
		tmp = fma(y, (y * 0.16666666666666666), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (cos(x) <= -0.02)
		tmp = Float64(-0.5 * Float64(x * x));
	else
		tmp = fma(y, Float64(y * 0.16666666666666666), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[Cos[x], $MachinePrecision], -0.02], N[(-0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision], N[(y * N[(y * 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 x) < -0.0200000000000000004

   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. lower-cos.f64 46.9

         \[\leadsto \color{blue}{\cos x} \]

   5. Simplified 46.9%

      \[\leadsto \color{blue}{\cos x} \]

   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. lower-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. lower-*.f64 28.0

         \[\leadsto \mathsf{fma}\left(-0.5, \color{blue}{x \cdot x}, 1\right) \]

   8. Simplified 28.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, x \cdot x, 1\right)} \]

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot {x}^{2}} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{-1}{2} \cdot {x}^{2}} \]

       2. unpow2 N/A

          \[\leadsto \frac{-1}{2} \cdot \color{blue}{\left(x \cdot x\right)} \]

       3. lower-*.f64 28.0

          \[\leadsto -0.5 \cdot \color{blue}{\left(x \cdot x\right)} \]

   11. Simplified 28.0%

       \[\leadsto \color{blue}{-0.5 \cdot \left(x \cdot x\right)} \]

   if -0.0200000000000000004 < (cos.f64 x)

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\cos x + {y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right) + \frac{1}{6} \cdot \cos x\right)} \]
   4. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto \color{blue}{\cos x \cdot 1} + {y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right) + \frac{1}{6} \cdot \cos x\right) \]

      2. *-commutative N/A

         \[\leadsto \cos x \cdot 1 + \color{blue}{\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right) + \frac{1}{6} \cdot \cos x\right) \cdot {y}^{2}} \]

      3. associate-*r* N/A

         \[\leadsto \cos x \cdot 1 + \left(\color{blue}{\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot \cos x} + \frac{1}{6} \cdot \cos x\right) \cdot {y}^{2} \]

      4. distribute-rgt-out N/A

         \[\leadsto \cos x \cdot 1 + \color{blue}{\left(\cos x \cdot \left(\frac{1}{120} \cdot {y}^{2} + \frac{1}{6}\right)\right)} \cdot {y}^{2} \]

      5. +-commutative N/A

         \[\leadsto \cos x \cdot 1 + \left(\cos x \cdot \color{blue}{\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)}\right) \cdot {y}^{2} \]

      6. associate-*l* N/A

         \[\leadsto \cos x \cdot 1 + \color{blue}{\cos x \cdot \left(\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) \cdot {y}^{2}\right)} \]

      7. *-commutative N/A

         \[\leadsto \cos x \cdot 1 + \cos x \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} \]

      8. distribute-lft-in N/A

         \[\leadsto \color{blue}{\cos x \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{\cos x \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} \]

      10. lower-cos.f64 N/A

          \[\leadsto \color{blue}{\cos x} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \cos x \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) + 1\right)} \]

      12. lower-fma.f64 N/A

          \[\leadsto \cos x \cdot \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{1}{6} + \frac{1}{120} \cdot {y}^{2}, 1\right)} \]

   5. Simplified 86.2%

      \[\leadsto \color{blue}{\cos x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]
   7. Step-by-step derivation

   8. Simplified 70.6%

      \[\leadsto \color{blue}{1} \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right) \]

   9. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1 + \frac{1}{6} \cdot {y}^{2}} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \color{blue}{\frac{1}{6} \cdot {y}^{2} + 1} \]

       2. *-commutative N/A

          \[\leadsto \color{blue}{{y}^{2} \cdot \frac{1}{6}} + 1 \]

       3. unpow2 N/A

          \[\leadsto \color{blue}{\left(y \cdot y\right)} \cdot \frac{1}{6} + 1 \]

       4. associate-*l* N/A

          \[\leadsto \color{blue}{y \cdot \left(y \cdot \frac{1}{6}\right)} + 1 \]

       5. *-commutative N/A

          \[\leadsto y \cdot \color{blue}{\left(\frac{1}{6} \cdot y\right)} + 1 \]

       6. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{1}{6} \cdot y, 1\right)} \]

       7. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(y, \color{blue}{y \cdot \frac{1}{6}}, 1\right) \]

       8. lower-*.f64 60.7

          \[\leadsto \mathsf{fma}\left(y, \color{blue}{y \cdot 0.16666666666666666}, 1\right) \]

   11. Simplified 60.7%

       \[\leadsto \color{blue}{\mathsf{fma}\left(y, y \cdot 0.16666666666666666, 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 25: 35.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos x \leq -0.02:\\ \;\;\;\;-0.5 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= (cos x) -0.02) (* -0.5 (* x x)) 1.0))

C

double code(double x, double y) {
	double tmp;
	if (cos(x) <= -0.02) {
		tmp = -0.5 * (x * x);
	} else {
		tmp = 1.0;
	}
	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) <= (-0.02d0)) then
        tmp = (-0.5d0) * (x * x)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (Math.cos(x) <= -0.02) {
		tmp = -0.5 * (x * x);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if math.cos(x) <= -0.02:
		tmp = -0.5 * (x * x)
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (cos(x) <= -0.02)
		tmp = Float64(-0.5 * Float64(x * x));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (cos(x) <= -0.02)
		tmp = -0.5 * (x * x);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[Cos[x], $MachinePrecision], -0.02], N[(-0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 x) < -0.0200000000000000004

   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. lower-cos.f64 46.9

         \[\leadsto \color{blue}{\cos x} \]

   5. Simplified 46.9%

      \[\leadsto \color{blue}{\cos x} \]

   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. lower-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. lower-*.f64 28.0

         \[\leadsto \mathsf{fma}\left(-0.5, \color{blue}{x \cdot x}, 1\right) \]

   8. Simplified 28.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, x \cdot x, 1\right)} \]

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot {x}^{2}} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{-1}{2} \cdot {x}^{2}} \]

       2. unpow2 N/A

          \[\leadsto \frac{-1}{2} \cdot \color{blue}{\left(x \cdot x\right)} \]

       3. lower-*.f64 28.0

          \[\leadsto -0.5 \cdot \color{blue}{\left(x \cdot x\right)} \]

   11. Simplified 28.0%

       \[\leadsto \color{blue}{-0.5 \cdot \left(x \cdot x\right)} \]

   if -0.0200000000000000004 < (cos.f64 x)

   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. lower-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

      \[\leadsto \color{blue}{1} \]
   7. Step-by-step derivation

   8. Simplified 33.7%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 26: 28.2% 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. lower-cos.f64 48.6

      \[\leadsto \color{blue}{\cos x} \]

5. Simplified 48.6%

   \[\leadsto \color{blue}{\cos x} \]

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1} \]
7. Step-by-step derivation

8. Simplified 24.4%

   \[\leadsto \color{blue}{1} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024212 
(FPCore (x y)
  :name "Linear.Quaternion:$csin from linear-1.19.1.3"
  :precision binary64
  (* (cos x) (/ (sinh y) y)))