Linear.Quaternion:$csin from linear-1.19.1.3

Percentage Accurate: 100.0% → 100.0%

Time: 13.2s

Alternatives: 28

Speedup: 1.0×

• 50.4% 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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y}{y}\\ t_1 := \cos x \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+204}:\\ \;\;\;\;t\_0 \cdot \left(-0.5 \cdot \left(x \cdot x\right)\right)\\ \mathbf{elif}\;t\_1 \leq 0.9999999999985344:\\ \;\;\;\;\cos x \cdot \frac{\mathsf{fma}\left(y, 0.16666666666666666 \cdot \left(y \cdot y\right), y\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sinh y) y)) (t_1 (* (cos x) t_0)))
   (if (<= t_1 -5e+204)
     (* t_0 (* -0.5 (* x x)))
     (if (<= t_1 0.9999999999985344)
       (* (cos x) (/ (fma y (* 0.16666666666666666 (* y y)) y) y))
       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 <= -5e+204) {
		tmp = t_0 * (-0.5 * (x * x));
	} else if (t_1 <= 0.9999999999985344) {
		tmp = cos(x) * (fma(y, (0.16666666666666666 * (y * y)), y) / y);
	} 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 <= -5e+204)
		tmp = Float64(t_0 * Float64(-0.5 * Float64(x * x)));
	elseif (t_1 <= 0.9999999999985344)
		tmp = Float64(cos(x) * Float64(fma(y, Float64(0.16666666666666666 * Float64(y * y)), y) / y));
	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, -5e+204], N[(t$95$0 * N[(-0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.9999999999985344], N[(N[Cos[x], $MachinePrecision] * N[(N[(y * N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (cos.f64 x) (/.f64 (sinh.f64 y) y)) < -5.00000000000000008e204

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 96.4

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

   5. Simplified 96.4%

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

   6. Taylor expanded in x around inf

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 96.4

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

   8. Simplified 96.4%

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

   if -5.00000000000000008e204 < (*.f64 (cos.f64 x) (/.f64 (sinh.f64 y) y)) < 0.999999999998534395

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

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

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

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

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 100.0

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

   5. Simplified 100.0%

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

   if 0.999999999998534395 < (*.f64 (cos.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 100.0%

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

      1. *-lft-identity N/A

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

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

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

      3. sinh-lowering-sinh.f64 100.0

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

   7. Applied egg-rr 100.0%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos x \cdot \frac{\sinh y}{y} \leq -5 \cdot 10^{+204}:\\ \;\;\;\;\frac{\sinh y}{y} \cdot \left(-0.5 \cdot \left(x \cdot x\right)\right)\\ \mathbf{elif}\;\cos x \cdot \frac{\sinh y}{y} \leq 0.9999999999985344:\\ \;\;\;\;\cos x \cdot \frac{\mathsf{fma}\left(y, 0.16666666666666666 \cdot \left(y \cdot y\right), y\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sinh y}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y}{y}\\ t_1 := \cos x \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+204}:\\ \;\;\;\;t\_0 \cdot \left(-0.5 \cdot \left(x \cdot x\right)\right)\\ \mathbf{elif}\;t\_1 \leq 1.00001:\\ \;\;\;\;\cos x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sinh y) y)) (t_1 (* (cos x) t_0)))
   (if (<= t_1 -5e+204)
     (* t_0 (* -0.5 (* x x)))
     (if (<= t_1 1.00001)
       (*
        (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 <= -5e+204) {
		tmp = t_0 * (-0.5 * (x * x));
	} else if (t_1 <= 1.00001) {
		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 <= -5e+204)
		tmp = Float64(t_0 * Float64(-0.5 * Float64(x * x)));
	elseif (t_1 <= 1.00001)
		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, -5e+204], N[(t$95$0 * N[(-0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1.00001], 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)) < -5.00000000000000008e204

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 96.4

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

   5. Simplified 96.4%

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

   6. Taylor expanded in x around inf

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 96.4

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

   8. Simplified 96.4%

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

   if -5.00000000000000008e204 < (*.f64 (cos.f64 x) (/.f64 (sinh.f64 y) y)) < 1.0000100000000001

   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. *-lowering-*.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. cos-lowering-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. accelerator-lowering-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 100.0%

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

   if 1.0000100000000001 < (*.f64 (cos.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 100.0%

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

      1. *-lft-identity N/A

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

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

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

      3. sinh-lowering-sinh.f64 100.0

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

   7. Applied egg-rr 100.0%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos x \cdot \frac{\sinh y}{y} \leq -5 \cdot 10^{+204}:\\ \;\;\;\;\frac{\sinh y}{y} \cdot \left(-0.5 \cdot \left(x \cdot x\right)\right)\\ \mathbf{elif}\;\cos x \cdot \frac{\sinh y}{y} \leq 1.00001:\\ \;\;\;\;\cos x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sinh y}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y}{y}\\ t_1 := \cos x \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+204}:\\ \;\;\;\;t\_0 \cdot \left(-0.5 \cdot \left(x \cdot x\right)\right)\\ \mathbf{elif}\;t\_1 \leq 0.9999999999985344:\\ \;\;\;\;\cos x \cdot \mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sinh y) y)) (t_1 (* (cos x) t_0)))
   (if (<= t_1 -5e+204)
     (* t_0 (* -0.5 (* x x)))
     (if (<= t_1 0.9999999999985344)
       (* (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 <= -5e+204) {
		tmp = t_0 * (-0.5 * (x * x));
	} else if (t_1 <= 0.9999999999985344) {
		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 <= -5e+204)
		tmp = Float64(t_0 * Float64(-0.5 * Float64(x * x)));
	elseif (t_1 <= 0.9999999999985344)
		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, -5e+204], N[(t$95$0 * N[(-0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.9999999999985344], 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)) < -5.00000000000000008e204

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 96.4

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

   5. Simplified 96.4%

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

   6. Taylor expanded in x around inf

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 96.4

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

   8. Simplified 96.4%

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

   if -5.00000000000000008e204 < (*.f64 (cos.f64 x) (/.f64 (sinh.f64 y) y)) < 0.999999999998534395

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-lft-identity N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-in N/A

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

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

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

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

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

      6. +-commutative N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 100.0

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

   5. Simplified 100.0%

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

   if 0.999999999998534395 < (*.f64 (cos.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 100.0%

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

      1. *-lft-identity N/A

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

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

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

      3. sinh-lowering-sinh.f64 100.0

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

   7. Applied egg-rr 100.0%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos x \cdot \frac{\sinh y}{y} \leq -5 \cdot 10^{+204}:\\ \;\;\;\;\frac{\sinh y}{y} \cdot \left(-0.5 \cdot \left(x \cdot x\right)\right)\\ \mathbf{elif}\;\cos x \cdot \frac{\sinh y}{y} \leq 0.9999999999985344:\\ \;\;\;\;\cos x \cdot \mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sinh y}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y}{y}\\ t_1 := \cos x \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+204}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right) \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right)\\ \mathbf{elif}\;t\_1 \leq 0.9999999999985344:\\ \;\;\;\;\cos x \cdot \mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sinh y) y)) (t_1 (* (cos x) t_0)))
   (if (<= t_1 -5e+204)
     (*
      (fma (* y y) (fma y (* y 0.008333333333333333) 0.16666666666666666) 1.0)
      (fma
       x
       (*
        x
        (fma
         (* x x)
         (fma (* x x) -0.001388888888888889 0.041666666666666664)
         -0.5))
       1.0))
     (if (<= t_1 0.9999999999985344)
       (* (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 <= -5e+204) {
		tmp = fma((y * y), fma(y, (y * 0.008333333333333333), 0.16666666666666666), 1.0) * fma(x, (x * fma((x * x), fma((x * x), -0.001388888888888889, 0.041666666666666664), -0.5)), 1.0);
	} else if (t_1 <= 0.9999999999985344) {
		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 <= -5e+204)
		tmp = Float64(fma(Float64(y * y), fma(y, Float64(y * 0.008333333333333333), 0.16666666666666666), 1.0) * fma(x, Float64(x * fma(Float64(x * x), fma(Float64(x * x), -0.001388888888888889, 0.041666666666666664), -0.5)), 1.0));
	elseif (t_1 <= 0.9999999999985344)
		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, -5e+204], N[(N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.001388888888888889 + 0.041666666666666664), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.9999999999985344], 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)) < -5.00000000000000008e204

   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. *-lowering-*.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. cos-lowering-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. accelerator-lowering-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 74.4%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)\right)} \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right) + 1\right)} \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      2. unpow2 N/A

         \[\leadsto \left(\color{blue}{\left(x \cdot x\right)} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right) + 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      3. associate-*l* N/A

         \[\leadsto \left(\color{blue}{x \cdot \left(x \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)\right)} + 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right), 1\right)} \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

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

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)}, 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(x, x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)}, 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(x, x \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) + \color{blue}{\frac{-1}{2}}\right), 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

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

         \[\leadsto \mathsf{fma}\left(x, x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}, \frac{-1}{2}\right)}, 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}, \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

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

          \[\leadsto \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}, \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{-1}{720} \cdot {x}^{2} + \frac{1}{24}}, \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{-1}{720}} + \frac{1}{24}, \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

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

          \[\leadsto \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{-1}{720}, \frac{1}{24}\right)}, \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{720}, \frac{1}{24}\right), \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      15. *-lowering-*.f64 92.7

          \[\leadsto \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right) \]

   8. Simplified 92.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right)} \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right) \]

   if -5.00000000000000008e204 < (*.f64 (cos.f64 x) (/.f64 (sinh.f64 y) y)) < 0.999999999998534395

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-lft-identity N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-in N/A

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

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

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

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

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

      6. +-commutative N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 100.0

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

   5. Simplified 100.0%

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

   if 0.999999999998534395 < (*.f64 (cos.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 100.0%

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

      1. *-lft-identity N/A

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

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

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

      3. sinh-lowering-sinh.f64 100.0

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

   7. Applied egg-rr 100.0%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos x \cdot \frac{\sinh y}{y} \leq -5 \cdot 10^{+204}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right) \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right)\\ \mathbf{elif}\;\cos x \cdot \frac{\sinh y}{y} \leq 0.9999999999985344:\\ \;\;\;\;\cos x \cdot \mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sinh y}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y}{y}\\ t_1 := \cos x \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+204}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right) \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right)\\ \mathbf{elif}\;t\_1 \leq 0.9999999999985344:\\ \;\;\;\;\cos x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sinh y) y)) (t_1 (* (cos x) t_0)))
   (if (<= t_1 -5e+204)
     (*
      (fma (* y y) (fma y (* y 0.008333333333333333) 0.16666666666666666) 1.0)
      (fma
       x
       (*
        x
        (fma
         (* x x)
         (fma (* x x) -0.001388888888888889 0.041666666666666664)
         -0.5))
       1.0))
     (if (<= t_1 0.9999999999985344) (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 <= -5e+204) {
		tmp = fma((y * y), fma(y, (y * 0.008333333333333333), 0.16666666666666666), 1.0) * fma(x, (x * fma((x * x), fma((x * x), -0.001388888888888889, 0.041666666666666664), -0.5)), 1.0);
	} else if (t_1 <= 0.9999999999985344) {
		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 <= -5e+204)
		tmp = Float64(fma(Float64(y * y), fma(y, Float64(y * 0.008333333333333333), 0.16666666666666666), 1.0) * fma(x, Float64(x * fma(Float64(x * x), fma(Float64(x * x), -0.001388888888888889, 0.041666666666666664), -0.5)), 1.0));
	elseif (t_1 <= 0.9999999999985344)
		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, -5e+204], N[(N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.001388888888888889 + 0.041666666666666664), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.9999999999985344], 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)) < -5.00000000000000008e204

   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. *-lowering-*.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. cos-lowering-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. accelerator-lowering-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 74.4%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)\right)} \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right) + 1\right)} \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      2. unpow2 N/A

         \[\leadsto \left(\color{blue}{\left(x \cdot x\right)} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right) + 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      3. associate-*l* N/A

         \[\leadsto \left(\color{blue}{x \cdot \left(x \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)\right)} + 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right), 1\right)} \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

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

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)}, 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(x, x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)}, 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(x, x \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) + \color{blue}{\frac{-1}{2}}\right), 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

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

         \[\leadsto \mathsf{fma}\left(x, x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}, \frac{-1}{2}\right)}, 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}, \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

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

          \[\leadsto \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}, \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{-1}{720} \cdot {x}^{2} + \frac{1}{24}}, \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{-1}{720}} + \frac{1}{24}, \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

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

          \[\leadsto \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{-1}{720}, \frac{1}{24}\right)}, \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{720}, \frac{1}{24}\right), \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      15. *-lowering-*.f64 92.7

          \[\leadsto \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right) \]

   8. Simplified 92.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right)} \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right) \]

   if -5.00000000000000008e204 < (*.f64 (cos.f64 x) (/.f64 (sinh.f64 y) y)) < 0.999999999998534395

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. cos-lowering-cos.f64 99.4

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

   5. Simplified 99.4%

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

   if 0.999999999998534395 < (*.f64 (cos.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 100.0%

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

      1. *-lft-identity N/A

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

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

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

      3. sinh-lowering-sinh.f64 100.0

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

   7. Applied egg-rr 100.0%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos x \cdot \frac{\sinh y}{y} \leq -5 \cdot 10^{+204}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right) \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right)\\ \mathbf{elif}\;\cos x \cdot \frac{\sinh y}{y} \leq 0.9999999999985344:\\ \;\;\;\;\cos x\\ \mathbf{else}:\\ \;\;\;\;\frac{\sinh y}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 94.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x \cdot \frac{\sinh y}{y}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+204}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right) \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right)\\ \mathbf{elif}\;t\_0 \leq 0.9999999999985344:\\ \;\;\;\;\cos x\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (cos x) (/ (sinh y) y))))
   (if (<= t_0 -5e+204)
     (*
      (fma (* y y) (fma y (* y 0.008333333333333333) 0.16666666666666666) 1.0)
      (fma
       x
       (*
        x
        (fma
         (* x x)
         (fma (* x x) -0.001388888888888889 0.041666666666666664)
         -0.5))
       1.0))
     (if (<= t_0 0.9999999999985344)
       (cos x)
       (/
        (fma
         (fma
          y
          (* y (fma y (* y 0.0001984126984126984) 0.008333333333333333))
          0.16666666666666666)
         (* y (* y y))
         y)
        y)))))

C

double code(double x, double y) {
	double t_0 = cos(x) * (sinh(y) / y);
	double tmp;
	if (t_0 <= -5e+204) {
		tmp = fma((y * y), fma(y, (y * 0.008333333333333333), 0.16666666666666666), 1.0) * fma(x, (x * fma((x * x), fma((x * x), -0.001388888888888889, 0.041666666666666664), -0.5)), 1.0);
	} else if (t_0 <= 0.9999999999985344) {
		tmp = cos(x);
	} else {
		tmp = fma(fma(y, (y * fma(y, (y * 0.0001984126984126984), 0.008333333333333333)), 0.16666666666666666), (y * (y * y)), y) / y;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) * Float64(sinh(y) / y))
	tmp = 0.0
	if (t_0 <= -5e+204)
		tmp = Float64(fma(Float64(y * y), fma(y, Float64(y * 0.008333333333333333), 0.16666666666666666), 1.0) * fma(x, Float64(x * fma(Float64(x * x), fma(Float64(x * x), -0.001388888888888889, 0.041666666666666664), -0.5)), 1.0));
	elseif (t_0 <= 0.9999999999985344)
		tmp = cos(x);
	else
		tmp = Float64(fma(fma(y, Float64(y * fma(y, Float64(y * 0.0001984126984126984), 0.008333333333333333)), 0.16666666666666666), Float64(y * Float64(y * y)), y) / y);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e+204], N[(N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.001388888888888889 + 0.041666666666666664), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.9999999999985344], N[Cos[x], $MachinePrecision], N[(N[(N[(y * N[(y * N[(y * N[(y * 0.0001984126984126984), $MachinePrecision] + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] / y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (cos.f64 x) (/.f64 (sinh.f64 y) y)) < -5.00000000000000008e204

   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. *-lowering-*.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. cos-lowering-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. accelerator-lowering-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 74.4%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)\right)} \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right) + 1\right)} \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      2. unpow2 N/A

         \[\leadsto \left(\color{blue}{\left(x \cdot x\right)} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right) + 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      3. associate-*l* N/A

         \[\leadsto \left(\color{blue}{x \cdot \left(x \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)\right)} + 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right), 1\right)} \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

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

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)}, 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(x, x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)}, 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(x, x \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) + \color{blue}{\frac{-1}{2}}\right), 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

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

         \[\leadsto \mathsf{fma}\left(x, x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}, \frac{-1}{2}\right)}, 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}, \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

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

          \[\leadsto \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}, \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{-1}{720} \cdot {x}^{2} + \frac{1}{24}}, \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{-1}{720}} + \frac{1}{24}, \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

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

          \[\leadsto \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{-1}{720}, \frac{1}{24}\right)}, \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{720}, \frac{1}{24}\right), \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      15. *-lowering-*.f64 92.7

          \[\leadsto \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right) \]

   8. Simplified 92.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right)} \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right) \]

   if -5.00000000000000008e204 < (*.f64 (cos.f64 x) (/.f64 (sinh.f64 y) y)) < 0.999999999998534395

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. cos-lowering-cos.f64 99.4

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

   5. Simplified 99.4%

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

   if 0.999999999998534395 < (*.f64 (cos.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-rgt-identity N/A

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

      7. *-inverses N/A

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

      8. associate-/l* N/A

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

      9. unpow2 N/A

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

      10. associate-/l* N/A

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

      11. pow-sqr N/A

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

      12. metadata-eval N/A

          \[\leadsto 1 \cdot \frac{\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot \frac{{y}^{\color{blue}{4}}}{y} + y}{y} \]

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

          \[\leadsto 1 \cdot \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right), \frac{{y}^{4}}{y}, y\right)}}{y} \]

   8. Simplified 90.6%

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

      1. *-lft-identity N/A

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

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

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

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

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

      4. associate-*l* N/A

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

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

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

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

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

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

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

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

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

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

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

      10. *-lowering-*.f64 90.6

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

   10. Applied egg-rr 90.6%

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

4. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos x \cdot \frac{\sinh y}{y} \leq -5 \cdot 10^{+204}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right) \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right)\\ \mathbf{elif}\;\cos x \cdot \frac{\sinh y}{y} \leq 0.9999999999985344:\\ \;\;\;\;\cos x\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 71.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x \cdot \frac{\sinh y}{y}\\ \mathbf{if}\;t\_0 \leq -0.05:\\ \;\;\;\;y \cdot \left(y \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.16666666666666666\right)\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot 0.008333333333333333, y, 0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right) \cdot \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (cos x) (/ (sinh y) y))))
   (if (<= t_0 -0.05)
     (* y (* y (fma (* x x) -0.08333333333333333 0.16666666666666666)))
     (if (<= t_0 2.0)
       (fma (* y y) (fma (* y 0.008333333333333333) y 0.16666666666666666) 1.0)
       (*
        (fma y (* y 0.0001984126984126984) 0.008333333333333333)
        (* (* y y) (* y y)))))))

C

double code(double x, double y) {
	double t_0 = cos(x) * (sinh(y) / y);
	double tmp;
	if (t_0 <= -0.05) {
		tmp = y * (y * fma((x * x), -0.08333333333333333, 0.16666666666666666));
	} else if (t_0 <= 2.0) {
		tmp = fma((y * y), fma((y * 0.008333333333333333), y, 0.16666666666666666), 1.0);
	} else {
		tmp = fma(y, (y * 0.0001984126984126984), 0.008333333333333333) * ((y * y) * (y * y));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) * Float64(sinh(y) / y))
	tmp = 0.0
	if (t_0 <= -0.05)
		tmp = Float64(y * Float64(y * fma(Float64(x * x), -0.08333333333333333, 0.16666666666666666)));
	elseif (t_0 <= 2.0)
		tmp = fma(Float64(y * y), fma(Float64(y * 0.008333333333333333), y, 0.16666666666666666), 1.0);
	else
		tmp = Float64(fma(y, Float64(y * 0.0001984126984126984), 0.008333333333333333) * Float64(Float64(y * y) * Float64(y * y)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.05], N[(y * N[(y * N[(N[(x * x), $MachinePrecision] * -0.08333333333333333 + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(N[(y * y), $MachinePrecision] * N[(N[(y * 0.008333333333333333), $MachinePrecision] * y + 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(y * N[(y * 0.0001984126984126984), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 45.3

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

   5. Simplified 45.3%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 38.9

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

   8. Simplified 38.9%

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

   9. Taylor expanded in y around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*l* N/A

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

       4. unpow2 N/A

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

       5. associate-*l* N/A

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

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

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

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

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

       8. +-commutative N/A

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

       9. distribute-rgt-in N/A

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

       10. *-commutative N/A

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

       11. associate-*l* N/A

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

       12. metadata-eval N/A

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

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

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

       14. unpow2 N/A

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

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

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

       16. metadata-eval 39.0

           \[\leadsto y \cdot \left(y \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{-0.08333333333333333}, 0.16666666666666666\right)\right) \]

   11. Simplified 39.0%

       \[\leadsto \color{blue}{y \cdot \left(y \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.16666666666666666\right)\right)} \]

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\cos x + {y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right) + \frac{1}{6} \cdot \cos x\right)} \]
   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. *-lowering-*.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. cos-lowering-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. accelerator-lowering-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 100.0%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

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

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

      3. unpow2 N/A

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

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

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 75.4

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

   8. Simplified 75.4%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

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

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

      4. *-lowering-*.f64 75.4

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

   10. Applied egg-rr 75.4%

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

   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 x around 0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

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

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

      3. unpow2 N/A

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

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

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

      5. +-commutative N/A

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

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

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

      7. unpow2 N/A

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

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

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

      9. +-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

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

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

      14. *-commutative N/A

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

      15. *-lowering-*.f64 79.8

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

   8. Simplified 79.8%

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

      1. distribute-lft-in N/A

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

      2. associate-+l+ N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

      10. *-lowering-*.f64 79.8

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

   10. Applied egg-rr 79.8%

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

   11. Taylor expanded in y around inf

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

       1. *-commutative N/A

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

       2. metadata-eval N/A

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

       3. pow-sqr N/A

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

       4. cube-prod N/A

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

       5. unpow2 N/A

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

       6. cube-unmult N/A

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

       7. pow-sqr N/A

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

       8. metadata-eval N/A

          \[\leadsto \left(\frac{1}{5040} + \frac{1}{120} \cdot \frac{1}{{y}^{2}}\right) \cdot \left({y}^{2} \cdot {y}^{\color{blue}{4}}\right) \]

       9. associate-*l* N/A

          \[\leadsto \color{blue}{\left(\left(\frac{1}{5040} + \frac{1}{120} \cdot \frac{1}{{y}^{2}}\right) \cdot {y}^{2}\right) \cdot {y}^{4}} \]

       10. *-commutative N/A

           \[\leadsto \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{5040} + \frac{1}{120} \cdot \frac{1}{{y}^{2}}\right)\right)} \cdot {y}^{4} \]

       11. +-commutative N/A

           \[\leadsto \left({y}^{2} \cdot \color{blue}{\left(\frac{1}{120} \cdot \frac{1}{{y}^{2}} + \frac{1}{5040}\right)}\right) \cdot {y}^{4} \]

       12. distribute-rgt-in N/A

           \[\leadsto \color{blue}{\left(\left(\frac{1}{120} \cdot \frac{1}{{y}^{2}}\right) \cdot {y}^{2} + \frac{1}{5040} \cdot {y}^{2}\right)} \cdot {y}^{4} \]

       13. associate-*l* N/A

           \[\leadsto \left(\color{blue}{\frac{1}{120} \cdot \left(\frac{1}{{y}^{2}} \cdot {y}^{2}\right)} + \frac{1}{5040} \cdot {y}^{2}\right) \cdot {y}^{4} \]

       14. lft-mult-inverse N/A

           \[\leadsto \left(\frac{1}{120} \cdot \color{blue}{1} + \frac{1}{5040} \cdot {y}^{2}\right) \cdot {y}^{4} \]

       15. metadata-eval N/A

           \[\leadsto \left(\color{blue}{\frac{1}{120}} + \frac{1}{5040} \cdot {y}^{2}\right) \cdot {y}^{4} \]

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

           \[\leadsto \color{blue}{\left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right) \cdot {y}^{4}} \]

   13. Simplified 79.8%

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

Alternative 9: 71.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x \cdot \frac{\sinh y}{y}\\ \mathbf{if}\;t\_0 \leq -0.05:\\ \;\;\;\;y \cdot \left(y \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.16666666666666666\right)\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot 0.008333333333333333, y, 0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;0.0001984126984126984 \cdot \left(\left(y \cdot y\right) \cdot \left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (cos x) (/ (sinh y) y))))
   (if (<= t_0 -0.05)
     (* y (* y (fma (* x x) -0.08333333333333333 0.16666666666666666)))
     (if (<= t_0 2.0)
       (fma (* y y) (fma (* y 0.008333333333333333) y 0.16666666666666666) 1.0)
       (* 0.0001984126984126984 (* (* y y) (* y (* y (* y y)))))))))

C

double code(double x, double y) {
	double t_0 = cos(x) * (sinh(y) / y);
	double tmp;
	if (t_0 <= -0.05) {
		tmp = y * (y * fma((x * x), -0.08333333333333333, 0.16666666666666666));
	} else if (t_0 <= 2.0) {
		tmp = fma((y * y), fma((y * 0.008333333333333333), y, 0.16666666666666666), 1.0);
	} else {
		tmp = 0.0001984126984126984 * ((y * y) * (y * (y * (y * y))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) * Float64(sinh(y) / y))
	tmp = 0.0
	if (t_0 <= -0.05)
		tmp = Float64(y * Float64(y * fma(Float64(x * x), -0.08333333333333333, 0.16666666666666666)));
	elseif (t_0 <= 2.0)
		tmp = fma(Float64(y * y), fma(Float64(y * 0.008333333333333333), y, 0.16666666666666666), 1.0);
	else
		tmp = Float64(0.0001984126984126984 * Float64(Float64(y * y) * Float64(y * Float64(y * Float64(y * y)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.05], N[(y * N[(y * N[(N[(x * x), $MachinePrecision] * -0.08333333333333333 + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(N[(y * y), $MachinePrecision] * N[(N[(y * 0.008333333333333333), $MachinePrecision] * y + 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision], N[(0.0001984126984126984 * N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 45.3

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

   5. Simplified 45.3%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 38.9

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

   8. Simplified 38.9%

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

   9. Taylor expanded in y around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*l* N/A

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

       4. unpow2 N/A

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

       5. associate-*l* N/A

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

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

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

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

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

       8. +-commutative N/A

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

       9. distribute-rgt-in N/A

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

       10. *-commutative N/A

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

       11. associate-*l* N/A

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

       12. metadata-eval N/A

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

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

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

       14. unpow2 N/A

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

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

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

       16. metadata-eval 39.0

           \[\leadsto y \cdot \left(y \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{-0.08333333333333333}, 0.16666666666666666\right)\right) \]

   11. Simplified 39.0%

       \[\leadsto \color{blue}{y \cdot \left(y \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.16666666666666666\right)\right)} \]

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\cos x + {y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right) + \frac{1}{6} \cdot \cos x\right)} \]
   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. *-lowering-*.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. cos-lowering-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. accelerator-lowering-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 100.0%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

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

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

      3. unpow2 N/A

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

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

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 75.4

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

   8. Simplified 75.4%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

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

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

      4. *-lowering-*.f64 75.4

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

   10. Applied egg-rr 75.4%

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

   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 x around 0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

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

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

      3. unpow2 N/A

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

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

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

      5. +-commutative N/A

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

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

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

      7. unpow2 N/A

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

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

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

      9. +-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

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

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

      14. *-commutative N/A

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

      15. *-lowering-*.f64 79.8

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

   8. Simplified 79.8%

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

   9. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{1}{5040} \cdot {y}^{6}} \]
   10. Step-by-step derivation

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

          \[\leadsto \color{blue}{\frac{1}{5040} \cdot {y}^{6}} \]

       2. metadata-eval N/A

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

       3. pow-sqr N/A

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

       4. cube-prod N/A

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

       5. unpow2 N/A

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

       6. cube-mult N/A

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

       7. pow-sqr N/A

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

       8. metadata-eval N/A

          \[\leadsto \frac{1}{5040} \cdot \left({y}^{2} \cdot {y}^{\color{blue}{4}}\right) \]

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

          \[\leadsto \frac{1}{5040} \cdot \color{blue}{\left({y}^{2} \cdot {y}^{4}\right)} \]

       10. unpow2 N/A

           \[\leadsto \frac{1}{5040} \cdot \left(\color{blue}{\left(y \cdot y\right)} \cdot {y}^{4}\right) \]

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

           \[\leadsto \frac{1}{5040} \cdot \left(\color{blue}{\left(y \cdot y\right)} \cdot {y}^{4}\right) \]

       12. /-rgt-identity N/A

           \[\leadsto \frac{1}{5040} \cdot \left(\left(y \cdot y\right) \cdot \color{blue}{\frac{{y}^{4}}{1}}\right) \]

       13. lft-mult-inverse N/A

           \[\leadsto \frac{1}{5040} \cdot \left(\left(y \cdot y\right) \cdot \frac{{y}^{4}}{\color{blue}{\frac{1}{{y}^{2}} \cdot {y}^{2}}}\right) \]

       14. associate-/l/ N/A

           \[\leadsto \frac{1}{5040} \cdot \left(\left(y \cdot y\right) \cdot \color{blue}{\frac{\frac{{y}^{4}}{{y}^{2}}}{\frac{1}{{y}^{2}}}}\right) \]

   11. Simplified 79.8%

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

Alternative 10: 67.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x \cdot \frac{\sinh y}{y}\\ \mathbf{if}\;t\_0 \leq -0.05:\\ \;\;\;\;y \cdot \left(y \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.16666666666666666\right)\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (cos x) (/ (sinh y) y))))
   (if (<= t_0 -0.05)
     (* y (* y (fma (* x x) -0.08333333333333333 0.16666666666666666)))
     (if (<= t_0 2.0)
       (fma (* y y) 0.16666666666666666 1.0)
       (* y (* y (fma (* y y) 0.008333333333333333 0.16666666666666666)))))))

C

double code(double x, double y) {
	double t_0 = cos(x) * (sinh(y) / y);
	double tmp;
	if (t_0 <= -0.05) {
		tmp = y * (y * fma((x * x), -0.08333333333333333, 0.16666666666666666));
	} else if (t_0 <= 2.0) {
		tmp = fma((y * y), 0.16666666666666666, 1.0);
	} else {
		tmp = y * (y * fma((y * y), 0.008333333333333333, 0.16666666666666666));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) * Float64(sinh(y) / y))
	tmp = 0.0
	if (t_0 <= -0.05)
		tmp = Float64(y * Float64(y * fma(Float64(x * x), -0.08333333333333333, 0.16666666666666666)));
	elseif (t_0 <= 2.0)
		tmp = fma(Float64(y * y), 0.16666666666666666, 1.0);
	else
		tmp = Float64(y * Float64(y * fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.05], N[(y * N[(y * N[(N[(x * x), $MachinePrecision] * -0.08333333333333333 + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(N[(y * y), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision], N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 45.3

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

   5. Simplified 45.3%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 38.9

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

   8. Simplified 38.9%

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

   9. Taylor expanded in y around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*l* N/A

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

       4. unpow2 N/A

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

       5. associate-*l* N/A

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

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

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

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

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

       8. +-commutative N/A

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

       9. distribute-rgt-in N/A

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

       10. *-commutative N/A

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

       11. associate-*l* N/A

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

       12. metadata-eval N/A

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

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

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

       14. unpow2 N/A

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

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

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

       16. metadata-eval 39.0

           \[\leadsto y \cdot \left(y \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{-0.08333333333333333}, 0.16666666666666666\right)\right) \]

   11. Simplified 39.0%

       \[\leadsto \color{blue}{y \cdot \left(y \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.16666666666666666\right)\right)} \]

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 75.4%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 74.8

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

   8. Simplified 74.8%

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

   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. *-lowering-*.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. cos-lowering-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. accelerator-lowering-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 72.0%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

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

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

      3. unpow2 N/A

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

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

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 72.0

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

   8. Simplified 72.0%

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

   9. Taylor expanded in y around inf

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

       1. distribute-rgt-in N/A

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

       2. associate-*l* N/A

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

       3. associate-*l/ N/A

          \[\leadsto \frac{1}{120} \cdot {y}^{4} + \frac{1}{6} \cdot \color{blue}{\frac{1 \cdot {y}^{4}}{{y}^{2}}} \]

       4. *-lft-identity N/A

          \[\leadsto \frac{1}{120} \cdot {y}^{4} + \frac{1}{6} \cdot \frac{\color{blue}{{y}^{4}}}{{y}^{2}} \]

       5. metadata-eval N/A

          \[\leadsto \frac{1}{120} \cdot {y}^{4} + \frac{1}{6} \cdot \frac{{y}^{\color{blue}{\left(3 + 1\right)}}}{{y}^{2}} \]

       6. pow-plus N/A

          \[\leadsto \frac{1}{120} \cdot {y}^{4} + \frac{1}{6} \cdot \frac{\color{blue}{{y}^{3} \cdot y}}{{y}^{2}} \]

       7. associate-*l/ N/A

          \[\leadsto \frac{1}{120} \cdot {y}^{4} + \frac{1}{6} \cdot \color{blue}{\left(\frac{{y}^{3}}{{y}^{2}} \cdot y\right)} \]

       8. *-lft-identity N/A

          \[\leadsto \frac{1}{120} \cdot {y}^{4} + \frac{1}{6} \cdot \left(\frac{\color{blue}{1 \cdot {y}^{3}}}{{y}^{2}} \cdot y\right) \]

       9. associate-*l/ N/A

          \[\leadsto \frac{1}{120} \cdot {y}^{4} + \frac{1}{6} \cdot \left(\color{blue}{\left(\frac{1}{{y}^{2}} \cdot {y}^{3}\right)} \cdot y\right) \]

       10. unpow3 N/A

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

       11. unpow2 N/A

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

       12. associate-*r* N/A

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

       13. lft-mult-inverse N/A

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

       14. *-lft-identity N/A

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

       15. unpow2 N/A

           \[\leadsto \frac{1}{120} \cdot {y}^{4} + \frac{1}{6} \cdot \color{blue}{{y}^{2}} \]

   11. Simplified 72.0%

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

Alternative 11: 67.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x \cdot \frac{\sinh y}{y}\\ \mathbf{if}\;t\_0 \leq -0.05:\\ \;\;\;\;y \cdot \left(y \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.16666666666666666\right)\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot \left(\left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (cos x) (/ (sinh y) y))))
   (if (<= t_0 -0.05)
     (* y (* y (fma (* x x) -0.08333333333333333 0.16666666666666666)))
     (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 <= -0.05) {
		tmp = y * (y * fma((x * x), -0.08333333333333333, 0.16666666666666666));
	} 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 <= -0.05)
		tmp = Float64(y * Float64(y * fma(Float64(x * x), -0.08333333333333333, 0.16666666666666666)));
	elseif (t_0 <= 2.0)
		tmp = fma(Float64(y * 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, -0.05], N[(y * N[(y * N[(N[(x * x), $MachinePrecision] * -0.08333333333333333 + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(N[(y * y), $MachinePrecision] * 0.16666666666666666 + 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)) < -0.050000000000000003

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 45.3

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

   5. Simplified 45.3%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 38.9

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

   8. Simplified 38.9%

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

   9. Taylor expanded in y around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*l* N/A

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

       4. unpow2 N/A

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

       5. associate-*l* N/A

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

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

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

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

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

       8. +-commutative N/A

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

       9. distribute-rgt-in N/A

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

       10. *-commutative N/A

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

       11. associate-*l* N/A

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

       12. metadata-eval N/A

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

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

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

       14. unpow2 N/A

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

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

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

       16. metadata-eval 39.0

           \[\leadsto y \cdot \left(y \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{-0.08333333333333333}, 0.16666666666666666\right)\right) \]

   11. Simplified 39.0%

       \[\leadsto \color{blue}{y \cdot \left(y \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.16666666666666666\right)\right)} \]

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 75.4%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 74.8

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

   8. Simplified 74.8%

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

   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. *-lowering-*.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. cos-lowering-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. accelerator-lowering-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 72.0%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

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

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

      3. unpow2 N/A

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

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

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 72.0

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

   8. Simplified 72.0%

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

   9. Taylor expanded in y around inf

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

       1. metadata-eval N/A

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

       2. pow-sqr N/A

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

       3. associate-*l* N/A

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

       4. unpow2 N/A

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

       5. associate-*r* N/A

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

       6. *-commutative N/A

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

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

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

       8. *-commutative N/A

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

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

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

       10. *-commutative N/A

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

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

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

       12. unpow2 N/A

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

       13. *-lowering-*.f64 72.0

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

   11. Simplified 72.0%

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

Alternative 12: 67.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x \cdot \frac{\sinh y}{y}\\ \mathbf{if}\;t\_0 \leq -0.05:\\ \;\;\;\;x \cdot \left(x \cdot \mathsf{fma}\left(y \cdot y, -0.08333333333333333, -0.5\right)\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot \left(\left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (cos x) (/ (sinh y) y))))
   (if (<= t_0 -0.05)
     (* 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 <= -0.05) {
		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 <= -0.05)
		tmp = Float64(x * Float64(x * fma(Float64(y * y), -0.08333333333333333, -0.5)));
	elseif (t_0 <= 2.0)
		tmp = fma(Float64(y * 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, -0.05], N[(x * N[(x * N[(N[(y * y), $MachinePrecision] * -0.08333333333333333 + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(N[(y * y), $MachinePrecision] * 0.16666666666666666 + 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)) < -0.050000000000000003

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 45.3

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

   5. Simplified 45.3%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 38.9

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

   8. Simplified 38.9%

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

   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. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. unpow2 N/A

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

       4. associate-*r* N/A

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

       5. *-commutative N/A

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

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

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

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

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

       9. +-commutative N/A

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

       10. distribute-rgt-in N/A

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

       11. *-commutative N/A

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

       12. associate-*r* N/A

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

       13. *-commutative N/A

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

       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. accelerator-lowering-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. *-lowering-*.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 38.9

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

   11. Simplified 38.9%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 75.4%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 74.8

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

   8. Simplified 74.8%

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

   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. *-lowering-*.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. cos-lowering-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. accelerator-lowering-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 72.0%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

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

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

      3. unpow2 N/A

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

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

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 72.0

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

   8. Simplified 72.0%

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

   9. Taylor expanded in y around inf

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

       1. metadata-eval N/A

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

       2. pow-sqr N/A

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

       3. associate-*l* N/A

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

       4. unpow2 N/A

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

       5. associate-*r* N/A

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

       6. *-commutative N/A

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

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

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

       8. *-commutative N/A

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

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

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

       10. *-commutative N/A

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

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

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

       12. unpow2 N/A

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

       13. *-lowering-*.f64 72.0

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

   11. Simplified 72.0%

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

Alternative 13: 63.5% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (cos x) (/ (sinh y) y))))
   (if (<= t_0 -0.05)
     (* -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 <= -0.05) {
		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 <= -0.05)
		tmp = Float64(-0.5 * Float64(x * x));
	elseif (t_0 <= 2.0)
		tmp = fma(Float64(y * 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, -0.05], N[(-0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(N[(y * y), $MachinePrecision] * 0.16666666666666666 + 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)) < -0.050000000000000003

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. cos-lowering-cos.f64 56.5

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

   5. Simplified 56.5%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 22.6

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

   8. Simplified 22.6%

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

   9. Taylor expanded in x around inf

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

       1. *-lowering-*.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. *-lowering-*.f64 22.6

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

   11. Simplified 22.6%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 75.4%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 74.8

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

   8. Simplified 74.8%

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

   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. *-lowering-*.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. cos-lowering-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. accelerator-lowering-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 72.0%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

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

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

      3. unpow2 N/A

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

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

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 72.0

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

   8. Simplified 72.0%

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

   9. Taylor expanded in y around inf

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

       1. metadata-eval N/A

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

       2. pow-sqr N/A

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

       3. associate-*l* N/A

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

       4. unpow2 N/A

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

       5. associate-*r* N/A

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

       6. *-commutative N/A

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

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

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

       8. *-commutative N/A

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

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

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

       10. *-commutative N/A

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

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

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

       12. unpow2 N/A

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

       13. *-lowering-*.f64 72.0

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

   11. Simplified 72.0%

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

Alternative 14: 73.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos x \cdot \frac{\sinh y}{y} \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right) \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cos x) (/ (sinh y) y)) -0.05)
   (*
    (fma (* y y) (fma y (* y 0.008333333333333333) 0.16666666666666666) 1.0)
    (fma
     x
     (*
      x
      (fma
       (* x x)
       (fma (* x x) -0.001388888888888889 0.041666666666666664)
       -0.5))
     1.0))
   (/
    (fma
     (fma
      y
      (* y (fma y (* y 0.0001984126984126984) 0.008333333333333333))
      0.16666666666666666)
     (* y (* y y))
     y)
    y)))

C

double code(double x, double y) {
	double tmp;
	if ((cos(x) * (sinh(y) / y)) <= -0.05) {
		tmp = fma((y * y), fma(y, (y * 0.008333333333333333), 0.16666666666666666), 1.0) * fma(x, (x * fma((x * x), fma((x * x), -0.001388888888888889, 0.041666666666666664), -0.5)), 1.0);
	} else {
		tmp = fma(fma(y, (y * fma(y, (y * 0.0001984126984126984), 0.008333333333333333)), 0.16666666666666666), (y * (y * y)), y) / y;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(cos(x) * Float64(sinh(y) / y)) <= -0.05)
		tmp = Float64(fma(Float64(y * y), fma(y, Float64(y * 0.008333333333333333), 0.16666666666666666), 1.0) * fma(x, Float64(x * fma(Float64(x * x), fma(Float64(x * x), -0.001388888888888889, 0.041666666666666664), -0.5)), 1.0));
	else
		tmp = Float64(fma(fma(y, Float64(y * fma(y, Float64(y * 0.0001984126984126984), 0.008333333333333333)), 0.16666666666666666), Float64(y * Float64(y * y)), y) / y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Cos[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], -0.05], N[(N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.001388888888888889 + 0.041666666666666664), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y * N[(y * N[(y * N[(y * 0.0001984126984126984), $MachinePrecision] + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in 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. *-lowering-*.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. cos-lowering-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. accelerator-lowering-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 88.5%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)\right)} \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right) + 1\right)} \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      2. unpow2 N/A

         \[\leadsto \left(\color{blue}{\left(x \cdot x\right)} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right) + 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      3. associate-*l* N/A

         \[\leadsto \left(\color{blue}{x \cdot \left(x \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)\right)} + 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right), 1\right)} \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

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

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)}, 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(x, x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)}, 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(x, x \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) + \color{blue}{\frac{-1}{2}}\right), 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

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

         \[\leadsto \mathsf{fma}\left(x, x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}, \frac{-1}{2}\right)}, 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}, \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

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

          \[\leadsto \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}, \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{-1}{720} \cdot {x}^{2} + \frac{1}{24}}, \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{-1}{720}} + \frac{1}{24}, \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

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

          \[\leadsto \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{-1}{720}, \frac{1}{24}\right)}, \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{720}, \frac{1}{24}\right), \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      15. *-lowering-*.f64 43.4

          \[\leadsto \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right) \]

   8. Simplified 43.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right)} \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right) \]

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 86.3%

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

   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

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-rgt-identity N/A

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

      7. *-inverses N/A

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

      8. associate-/l* N/A

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

      9. unpow2 N/A

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

      10. associate-/l* N/A

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

      11. pow-sqr N/A

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

      12. metadata-eval N/A

          \[\leadsto 1 \cdot \frac{\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot \frac{{y}^{\color{blue}{4}}}{y} + y}{y} \]

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

          \[\leadsto 1 \cdot \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right), \frac{{y}^{4}}{y}, y\right)}}{y} \]

   8. Simplified 78.7%

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

      1. *-lft-identity N/A

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

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

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

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

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

      4. associate-*l* N/A

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

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

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

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

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

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

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

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

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

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

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

      10. *-lowering-*.f64 78.7

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

   10. Applied egg-rr 78.7%

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

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos x \cdot \frac{\sinh y}{y} \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right) \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 72.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos x \cdot \frac{\sinh y}{y} \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right) \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cos x) (/ (sinh y) y)) -0.05)
   (*
    (fma 0.16666666666666666 (* y y) 1.0)
    (fma
     x
     (*
      x
      (fma
       (* x x)
       (fma (* x x) -0.001388888888888889 0.041666666666666664)
       -0.5))
     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) * (sinh(y) / y)) <= -0.05) {
		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 {
		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)) <= -0.05)
		tmp = Float64(fma(0.16666666666666666, Float64(y * y), 1.0) * fma(x, Float64(x * fma(Float64(x * x), fma(Float64(x * x), -0.001388888888888889, 0.041666666666666664), -0.5)), 1.0));
	else
		tmp = fma(Float64(y * y), fma(Float64(y * y), fma(y, Float64(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], -0.05], N[(N[(0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.001388888888888889 + 0.041666666666666664), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * 0.0001984126984126984), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-lft-identity N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-in N/A

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

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

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

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

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

      6. +-commutative N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 80.3

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

   5. Simplified 80.3%

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

   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. accelerator-lowering-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. *-lowering-*.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. accelerator-lowering-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. *-lowering-*.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. *-commutative N/A

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

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

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 43.4

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

   8. Simplified 43.4%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 86.3%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

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

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

      3. unpow2 N/A

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

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

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

      5. +-commutative N/A

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

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

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

      7. unpow2 N/A

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

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

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

      9. +-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

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

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

      14. *-commutative N/A

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

      15. *-lowering-*.f64 77.4

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

   8. Simplified 77.4%

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

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos x \cdot \frac{\sinh y}{y} \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right) \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 72.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)\\ \mathbf{if}\;\cos x \cdot \frac{\sinh y}{y} \leq -0.05:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(x, x \cdot -0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (fma
          (* y y)
          (fma
           (* y y)
           (fma y (* y 0.0001984126984126984) 0.008333333333333333)
           0.16666666666666666)
          1.0)))
   (if (<= (* (cos x) (/ (sinh y) y)) -0.05)
     (* t_0 (fma x (* x -0.5) 1.0))
     t_0)))

C

double code(double x, double y) {
	double t_0 = fma((y * y), fma((y * y), fma(y, (y * 0.0001984126984126984), 0.008333333333333333), 0.16666666666666666), 1.0);
	double tmp;
	if ((cos(x) * (sinh(y) / y)) <= -0.05) {
		tmp = t_0 * fma(x, (x * -0.5), 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(Float64(y * y), fma(Float64(y * y), fma(y, Float64(y * 0.0001984126984126984), 0.008333333333333333), 0.16666666666666666), 1.0)
	tmp = 0.0
	if (Float64(cos(x) * Float64(sinh(y) / y)) <= -0.05)
		tmp = Float64(t_0 * fma(x, Float64(x * -0.5), 1.0));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * 0.0001984126984126984), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[N[(N[Cos[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], -0.05], N[(t$95$0 * N[(x * N[(x * -0.5), $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)) < -0.050000000000000003

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(\frac{-1}{2} \cdot x\right) \cdot x} + 1\right) \cdot \frac{\sinh y}{y} \]

      4. *-commutative N/A

         \[\leadsto \left(\color{blue}{x \cdot \left(\frac{-1}{2} \cdot x\right)} + 1\right) \cdot \frac{\sinh y}{y} \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{-1}{2} \cdot x, 1\right)} \cdot \frac{\sinh y}{y} \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{-1}{2}}, 1\right) \cdot \frac{\sinh y}{y} \]

      7. *-lowering-*.f64 45.3

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot -0.5}, 1\right) \cdot \frac{\sinh y}{y} \]

   5. Simplified 45.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot -0.5, 1\right)} \cdot \frac{\sinh y}{y} \]

   6. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{-1}{2}, 1\right) \cdot \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{-1}{2}, 1\right) \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) + 1\right)} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{-1}{2}, 1\right) \cdot \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right), 1\right)} \]

      3. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{-1}{2}, 1\right) \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right), 1\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{-1}{2}, 1\right) \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right), 1\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{-1}{2}, 1\right) \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{{y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right) + \frac{1}{6}}, 1\right) \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{-1}{2}, 1\right) \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}, \frac{1}{6}\right)}, 1\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{-1}{2}, 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}, \frac{1}{6}\right), 1\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{-1}{2}, 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}, \frac{1}{6}\right), 1\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{-1}{2}, 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \color{blue}{\frac{1}{5040} \cdot {y}^{2} + \frac{1}{120}}, \frac{1}{6}\right), 1\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{-1}{2}, 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \frac{1}{5040} \cdot \color{blue}{\left(y \cdot y\right)} + \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      11. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{-1}{2}, 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \color{blue}{\left(\frac{1}{5040} \cdot y\right) \cdot y} + \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{-1}{2}, 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \color{blue}{y \cdot \left(\frac{1}{5040} \cdot y\right)} + \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      13. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{-1}{2}, 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \color{blue}{\mathsf{fma}\left(y, \frac{1}{5040} \cdot y, \frac{1}{120}\right)}, \frac{1}{6}\right), 1\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{-1}{2}, 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, \color{blue}{y \cdot \frac{1}{5040}}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      15. *-lowering-*.f64 42.1

          \[\leadsto \mathsf{fma}\left(x, x \cdot -0.5, 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, \color{blue}{y \cdot 0.0001984126984126984}, 0.008333333333333333\right), 0.16666666666666666\right), 1\right) \]

   8. Simplified 42.1%

      \[\leadsto \mathsf{fma}\left(x, x \cdot -0.5, 1\right) \cdot \color{blue}{\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)} \]

   if -0.050000000000000003 < (*.f64 (cos.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \cdot \frac{\sinh y}{y} \]
   4. Step-by-step derivation

   5. Simplified 86.3%

      \[\leadsto \color{blue}{1} \cdot \frac{\sinh y}{y} \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) + 1} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right), 1\right)} \]

      3. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right), 1\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right), 1\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(y \cdot y, \color{blue}{{y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right) + \frac{1}{6}}, 1\right) \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(y \cdot y, \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}, \frac{1}{6}\right)}, 1\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}, \frac{1}{6}\right), 1\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}, \frac{1}{6}\right), 1\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \color{blue}{\frac{1}{5040} \cdot {y}^{2} + \frac{1}{120}}, \frac{1}{6}\right), 1\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \frac{1}{5040} \cdot \color{blue}{\left(y \cdot y\right)} + \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      11. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \color{blue}{\left(\frac{1}{5040} \cdot y\right) \cdot y} + \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \color{blue}{y \cdot \left(\frac{1}{5040} \cdot y\right)} + \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      13. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \color{blue}{\mathsf{fma}\left(y, \frac{1}{5040} \cdot y, \frac{1}{120}\right)}, \frac{1}{6}\right), 1\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, \color{blue}{y \cdot \frac{1}{5040}}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      15. *-lowering-*.f64 77.4

          \[\leadsto \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, \color{blue}{y \cdot 0.0001984126984126984}, 0.008333333333333333\right), 0.16666666666666666\right), 1\right) \]

   8. Simplified 77.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos x \cdot \frac{\sinh y}{y} \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right) \cdot \mathsf{fma}\left(x, x \cdot -0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 71.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos x \cdot \frac{\sinh y}{y} \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot -0.5, 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cos x) (/ (sinh y) y)) -0.05)
   (*
    (fma x (* x -0.5) 1.0)
    (fma (* y y) (fma (* y y) 0.008333333333333333 0.16666666666666666) 1.0))
   (fma
    (* y y)
    (fma
     (* y y)
     (fma y (* y 0.0001984126984126984) 0.008333333333333333)
     0.16666666666666666)
    1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((cos(x) * (sinh(y) / y)) <= -0.05) {
		tmp = fma(x, (x * -0.5), 1.0) * fma((y * y), fma((y * y), 0.008333333333333333, 0.16666666666666666), 1.0);
	} else {
		tmp = fma((y * y), fma((y * y), fma(y, (y * 0.0001984126984126984), 0.008333333333333333), 0.16666666666666666), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(cos(x) * Float64(sinh(y) / y)) <= -0.05)
		tmp = Float64(fma(x, Float64(x * -0.5), 1.0) * fma(Float64(y * y), fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666), 1.0));
	else
		tmp = fma(Float64(y * y), fma(Float64(y * y), fma(y, Float64(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], -0.05], N[(N[(x * N[(x * -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * 0.0001984126984126984), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cos.f64 x) (/.f64 (sinh.f64 y) y)) < -0.050000000000000003

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + \frac{-1}{2} \cdot {x}^{2}\right)} \cdot \frac{\sinh y}{y} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{2} \cdot {x}^{2} + 1\right)} \cdot \frac{\sinh y}{y} \]

      2. unpow2 N/A

         \[\leadsto \left(\frac{-1}{2} \cdot \color{blue}{\left(x \cdot x\right)} + 1\right) \cdot \frac{\sinh y}{y} \]

      3. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(\frac{-1}{2} \cdot x\right) \cdot x} + 1\right) \cdot \frac{\sinh y}{y} \]

      4. *-commutative N/A

         \[\leadsto \left(\color{blue}{x \cdot \left(\frac{-1}{2} \cdot x\right)} + 1\right) \cdot \frac{\sinh y}{y} \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{-1}{2} \cdot x, 1\right)} \cdot \frac{\sinh y}{y} \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{-1}{2}}, 1\right) \cdot \frac{\sinh y}{y} \]

      7. *-lowering-*.f64 45.3

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot -0.5}, 1\right) \cdot \frac{\sinh y}{y} \]

   5. Simplified 45.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot -0.5, 1\right)} \cdot \frac{\sinh y}{y} \]

   6. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{-1}{2}, 1\right) \cdot \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{-1}{2}, 1\right) \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) + 1\right)} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{-1}{2}, 1\right) \cdot \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{1}{6} + \frac{1}{120} \cdot {y}^{2}, 1\right)} \]

      3. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{-1}{2}, 1\right) \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{6} + \frac{1}{120} \cdot {y}^{2}, 1\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{-1}{2}, 1\right) \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{6} + \frac{1}{120} \cdot {y}^{2}, 1\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{-1}{2}, 1\right) \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{\frac{1}{120} \cdot {y}^{2} + \frac{1}{6}}, 1\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{-1}{2}, 1\right) \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{{y}^{2} \cdot \frac{1}{120}} + \frac{1}{6}, 1\right) \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{-1}{2}, 1\right) \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{1}{120}, \frac{1}{6}\right)}, 1\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{-1}{2}, 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      9. *-lowering-*.f64 42.1

         \[\leadsto \mathsf{fma}\left(x, x \cdot -0.5, 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, 0.008333333333333333, 0.16666666666666666\right), 1\right) \]

   8. Simplified 42.1%

      \[\leadsto \mathsf{fma}\left(x, x \cdot -0.5, 1\right) \cdot \color{blue}{\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), 1\right)} \]

   if -0.050000000000000003 < (*.f64 (cos.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \cdot \frac{\sinh y}{y} \]
   4. Step-by-step derivation

   5. Simplified 86.3%

      \[\leadsto \color{blue}{1} \cdot \frac{\sinh y}{y} \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) + 1} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right), 1\right)} \]

      3. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right), 1\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right), 1\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(y \cdot y, \color{blue}{{y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right) + \frac{1}{6}}, 1\right) \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(y \cdot y, \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}, \frac{1}{6}\right)}, 1\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}, \frac{1}{6}\right), 1\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}, \frac{1}{6}\right), 1\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \color{blue}{\frac{1}{5040} \cdot {y}^{2} + \frac{1}{120}}, \frac{1}{6}\right), 1\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \frac{1}{5040} \cdot \color{blue}{\left(y \cdot y\right)} + \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      11. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \color{blue}{\left(\frac{1}{5040} \cdot y\right) \cdot y} + \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \color{blue}{y \cdot \left(\frac{1}{5040} \cdot y\right)} + \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      13. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \color{blue}{\mathsf{fma}\left(y, \frac{1}{5040} \cdot y, \frac{1}{120}\right)}, \frac{1}{6}\right), 1\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, \color{blue}{y \cdot \frac{1}{5040}}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      15. *-lowering-*.f64 77.4

          \[\leadsto \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, \color{blue}{y \cdot 0.0001984126984126984}, 0.008333333333333333\right), 0.16666666666666666\right), 1\right) \]

   8. Simplified 77.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 70.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos x \cdot \frac{\sinh y}{y} \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cos x) (/ (sinh y) y)) -0.05)
   (fma
    x
    (*
     x
     (fma
      (* x x)
      (fma (* x x) -0.001388888888888889 0.041666666666666664)
      -0.5))
    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) * (sinh(y) / y)) <= -0.05) {
		tmp = fma(x, (x * fma((x * x), fma((x * x), -0.001388888888888889, 0.041666666666666664), -0.5)), 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 (Float64(cos(x) * Float64(sinh(y) / y)) <= -0.05)
		tmp = fma(x, Float64(x * fma(Float64(x * x), fma(Float64(x * x), -0.001388888888888889, 0.041666666666666664), -0.5)), 1.0);
	else
		tmp = fma(Float64(y * y), fma(Float64(y * y), fma(y, Float64(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], -0.05], N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.001388888888888889 + 0.041666666666666664), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * 0.0001984126984126984), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cos.f64 x) (/.f64 (sinh.f64 y) y)) < -0.050000000000000003

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\cos x} \]
   4. Step-by-step derivation

      1. cos-lowering-cos.f64 56.5

         \[\leadsto \color{blue}{\cos x} \]

   5. Simplified 56.5%

      \[\leadsto \color{blue}{\cos x} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right) + 1} \]

      2. unpow2 N/A

         \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right) + 1 \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{x \cdot \left(x \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)\right)} + 1 \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right), 1\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)}, 1\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(x, x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)}, 1\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(x, x \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) + \color{blue}{\frac{-1}{2}}\right), 1\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}, \frac{-1}{2}\right)}, 1\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}, \frac{-1}{2}\right), 1\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}, \frac{-1}{2}\right), 1\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{-1}{720} \cdot {x}^{2} + \frac{1}{24}}, \frac{-1}{2}\right), 1\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{-1}{720}} + \frac{1}{24}, \frac{-1}{2}\right), 1\right) \]

      13. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{-1}{720}, \frac{1}{24}\right)}, \frac{-1}{2}\right), 1\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{720}, \frac{1}{24}\right), \frac{-1}{2}\right), 1\right) \]

      15. *-lowering-*.f64 40.1

          \[\leadsto \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right) \]

   8. Simplified 40.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right)} \]

   if -0.050000000000000003 < (*.f64 (cos.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \cdot \frac{\sinh y}{y} \]
   4. Step-by-step derivation

   5. Simplified 86.3%

      \[\leadsto \color{blue}{1} \cdot \frac{\sinh y}{y} \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) + 1} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right), 1\right)} \]

      3. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right), 1\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right), 1\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(y \cdot y, \color{blue}{{y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right) + \frac{1}{6}}, 1\right) \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(y \cdot y, \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}, \frac{1}{6}\right)}, 1\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}, \frac{1}{6}\right), 1\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}, \frac{1}{6}\right), 1\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \color{blue}{\frac{1}{5040} \cdot {y}^{2} + \frac{1}{120}}, \frac{1}{6}\right), 1\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \frac{1}{5040} \cdot \color{blue}{\left(y \cdot y\right)} + \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      11. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \color{blue}{\left(\frac{1}{5040} \cdot y\right) \cdot y} + \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \color{blue}{y \cdot \left(\frac{1}{5040} \cdot y\right)} + \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      13. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \color{blue}{\mathsf{fma}\left(y, \frac{1}{5040} \cdot y, \frac{1}{120}\right)}, \frac{1}{6}\right), 1\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, \color{blue}{y \cdot \frac{1}{5040}}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      15. *-lowering-*.f64 77.4

          \[\leadsto \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, \color{blue}{y \cdot 0.0001984126984126984}, 0.008333333333333333\right), 0.16666666666666666\right), 1\right) \]

   8. Simplified 77.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 70.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos x \cdot \frac{\sinh y}{y} \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right), 0.0001984126984126984, 0.16666666666666666\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cos x) (/ (sinh y) y)) -0.05)
   (fma
    x
    (*
     x
     (fma
      (* x x)
      (fma (* x x) -0.001388888888888889 0.041666666666666664)
      -0.5))
    1.0)
   (fma
    (* y y)
    (fma (* (* y y) (* y y)) 0.0001984126984126984 0.16666666666666666)
    1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((cos(x) * (sinh(y) / y)) <= -0.05) {
		tmp = fma(x, (x * fma((x * x), fma((x * x), -0.001388888888888889, 0.041666666666666664), -0.5)), 1.0);
	} else {
		tmp = fma((y * y), fma(((y * y) * (y * y)), 0.0001984126984126984, 0.16666666666666666), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(cos(x) * Float64(sinh(y) / y)) <= -0.05)
		tmp = fma(x, Float64(x * fma(Float64(x * x), fma(Float64(x * x), -0.001388888888888889, 0.041666666666666664), -0.5)), 1.0);
	else
		tmp = fma(Float64(y * y), fma(Float64(Float64(y * y) * Float64(y * y)), 0.0001984126984126984, 0.16666666666666666), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Cos[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], -0.05], N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.001388888888888889 + 0.041666666666666664), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(y * y), $MachinePrecision] * N[(N[(N[(y * y), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision] * 0.0001984126984126984 + 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cos.f64 x) (/.f64 (sinh.f64 y) y)) < -0.050000000000000003

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\cos x} \]
   4. Step-by-step derivation

      1. cos-lowering-cos.f64 56.5

         \[\leadsto \color{blue}{\cos x} \]

   5. Simplified 56.5%

      \[\leadsto \color{blue}{\cos x} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right) + 1} \]

      2. unpow2 N/A

         \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right) + 1 \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{x \cdot \left(x \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)\right)} + 1 \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right), 1\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)}, 1\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(x, x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)}, 1\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(x, x \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) + \color{blue}{\frac{-1}{2}}\right), 1\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}, \frac{-1}{2}\right)}, 1\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}, \frac{-1}{2}\right), 1\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}, \frac{-1}{2}\right), 1\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{-1}{720} \cdot {x}^{2} + \frac{1}{24}}, \frac{-1}{2}\right), 1\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{-1}{720}} + \frac{1}{24}, \frac{-1}{2}\right), 1\right) \]

      13. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{-1}{720}, \frac{1}{24}\right)}, \frac{-1}{2}\right), 1\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{720}, \frac{1}{24}\right), \frac{-1}{2}\right), 1\right) \]

      15. *-lowering-*.f64 40.1

          \[\leadsto \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right) \]

   8. Simplified 40.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right)} \]

   if -0.050000000000000003 < (*.f64 (cos.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \cdot \frac{\sinh y}{y} \]
   4. Step-by-step derivation

   5. Simplified 86.3%

      \[\leadsto \color{blue}{1} \cdot \frac{\sinh y}{y} \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) + 1} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right), 1\right)} \]

      3. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right), 1\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right), 1\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(y \cdot y, \color{blue}{{y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right) + \frac{1}{6}}, 1\right) \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(y \cdot y, \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}, \frac{1}{6}\right)}, 1\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}, \frac{1}{6}\right), 1\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}, \frac{1}{6}\right), 1\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \color{blue}{\frac{1}{5040} \cdot {y}^{2} + \frac{1}{120}}, \frac{1}{6}\right), 1\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \frac{1}{5040} \cdot \color{blue}{\left(y \cdot y\right)} + \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      11. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \color{blue}{\left(\frac{1}{5040} \cdot y\right) \cdot y} + \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \color{blue}{y \cdot \left(\frac{1}{5040} \cdot y\right)} + \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      13. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \color{blue}{\mathsf{fma}\left(y, \frac{1}{5040} \cdot y, \frac{1}{120}\right)}, \frac{1}{6}\right), 1\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, \color{blue}{y \cdot \frac{1}{5040}}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      15. *-lowering-*.f64 77.4

          \[\leadsto \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, \color{blue}{y \cdot 0.0001984126984126984}, 0.008333333333333333\right), 0.16666666666666666\right), 1\right) \]

   8. Simplified 77.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)} \]
   9. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \mathsf{fma}\left(y \cdot y, \color{blue}{\left(\left(y \cdot y\right) \cdot \left(y \cdot \left(y \cdot \frac{1}{5040}\right)\right) + \left(y \cdot y\right) \cdot \frac{1}{120}\right)} + \frac{1}{6}, 1\right) \]

      2. associate-+l+ N/A

         \[\leadsto \mathsf{fma}\left(y \cdot y, \color{blue}{\left(y \cdot y\right) \cdot \left(y \cdot \left(y \cdot \frac{1}{5040}\right)\right) + \left(\left(y \cdot y\right) \cdot \frac{1}{120} + \frac{1}{6}\right)}, 1\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(y \cdot y, \left(y \cdot y\right) \cdot \color{blue}{\left(\left(y \cdot y\right) \cdot \frac{1}{5040}\right)} + \left(\left(y \cdot y\right) \cdot \frac{1}{120} + \frac{1}{6}\right), 1\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(y \cdot y, \color{blue}{\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \frac{1}{5040}} + \left(\left(y \cdot y\right) \cdot \frac{1}{120} + \frac{1}{6}\right), 1\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(y \cdot y, \color{blue}{\mathsf{fma}\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right), \frac{1}{5040}, \left(y \cdot y\right) \cdot \frac{1}{120} + \frac{1}{6}\right)}, 1\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{\left(y \cdot y\right) \cdot \left(y \cdot y\right)}, \frac{1}{5040}, \left(y \cdot y\right) \cdot \frac{1}{120} + \frac{1}{6}\right), 1\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{\left(y \cdot y\right)} \cdot \left(y \cdot y\right), \frac{1}{5040}, \left(y \cdot y\right) \cdot \frac{1}{120} + \frac{1}{6}\right), 1\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\left(y \cdot y\right) \cdot \color{blue}{\left(y \cdot y\right)}, \frac{1}{5040}, \left(y \cdot y\right) \cdot \frac{1}{120} + \frac{1}{6}\right), 1\right) \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right), \frac{1}{5040}, \color{blue}{\mathsf{fma}\left(y \cdot y, \frac{1}{120}, \frac{1}{6}\right)}\right), 1\right) \]

      10. *-lowering-*.f64 77.4

          \[\leadsto \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right), 0.0001984126984126984, \mathsf{fma}\left(\color{blue}{y \cdot y}, 0.008333333333333333, 0.16666666666666666\right)\right), 1\right) \]

   10. Applied egg-rr 77.4%

       \[\leadsto \mathsf{fma}\left(y \cdot y, \color{blue}{\mathsf{fma}\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right), 0.0001984126984126984, \mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right)\right)}, 1\right) \]

   11. Taylor expanded in y around 0

       \[\leadsto \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right), \frac{1}{5040}, \color{blue}{\frac{1}{6}}\right), 1\right) \]
   12. Step-by-step derivation

   13. Simplified 77.0%

       \[\leadsto \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right), 0.0001984126984126984, \color{blue}{0.16666666666666666}\right), 1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 71.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos x \cdot \frac{\sinh y}{y} \leq -0.05:\\ \;\;\;\;y \cdot \left(y \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right), 0.0001984126984126984, 0.16666666666666666\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cos x) (/ (sinh y) y)) -0.05)
   (* y (* y (fma (* x x) -0.08333333333333333 0.16666666666666666)))
   (fma
    (* y y)
    (fma (* (* y y) (* y y)) 0.0001984126984126984 0.16666666666666666)
    1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((cos(x) * (sinh(y) / y)) <= -0.05) {
		tmp = y * (y * fma((x * x), -0.08333333333333333, 0.16666666666666666));
	} else {
		tmp = fma((y * y), fma(((y * y) * (y * y)), 0.0001984126984126984, 0.16666666666666666), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(cos(x) * Float64(sinh(y) / y)) <= -0.05)
		tmp = Float64(y * Float64(y * fma(Float64(x * x), -0.08333333333333333, 0.16666666666666666)));
	else
		tmp = fma(Float64(y * y), fma(Float64(Float64(y * y) * Float64(y * y)), 0.0001984126984126984, 0.16666666666666666), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Cos[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], -0.05], N[(y * N[(y * N[(N[(x * x), $MachinePrecision] * -0.08333333333333333 + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * y), $MachinePrecision] * N[(N[(N[(y * y), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision] * 0.0001984126984126984 + 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cos.f64 x) (/.f64 (sinh.f64 y) y)) < -0.050000000000000003

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + \frac{-1}{2} \cdot {x}^{2}\right)} \cdot \frac{\sinh y}{y} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{2} \cdot {x}^{2} + 1\right)} \cdot \frac{\sinh y}{y} \]

      2. unpow2 N/A

         \[\leadsto \left(\frac{-1}{2} \cdot \color{blue}{\left(x \cdot x\right)} + 1\right) \cdot \frac{\sinh y}{y} \]

      3. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(\frac{-1}{2} \cdot x\right) \cdot x} + 1\right) \cdot \frac{\sinh y}{y} \]

      4. *-commutative N/A

         \[\leadsto \left(\color{blue}{x \cdot \left(\frac{-1}{2} \cdot x\right)} + 1\right) \cdot \frac{\sinh y}{y} \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{-1}{2} \cdot x, 1\right)} \cdot \frac{\sinh y}{y} \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{-1}{2}}, 1\right) \cdot \frac{\sinh y}{y} \]

      7. *-lowering-*.f64 45.3

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot -0.5}, 1\right) \cdot \frac{\sinh y}{y} \]

   5. Simplified 45.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot -0.5, 1\right)} \cdot \frac{\sinh y}{y} \]

   6. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{-1}{2}, 1\right) \cdot \color{blue}{\left(1 + \frac{1}{6} \cdot {y}^{2}\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{-1}{2}, 1\right) \cdot \color{blue}{\left(\frac{1}{6} \cdot {y}^{2} + 1\right)} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{-1}{2}, 1\right) \cdot \left(\color{blue}{{y}^{2} \cdot \frac{1}{6}} + 1\right) \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{-1}{2}, 1\right) \cdot \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{1}{6}, 1\right)} \]

      4. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{-1}{2}, 1\right) \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{6}, 1\right) \]

      5. *-lowering-*.f64 38.9

         \[\leadsto \mathsf{fma}\left(x, x \cdot -0.5, 1\right) \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, 0.16666666666666666, 1\right) \]

   8. Simplified 38.9%

      \[\leadsto \mathsf{fma}\left(x, x \cdot -0.5, 1\right) \cdot \color{blue}{\mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right)} \]

   9. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{1}{6} \cdot \left({y}^{2} \cdot \left(1 + \frac{-1}{2} \cdot {x}^{2}\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \left(1 + \frac{-1}{2} \cdot {x}^{2}\right)} \]

       2. *-commutative N/A

          \[\leadsto \color{blue}{\left({y}^{2} \cdot \frac{1}{6}\right)} \cdot \left(1 + \frac{-1}{2} \cdot {x}^{2}\right) \]

       3. associate-*l* N/A

          \[\leadsto \color{blue}{{y}^{2} \cdot \left(\frac{1}{6} \cdot \left(1 + \frac{-1}{2} \cdot {x}^{2}\right)\right)} \]

       4. unpow2 N/A

          \[\leadsto \color{blue}{\left(y \cdot y\right)} \cdot \left(\frac{1}{6} \cdot \left(1 + \frac{-1}{2} \cdot {x}^{2}\right)\right) \]

       5. associate-*l* N/A

          \[\leadsto \color{blue}{y \cdot \left(y \cdot \left(\frac{1}{6} \cdot \left(1 + \frac{-1}{2} \cdot {x}^{2}\right)\right)\right)} \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{y \cdot \left(y \cdot \left(\frac{1}{6} \cdot \left(1 + \frac{-1}{2} \cdot {x}^{2}\right)\right)\right)} \]

       7. *-lowering-*.f64 N/A

          \[\leadsto y \cdot \color{blue}{\left(y \cdot \left(\frac{1}{6} \cdot \left(1 + \frac{-1}{2} \cdot {x}^{2}\right)\right)\right)} \]

       8. +-commutative N/A

          \[\leadsto y \cdot \left(y \cdot \left(\frac{1}{6} \cdot \color{blue}{\left(\frac{-1}{2} \cdot {x}^{2} + 1\right)}\right)\right) \]

       9. distribute-rgt-in N/A

          \[\leadsto y \cdot \left(y \cdot \color{blue}{\left(\left(\frac{-1}{2} \cdot {x}^{2}\right) \cdot \frac{1}{6} + 1 \cdot \frac{1}{6}\right)}\right) \]

       10. *-commutative N/A

           \[\leadsto y \cdot \left(y \cdot \left(\color{blue}{\left({x}^{2} \cdot \frac{-1}{2}\right)} \cdot \frac{1}{6} + 1 \cdot \frac{1}{6}\right)\right) \]

       11. associate-*l* N/A

           \[\leadsto y \cdot \left(y \cdot \left(\color{blue}{{x}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{1}{6}\right)} + 1 \cdot \frac{1}{6}\right)\right) \]

       12. metadata-eval N/A

           \[\leadsto y \cdot \left(y \cdot \left({x}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{1}{6}\right) + \color{blue}{\frac{1}{6}}\right)\right) \]

       13. accelerator-lowering-fma.f64 N/A

           \[\leadsto y \cdot \left(y \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{-1}{2} \cdot \frac{1}{6}, \frac{1}{6}\right)}\right) \]

       14. unpow2 N/A

           \[\leadsto y \cdot \left(y \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{2} \cdot \frac{1}{6}, \frac{1}{6}\right)\right) \]

       15. *-lowering-*.f64 N/A

           \[\leadsto y \cdot \left(y \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{2} \cdot \frac{1}{6}, \frac{1}{6}\right)\right) \]

       16. metadata-eval 39.0

           \[\leadsto y \cdot \left(y \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{-0.08333333333333333}, 0.16666666666666666\right)\right) \]

   11. Simplified 39.0%

       \[\leadsto \color{blue}{y \cdot \left(y \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.16666666666666666\right)\right)} \]

   if -0.050000000000000003 < (*.f64 (cos.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \cdot \frac{\sinh y}{y} \]
   4. Step-by-step derivation

   5. Simplified 86.3%

      \[\leadsto \color{blue}{1} \cdot \frac{\sinh y}{y} \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) + 1} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right), 1\right)} \]

      3. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right), 1\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right), 1\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(y \cdot y, \color{blue}{{y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right) + \frac{1}{6}}, 1\right) \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(y \cdot y, \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}, \frac{1}{6}\right)}, 1\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}, \frac{1}{6}\right), 1\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}, \frac{1}{6}\right), 1\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \color{blue}{\frac{1}{5040} \cdot {y}^{2} + \frac{1}{120}}, \frac{1}{6}\right), 1\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \frac{1}{5040} \cdot \color{blue}{\left(y \cdot y\right)} + \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      11. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \color{blue}{\left(\frac{1}{5040} \cdot y\right) \cdot y} + \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \color{blue}{y \cdot \left(\frac{1}{5040} \cdot y\right)} + \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      13. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \color{blue}{\mathsf{fma}\left(y, \frac{1}{5040} \cdot y, \frac{1}{120}\right)}, \frac{1}{6}\right), 1\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, \color{blue}{y \cdot \frac{1}{5040}}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      15. *-lowering-*.f64 77.4

          \[\leadsto \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, \color{blue}{y \cdot 0.0001984126984126984}, 0.008333333333333333\right), 0.16666666666666666\right), 1\right) \]

   8. Simplified 77.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)} \]
   9. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \mathsf{fma}\left(y \cdot y, \color{blue}{\left(\left(y \cdot y\right) \cdot \left(y \cdot \left(y \cdot \frac{1}{5040}\right)\right) + \left(y \cdot y\right) \cdot \frac{1}{120}\right)} + \frac{1}{6}, 1\right) \]

      2. associate-+l+ N/A

         \[\leadsto \mathsf{fma}\left(y \cdot y, \color{blue}{\left(y \cdot y\right) \cdot \left(y \cdot \left(y \cdot \frac{1}{5040}\right)\right) + \left(\left(y \cdot y\right) \cdot \frac{1}{120} + \frac{1}{6}\right)}, 1\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(y \cdot y, \left(y \cdot y\right) \cdot \color{blue}{\left(\left(y \cdot y\right) \cdot \frac{1}{5040}\right)} + \left(\left(y \cdot y\right) \cdot \frac{1}{120} + \frac{1}{6}\right), 1\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(y \cdot y, \color{blue}{\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \frac{1}{5040}} + \left(\left(y \cdot y\right) \cdot \frac{1}{120} + \frac{1}{6}\right), 1\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(y \cdot y, \color{blue}{\mathsf{fma}\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right), \frac{1}{5040}, \left(y \cdot y\right) \cdot \frac{1}{120} + \frac{1}{6}\right)}, 1\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{\left(y \cdot y\right) \cdot \left(y \cdot y\right)}, \frac{1}{5040}, \left(y \cdot y\right) \cdot \frac{1}{120} + \frac{1}{6}\right), 1\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{\left(y \cdot y\right)} \cdot \left(y \cdot y\right), \frac{1}{5040}, \left(y \cdot y\right) \cdot \frac{1}{120} + \frac{1}{6}\right), 1\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\left(y \cdot y\right) \cdot \color{blue}{\left(y \cdot y\right)}, \frac{1}{5040}, \left(y \cdot y\right) \cdot \frac{1}{120} + \frac{1}{6}\right), 1\right) \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right), \frac{1}{5040}, \color{blue}{\mathsf{fma}\left(y \cdot y, \frac{1}{120}, \frac{1}{6}\right)}\right), 1\right) \]

      10. *-lowering-*.f64 77.4

          \[\leadsto \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right), 0.0001984126984126984, \mathsf{fma}\left(\color{blue}{y \cdot y}, 0.008333333333333333, 0.16666666666666666\right)\right), 1\right) \]

   10. Applied egg-rr 77.4%

       \[\leadsto \mathsf{fma}\left(y \cdot y, \color{blue}{\mathsf{fma}\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right), 0.0001984126984126984, \mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right)\right)}, 1\right) \]

   11. Taylor expanded in y around 0

       \[\leadsto \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right), \frac{1}{5040}, \color{blue}{\frac{1}{6}}\right), 1\right) \]
   12. Step-by-step derivation

   13. Simplified 77.0%

       \[\leadsto \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right), 0.0001984126984126984, \color{blue}{0.16666666666666666}\right), 1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 70.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos x \cdot \frac{\sinh y}{y} \leq -0.05:\\ \;\;\;\;y \cdot \left(y \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, y \cdot \left(y \cdot \mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right)\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cos x) (/ (sinh y) y)) -0.05)
   (* y (* y (fma (* x x) -0.08333333333333333 0.16666666666666666)))
   (fma
    (* y y)
    (* y (* y (fma y (* y 0.0001984126984126984) 0.008333333333333333)))
    1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((cos(x) * (sinh(y) / y)) <= -0.05) {
		tmp = y * (y * fma((x * x), -0.08333333333333333, 0.16666666666666666));
	} else {
		tmp = fma((y * y), (y * (y * fma(y, (y * 0.0001984126984126984), 0.008333333333333333))), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(cos(x) * Float64(sinh(y) / y)) <= -0.05)
		tmp = Float64(y * Float64(y * fma(Float64(x * x), -0.08333333333333333, 0.16666666666666666)));
	else
		tmp = fma(Float64(y * y), Float64(y * Float64(y * fma(y, Float64(y * 0.0001984126984126984), 0.008333333333333333))), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Cos[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], -0.05], N[(y * N[(y * N[(N[(x * x), $MachinePrecision] * -0.08333333333333333 + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * N[(y * N[(y * 0.0001984126984126984), $MachinePrecision] + 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cos.f64 x) (/.f64 (sinh.f64 y) y)) < -0.050000000000000003

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + \frac{-1}{2} \cdot {x}^{2}\right)} \cdot \frac{\sinh y}{y} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{2} \cdot {x}^{2} + 1\right)} \cdot \frac{\sinh y}{y} \]

      2. unpow2 N/A

         \[\leadsto \left(\frac{-1}{2} \cdot \color{blue}{\left(x \cdot x\right)} + 1\right) \cdot \frac{\sinh y}{y} \]

      3. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(\frac{-1}{2} \cdot x\right) \cdot x} + 1\right) \cdot \frac{\sinh y}{y} \]

      4. *-commutative N/A

         \[\leadsto \left(\color{blue}{x \cdot \left(\frac{-1}{2} \cdot x\right)} + 1\right) \cdot \frac{\sinh y}{y} \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{-1}{2} \cdot x, 1\right)} \cdot \frac{\sinh y}{y} \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{-1}{2}}, 1\right) \cdot \frac{\sinh y}{y} \]

      7. *-lowering-*.f64 45.3

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot -0.5}, 1\right) \cdot \frac{\sinh y}{y} \]

   5. Simplified 45.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot -0.5, 1\right)} \cdot \frac{\sinh y}{y} \]

   6. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{-1}{2}, 1\right) \cdot \color{blue}{\left(1 + \frac{1}{6} \cdot {y}^{2}\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{-1}{2}, 1\right) \cdot \color{blue}{\left(\frac{1}{6} \cdot {y}^{2} + 1\right)} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{-1}{2}, 1\right) \cdot \left(\color{blue}{{y}^{2} \cdot \frac{1}{6}} + 1\right) \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{-1}{2}, 1\right) \cdot \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{1}{6}, 1\right)} \]

      4. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{-1}{2}, 1\right) \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{6}, 1\right) \]

      5. *-lowering-*.f64 38.9

         \[\leadsto \mathsf{fma}\left(x, x \cdot -0.5, 1\right) \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, 0.16666666666666666, 1\right) \]

   8. Simplified 38.9%

      \[\leadsto \mathsf{fma}\left(x, x \cdot -0.5, 1\right) \cdot \color{blue}{\mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right)} \]

   9. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{1}{6} \cdot \left({y}^{2} \cdot \left(1 + \frac{-1}{2} \cdot {x}^{2}\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \left(1 + \frac{-1}{2} \cdot {x}^{2}\right)} \]

       2. *-commutative N/A

          \[\leadsto \color{blue}{\left({y}^{2} \cdot \frac{1}{6}\right)} \cdot \left(1 + \frac{-1}{2} \cdot {x}^{2}\right) \]

       3. associate-*l* N/A

          \[\leadsto \color{blue}{{y}^{2} \cdot \left(\frac{1}{6} \cdot \left(1 + \frac{-1}{2} \cdot {x}^{2}\right)\right)} \]

       4. unpow2 N/A

          \[\leadsto \color{blue}{\left(y \cdot y\right)} \cdot \left(\frac{1}{6} \cdot \left(1 + \frac{-1}{2} \cdot {x}^{2}\right)\right) \]

       5. associate-*l* N/A

          \[\leadsto \color{blue}{y \cdot \left(y \cdot \left(\frac{1}{6} \cdot \left(1 + \frac{-1}{2} \cdot {x}^{2}\right)\right)\right)} \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{y \cdot \left(y \cdot \left(\frac{1}{6} \cdot \left(1 + \frac{-1}{2} \cdot {x}^{2}\right)\right)\right)} \]

       7. *-lowering-*.f64 N/A

          \[\leadsto y \cdot \color{blue}{\left(y \cdot \left(\frac{1}{6} \cdot \left(1 + \frac{-1}{2} \cdot {x}^{2}\right)\right)\right)} \]

       8. +-commutative N/A

          \[\leadsto y \cdot \left(y \cdot \left(\frac{1}{6} \cdot \color{blue}{\left(\frac{-1}{2} \cdot {x}^{2} + 1\right)}\right)\right) \]

       9. distribute-rgt-in N/A

          \[\leadsto y \cdot \left(y \cdot \color{blue}{\left(\left(\frac{-1}{2} \cdot {x}^{2}\right) \cdot \frac{1}{6} + 1 \cdot \frac{1}{6}\right)}\right) \]

       10. *-commutative N/A

           \[\leadsto y \cdot \left(y \cdot \left(\color{blue}{\left({x}^{2} \cdot \frac{-1}{2}\right)} \cdot \frac{1}{6} + 1 \cdot \frac{1}{6}\right)\right) \]

       11. associate-*l* N/A

           \[\leadsto y \cdot \left(y \cdot \left(\color{blue}{{x}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{1}{6}\right)} + 1 \cdot \frac{1}{6}\right)\right) \]

       12. metadata-eval N/A

           \[\leadsto y \cdot \left(y \cdot \left({x}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{1}{6}\right) + \color{blue}{\frac{1}{6}}\right)\right) \]

       13. accelerator-lowering-fma.f64 N/A

           \[\leadsto y \cdot \left(y \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{-1}{2} \cdot \frac{1}{6}, \frac{1}{6}\right)}\right) \]

       14. unpow2 N/A

           \[\leadsto y \cdot \left(y \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{2} \cdot \frac{1}{6}, \frac{1}{6}\right)\right) \]

       15. *-lowering-*.f64 N/A

           \[\leadsto y \cdot \left(y \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{2} \cdot \frac{1}{6}, \frac{1}{6}\right)\right) \]

       16. metadata-eval 39.0

           \[\leadsto y \cdot \left(y \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{-0.08333333333333333}, 0.16666666666666666\right)\right) \]

   11. Simplified 39.0%

       \[\leadsto \color{blue}{y \cdot \left(y \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.16666666666666666\right)\right)} \]

   if -0.050000000000000003 < (*.f64 (cos.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \cdot \frac{\sinh y}{y} \]
   4. Step-by-step derivation

   5. Simplified 86.3%

      \[\leadsto \color{blue}{1} \cdot \frac{\sinh y}{y} \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) + 1} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right), 1\right)} \]

      3. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right), 1\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right), 1\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(y \cdot y, \color{blue}{{y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right) + \frac{1}{6}}, 1\right) \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(y \cdot y, \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}, \frac{1}{6}\right)}, 1\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}, \frac{1}{6}\right), 1\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}, \frac{1}{6}\right), 1\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \color{blue}{\frac{1}{5040} \cdot {y}^{2} + \frac{1}{120}}, \frac{1}{6}\right), 1\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \frac{1}{5040} \cdot \color{blue}{\left(y \cdot y\right)} + \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      11. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \color{blue}{\left(\frac{1}{5040} \cdot y\right) \cdot y} + \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \color{blue}{y \cdot \left(\frac{1}{5040} \cdot y\right)} + \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      13. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \color{blue}{\mathsf{fma}\left(y, \frac{1}{5040} \cdot y, \frac{1}{120}\right)}, \frac{1}{6}\right), 1\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, \color{blue}{y \cdot \frac{1}{5040}}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      15. *-lowering-*.f64 77.4

          \[\leadsto \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, \color{blue}{y \cdot 0.0001984126984126984}, 0.008333333333333333\right), 0.16666666666666666\right), 1\right) \]

   8. Simplified 77.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)} \]
   9. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \mathsf{fma}\left(y \cdot y, \color{blue}{\left(\left(y \cdot y\right) \cdot \left(y \cdot \left(y \cdot \frac{1}{5040}\right)\right) + \left(y \cdot y\right) \cdot \frac{1}{120}\right)} + \frac{1}{6}, 1\right) \]

      2. associate-+l+ N/A

         \[\leadsto \mathsf{fma}\left(y \cdot y, \color{blue}{\left(y \cdot y\right) \cdot \left(y \cdot \left(y \cdot \frac{1}{5040}\right)\right) + \left(\left(y \cdot y\right) \cdot \frac{1}{120} + \frac{1}{6}\right)}, 1\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(y \cdot y, \left(y \cdot y\right) \cdot \color{blue}{\left(\left(y \cdot y\right) \cdot \frac{1}{5040}\right)} + \left(\left(y \cdot y\right) \cdot \frac{1}{120} + \frac{1}{6}\right), 1\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(y \cdot y, \color{blue}{\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \frac{1}{5040}} + \left(\left(y \cdot y\right) \cdot \frac{1}{120} + \frac{1}{6}\right), 1\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(y \cdot y, \color{blue}{\mathsf{fma}\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right), \frac{1}{5040}, \left(y \cdot y\right) \cdot \frac{1}{120} + \frac{1}{6}\right)}, 1\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{\left(y \cdot y\right) \cdot \left(y \cdot y\right)}, \frac{1}{5040}, \left(y \cdot y\right) \cdot \frac{1}{120} + \frac{1}{6}\right), 1\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{\left(y \cdot y\right)} \cdot \left(y \cdot y\right), \frac{1}{5040}, \left(y \cdot y\right) \cdot \frac{1}{120} + \frac{1}{6}\right), 1\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\left(y \cdot y\right) \cdot \color{blue}{\left(y \cdot y\right)}, \frac{1}{5040}, \left(y \cdot y\right) \cdot \frac{1}{120} + \frac{1}{6}\right), 1\right) \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right), \frac{1}{5040}, \color{blue}{\mathsf{fma}\left(y \cdot y, \frac{1}{120}, \frac{1}{6}\right)}\right), 1\right) \]

      10. *-lowering-*.f64 77.4

          \[\leadsto \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right), 0.0001984126984126984, \mathsf{fma}\left(\color{blue}{y \cdot y}, 0.008333333333333333, 0.16666666666666666\right)\right), 1\right) \]

   10. Applied egg-rr 77.4%

       \[\leadsto \mathsf{fma}\left(y \cdot y, \color{blue}{\mathsf{fma}\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right), 0.0001984126984126984, \mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right)\right)}, 1\right) \]

   11. Taylor expanded in y around inf

       \[\leadsto \mathsf{fma}\left(y \cdot y, \color{blue}{{y}^{4} \cdot \left(\frac{1}{5040} + \frac{1}{120} \cdot \frac{1}{{y}^{2}}\right)}, 1\right) \]
   12. Step-by-step derivation

       1. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(y \cdot y, {y}^{\color{blue}{\left(2 \cdot 2\right)}} \cdot \left(\frac{1}{5040} + \frac{1}{120} \cdot \frac{1}{{y}^{2}}\right), 1\right) \]

       2. pow-sqr N/A

          \[\leadsto \mathsf{fma}\left(y \cdot y, \color{blue}{\left({y}^{2} \cdot {y}^{2}\right)} \cdot \left(\frac{1}{5040} + \frac{1}{120} \cdot \frac{1}{{y}^{2}}\right), 1\right) \]

       3. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(y \cdot y, \color{blue}{{y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{5040} + \frac{1}{120} \cdot \frac{1}{{y}^{2}}\right)\right)}, 1\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(y \cdot y, \color{blue}{\left(y \cdot y\right)} \cdot \left({y}^{2} \cdot \left(\frac{1}{5040} + \frac{1}{120} \cdot \frac{1}{{y}^{2}}\right)\right), 1\right) \]

       5. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(y \cdot y, \left(y \cdot y\right) \cdot \left({y}^{2} \cdot \color{blue}{\left(\frac{1}{120} \cdot \frac{1}{{y}^{2}} + \frac{1}{5040}\right)}\right), 1\right) \]

       6. distribute-rgt-in N/A

          \[\leadsto \mathsf{fma}\left(y \cdot y, \left(y \cdot y\right) \cdot \color{blue}{\left(\left(\frac{1}{120} \cdot \frac{1}{{y}^{2}}\right) \cdot {y}^{2} + \frac{1}{5040} \cdot {y}^{2}\right)}, 1\right) \]

       7. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(y \cdot y, \left(y \cdot y\right) \cdot \left(\color{blue}{\frac{1}{120} \cdot \left(\frac{1}{{y}^{2}} \cdot {y}^{2}\right)} + \frac{1}{5040} \cdot {y}^{2}\right), 1\right) \]

       8. lft-mult-inverse N/A

          \[\leadsto \mathsf{fma}\left(y \cdot y, \left(y \cdot y\right) \cdot \left(\frac{1}{120} \cdot \color{blue}{1} + \frac{1}{5040} \cdot {y}^{2}\right), 1\right) \]

       9. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(y \cdot y, \left(y \cdot y\right) \cdot \left(\color{blue}{\frac{1}{120}} + \frac{1}{5040} \cdot {y}^{2}\right), 1\right) \]

       10. associate-*l* N/A

           \[\leadsto \mathsf{fma}\left(y \cdot y, \color{blue}{y \cdot \left(y \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)}, 1\right) \]

       11. *-commutative N/A

           \[\leadsto \mathsf{fma}\left(y \cdot y, y \cdot \color{blue}{\left(\left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right) \cdot y\right)}, 1\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(y \cdot y, \color{blue}{y \cdot \left(\left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right) \cdot y\right)}, 1\right) \]

       13. *-commutative N/A

           \[\leadsto \mathsf{fma}\left(y \cdot y, y \cdot \color{blue}{\left(y \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)}, 1\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(y \cdot y, y \cdot \color{blue}{\left(y \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)}, 1\right) \]

       15. +-commutative N/A

           \[\leadsto \mathsf{fma}\left(y \cdot y, y \cdot \left(y \cdot \color{blue}{\left(\frac{1}{5040} \cdot {y}^{2} + \frac{1}{120}\right)}\right), 1\right) \]

       16. unpow2 N/A

           \[\leadsto \mathsf{fma}\left(y \cdot y, y \cdot \left(y \cdot \left(\frac{1}{5040} \cdot \color{blue}{\left(y \cdot y\right)} + \frac{1}{120}\right)\right), 1\right) \]

       17. associate-*r* N/A

           \[\leadsto \mathsf{fma}\left(y \cdot y, y \cdot \left(y \cdot \left(\color{blue}{\left(\frac{1}{5040} \cdot y\right) \cdot y} + \frac{1}{120}\right)\right), 1\right) \]

       18. *-commutative N/A

           \[\leadsto \mathsf{fma}\left(y \cdot y, y \cdot \left(y \cdot \left(\color{blue}{y \cdot \left(\frac{1}{5040} \cdot y\right)} + \frac{1}{120}\right)\right), 1\right) \]

       19. accelerator-lowering-fma.f64 N/A

           \[\leadsto \mathsf{fma}\left(y \cdot y, y \cdot \left(y \cdot \color{blue}{\mathsf{fma}\left(y, \frac{1}{5040} \cdot y, \frac{1}{120}\right)}\right), 1\right) \]

       20. *-commutative N/A

           \[\leadsto \mathsf{fma}\left(y \cdot y, y \cdot \left(y \cdot \mathsf{fma}\left(y, \color{blue}{y \cdot \frac{1}{5040}}, \frac{1}{120}\right)\right), 1\right) \]

       21. *-lowering-*.f64 76.2

           \[\leadsto \mathsf{fma}\left(y \cdot y, y \cdot \left(y \cdot \mathsf{fma}\left(y, \color{blue}{y \cdot 0.0001984126984126984}, 0.008333333333333333\right)\right), 1\right) \]

   13. Simplified 76.2%

       \[\leadsto \mathsf{fma}\left(y \cdot y, \color{blue}{y \cdot \left(y \cdot \mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right)\right)}, 1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 22: 70.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos x \cdot \frac{\sinh y}{y} \leq -0.05:\\ \;\;\;\;y \cdot \left(y \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, \left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot 0.0001984126984126984\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cos x) (/ (sinh y) y)) -0.05)
   (* y (* y (fma (* x x) -0.08333333333333333 0.16666666666666666)))
   (fma (* y y) (* (* y y) (* (* y y) 0.0001984126984126984)) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((cos(x) * (sinh(y) / y)) <= -0.05) {
		tmp = y * (y * fma((x * x), -0.08333333333333333, 0.16666666666666666));
	} else {
		tmp = fma((y * y), ((y * y) * ((y * y) * 0.0001984126984126984)), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(cos(x) * Float64(sinh(y) / y)) <= -0.05)
		tmp = Float64(y * Float64(y * fma(Float64(x * x), -0.08333333333333333, 0.16666666666666666)));
	else
		tmp = fma(Float64(y * y), Float64(Float64(y * y) * Float64(Float64(y * y) * 0.0001984126984126984)), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Cos[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], -0.05], N[(y * N[(y * N[(N[(x * x), $MachinePrecision] * -0.08333333333333333 + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cos.f64 x) (/.f64 (sinh.f64 y) y)) < -0.050000000000000003

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + \frac{-1}{2} \cdot {x}^{2}\right)} \cdot \frac{\sinh y}{y} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{2} \cdot {x}^{2} + 1\right)} \cdot \frac{\sinh y}{y} \]

      2. unpow2 N/A

         \[\leadsto \left(\frac{-1}{2} \cdot \color{blue}{\left(x \cdot x\right)} + 1\right) \cdot \frac{\sinh y}{y} \]

      3. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(\frac{-1}{2} \cdot x\right) \cdot x} + 1\right) \cdot \frac{\sinh y}{y} \]

      4. *-commutative N/A

         \[\leadsto \left(\color{blue}{x \cdot \left(\frac{-1}{2} \cdot x\right)} + 1\right) \cdot \frac{\sinh y}{y} \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{-1}{2} \cdot x, 1\right)} \cdot \frac{\sinh y}{y} \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{-1}{2}}, 1\right) \cdot \frac{\sinh y}{y} \]

      7. *-lowering-*.f64 45.3

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot -0.5}, 1\right) \cdot \frac{\sinh y}{y} \]

   5. Simplified 45.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot -0.5, 1\right)} \cdot \frac{\sinh y}{y} \]

   6. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{-1}{2}, 1\right) \cdot \color{blue}{\left(1 + \frac{1}{6} \cdot {y}^{2}\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{-1}{2}, 1\right) \cdot \color{blue}{\left(\frac{1}{6} \cdot {y}^{2} + 1\right)} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{-1}{2}, 1\right) \cdot \left(\color{blue}{{y}^{2} \cdot \frac{1}{6}} + 1\right) \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{-1}{2}, 1\right) \cdot \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{1}{6}, 1\right)} \]

      4. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{-1}{2}, 1\right) \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{6}, 1\right) \]

      5. *-lowering-*.f64 38.9

         \[\leadsto \mathsf{fma}\left(x, x \cdot -0.5, 1\right) \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, 0.16666666666666666, 1\right) \]

   8. Simplified 38.9%

      \[\leadsto \mathsf{fma}\left(x, x \cdot -0.5, 1\right) \cdot \color{blue}{\mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right)} \]

   9. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{1}{6} \cdot \left({y}^{2} \cdot \left(1 + \frac{-1}{2} \cdot {x}^{2}\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \left(1 + \frac{-1}{2} \cdot {x}^{2}\right)} \]

       2. *-commutative N/A

          \[\leadsto \color{blue}{\left({y}^{2} \cdot \frac{1}{6}\right)} \cdot \left(1 + \frac{-1}{2} \cdot {x}^{2}\right) \]

       3. associate-*l* N/A

          \[\leadsto \color{blue}{{y}^{2} \cdot \left(\frac{1}{6} \cdot \left(1 + \frac{-1}{2} \cdot {x}^{2}\right)\right)} \]

       4. unpow2 N/A

          \[\leadsto \color{blue}{\left(y \cdot y\right)} \cdot \left(\frac{1}{6} \cdot \left(1 + \frac{-1}{2} \cdot {x}^{2}\right)\right) \]

       5. associate-*l* N/A

          \[\leadsto \color{blue}{y \cdot \left(y \cdot \left(\frac{1}{6} \cdot \left(1 + \frac{-1}{2} \cdot {x}^{2}\right)\right)\right)} \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{y \cdot \left(y \cdot \left(\frac{1}{6} \cdot \left(1 + \frac{-1}{2} \cdot {x}^{2}\right)\right)\right)} \]

       7. *-lowering-*.f64 N/A

          \[\leadsto y \cdot \color{blue}{\left(y \cdot \left(\frac{1}{6} \cdot \left(1 + \frac{-1}{2} \cdot {x}^{2}\right)\right)\right)} \]

       8. +-commutative N/A

          \[\leadsto y \cdot \left(y \cdot \left(\frac{1}{6} \cdot \color{blue}{\left(\frac{-1}{2} \cdot {x}^{2} + 1\right)}\right)\right) \]

       9. distribute-rgt-in N/A

          \[\leadsto y \cdot \left(y \cdot \color{blue}{\left(\left(\frac{-1}{2} \cdot {x}^{2}\right) \cdot \frac{1}{6} + 1 \cdot \frac{1}{6}\right)}\right) \]

       10. *-commutative N/A

           \[\leadsto y \cdot \left(y \cdot \left(\color{blue}{\left({x}^{2} \cdot \frac{-1}{2}\right)} \cdot \frac{1}{6} + 1 \cdot \frac{1}{6}\right)\right) \]

       11. associate-*l* N/A

           \[\leadsto y \cdot \left(y \cdot \left(\color{blue}{{x}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{1}{6}\right)} + 1 \cdot \frac{1}{6}\right)\right) \]

       12. metadata-eval N/A

           \[\leadsto y \cdot \left(y \cdot \left({x}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{1}{6}\right) + \color{blue}{\frac{1}{6}}\right)\right) \]

       13. accelerator-lowering-fma.f64 N/A

           \[\leadsto y \cdot \left(y \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{-1}{2} \cdot \frac{1}{6}, \frac{1}{6}\right)}\right) \]

       14. unpow2 N/A

           \[\leadsto y \cdot \left(y \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{2} \cdot \frac{1}{6}, \frac{1}{6}\right)\right) \]

       15. *-lowering-*.f64 N/A

           \[\leadsto y \cdot \left(y \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{2} \cdot \frac{1}{6}, \frac{1}{6}\right)\right) \]

       16. metadata-eval 39.0

           \[\leadsto y \cdot \left(y \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{-0.08333333333333333}, 0.16666666666666666\right)\right) \]

   11. Simplified 39.0%

       \[\leadsto \color{blue}{y \cdot \left(y \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.16666666666666666\right)\right)} \]

   if -0.050000000000000003 < (*.f64 (cos.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \cdot \frac{\sinh y}{y} \]
   4. Step-by-step derivation

   5. Simplified 86.3%

      \[\leadsto \color{blue}{1} \cdot \frac{\sinh y}{y} \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) + 1} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right), 1\right)} \]

      3. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right), 1\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right), 1\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(y \cdot y, \color{blue}{{y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right) + \frac{1}{6}}, 1\right) \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(y \cdot y, \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}, \frac{1}{6}\right)}, 1\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}, \frac{1}{6}\right), 1\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}, \frac{1}{6}\right), 1\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \color{blue}{\frac{1}{5040} \cdot {y}^{2} + \frac{1}{120}}, \frac{1}{6}\right), 1\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \frac{1}{5040} \cdot \color{blue}{\left(y \cdot y\right)} + \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      11. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \color{blue}{\left(\frac{1}{5040} \cdot y\right) \cdot y} + \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \color{blue}{y \cdot \left(\frac{1}{5040} \cdot y\right)} + \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      13. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \color{blue}{\mathsf{fma}\left(y, \frac{1}{5040} \cdot y, \frac{1}{120}\right)}, \frac{1}{6}\right), 1\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, \color{blue}{y \cdot \frac{1}{5040}}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      15. *-lowering-*.f64 77.4

          \[\leadsto \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, \color{blue}{y \cdot 0.0001984126984126984}, 0.008333333333333333\right), 0.16666666666666666\right), 1\right) \]

   8. Simplified 77.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)} \]

   9. Taylor expanded in y around inf

      \[\leadsto \mathsf{fma}\left(y \cdot y, \color{blue}{\frac{1}{5040} \cdot {y}^{4}}, 1\right) \]
   10. Step-by-step derivation

       1. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(y \cdot y, \frac{1}{5040} \cdot {y}^{\color{blue}{\left(2 \cdot 2\right)}}, 1\right) \]

       2. pow-sqr N/A

          \[\leadsto \mathsf{fma}\left(y \cdot y, \frac{1}{5040} \cdot \color{blue}{\left({y}^{2} \cdot {y}^{2}\right)}, 1\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(y \cdot y, \color{blue}{\left(\frac{1}{5040} \cdot {y}^{2}\right) \cdot {y}^{2}}, 1\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(y \cdot y, \color{blue}{{y}^{2} \cdot \left(\frac{1}{5040} \cdot {y}^{2}\right)}, 1\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(y \cdot y, \color{blue}{{y}^{2} \cdot \left(\frac{1}{5040} \cdot {y}^{2}\right)}, 1\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(y \cdot y, \color{blue}{\left(y \cdot y\right)} \cdot \left(\frac{1}{5040} \cdot {y}^{2}\right), 1\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(y \cdot y, \color{blue}{\left(y \cdot y\right)} \cdot \left(\frac{1}{5040} \cdot {y}^{2}\right), 1\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(y \cdot y, \left(y \cdot y\right) \cdot \color{blue}{\left({y}^{2} \cdot \frac{1}{5040}\right)}, 1\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(y \cdot y, \left(y \cdot y\right) \cdot \color{blue}{\left({y}^{2} \cdot \frac{1}{5040}\right)}, 1\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{fma}\left(y \cdot y, \left(y \cdot y\right) \cdot \left(\color{blue}{\left(y \cdot y\right)} \cdot \frac{1}{5040}\right), 1\right) \]

       11. *-lowering-*.f64 76.2

           \[\leadsto \mathsf{fma}\left(y \cdot y, \left(y \cdot y\right) \cdot \left(\color{blue}{\left(y \cdot y\right)} \cdot 0.0001984126984126984\right), 1\right) \]

   11. Simplified 76.2%

       \[\leadsto \mathsf{fma}\left(y \cdot y, \color{blue}{\left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot 0.0001984126984126984\right)}, 1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 23: 67.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos x \cdot \frac{\sinh y}{y} \leq -0.05:\\ \;\;\;\;y \cdot \left(y \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot 0.008333333333333333, y, 0.16666666666666666\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cos x) (/ (sinh y) y)) -0.05)
   (* y (* y (fma (* x x) -0.08333333333333333 0.16666666666666666)))
   (fma (* y y) (fma (* y 0.008333333333333333) y 0.16666666666666666) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((cos(x) * (sinh(y) / y)) <= -0.05) {
		tmp = y * (y * fma((x * x), -0.08333333333333333, 0.16666666666666666));
	} else {
		tmp = fma((y * y), fma((y * 0.008333333333333333), y, 0.16666666666666666), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(cos(x) * Float64(sinh(y) / y)) <= -0.05)
		tmp = Float64(y * Float64(y * fma(Float64(x * x), -0.08333333333333333, 0.16666666666666666)));
	else
		tmp = fma(Float64(y * y), fma(Float64(y * 0.008333333333333333), y, 0.16666666666666666), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Cos[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], -0.05], N[(y * N[(y * N[(N[(x * x), $MachinePrecision] * -0.08333333333333333 + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * y), $MachinePrecision] * N[(N[(y * 0.008333333333333333), $MachinePrecision] * y + 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cos.f64 x) (/.f64 (sinh.f64 y) y)) < -0.050000000000000003

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + \frac{-1}{2} \cdot {x}^{2}\right)} \cdot \frac{\sinh y}{y} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{2} \cdot {x}^{2} + 1\right)} \cdot \frac{\sinh y}{y} \]

      2. unpow2 N/A

         \[\leadsto \left(\frac{-1}{2} \cdot \color{blue}{\left(x \cdot x\right)} + 1\right) \cdot \frac{\sinh y}{y} \]

      3. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(\frac{-1}{2} \cdot x\right) \cdot x} + 1\right) \cdot \frac{\sinh y}{y} \]

      4. *-commutative N/A

         \[\leadsto \left(\color{blue}{x \cdot \left(\frac{-1}{2} \cdot x\right)} + 1\right) \cdot \frac{\sinh y}{y} \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{-1}{2} \cdot x, 1\right)} \cdot \frac{\sinh y}{y} \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{-1}{2}}, 1\right) \cdot \frac{\sinh y}{y} \]

      7. *-lowering-*.f64 45.3

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot -0.5}, 1\right) \cdot \frac{\sinh y}{y} \]

   5. Simplified 45.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot -0.5, 1\right)} \cdot \frac{\sinh y}{y} \]

   6. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{-1}{2}, 1\right) \cdot \color{blue}{\left(1 + \frac{1}{6} \cdot {y}^{2}\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{-1}{2}, 1\right) \cdot \color{blue}{\left(\frac{1}{6} \cdot {y}^{2} + 1\right)} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{-1}{2}, 1\right) \cdot \left(\color{blue}{{y}^{2} \cdot \frac{1}{6}} + 1\right) \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{-1}{2}, 1\right) \cdot \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{1}{6}, 1\right)} \]

      4. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{-1}{2}, 1\right) \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{6}, 1\right) \]

      5. *-lowering-*.f64 38.9

         \[\leadsto \mathsf{fma}\left(x, x \cdot -0.5, 1\right) \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, 0.16666666666666666, 1\right) \]

   8. Simplified 38.9%

      \[\leadsto \mathsf{fma}\left(x, x \cdot -0.5, 1\right) \cdot \color{blue}{\mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right)} \]

   9. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{1}{6} \cdot \left({y}^{2} \cdot \left(1 + \frac{-1}{2} \cdot {x}^{2}\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \left(1 + \frac{-1}{2} \cdot {x}^{2}\right)} \]

       2. *-commutative N/A

          \[\leadsto \color{blue}{\left({y}^{2} \cdot \frac{1}{6}\right)} \cdot \left(1 + \frac{-1}{2} \cdot {x}^{2}\right) \]

       3. associate-*l* N/A

          \[\leadsto \color{blue}{{y}^{2} \cdot \left(\frac{1}{6} \cdot \left(1 + \frac{-1}{2} \cdot {x}^{2}\right)\right)} \]

       4. unpow2 N/A

          \[\leadsto \color{blue}{\left(y \cdot y\right)} \cdot \left(\frac{1}{6} \cdot \left(1 + \frac{-1}{2} \cdot {x}^{2}\right)\right) \]

       5. associate-*l* N/A

          \[\leadsto \color{blue}{y \cdot \left(y \cdot \left(\frac{1}{6} \cdot \left(1 + \frac{-1}{2} \cdot {x}^{2}\right)\right)\right)} \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{y \cdot \left(y \cdot \left(\frac{1}{6} \cdot \left(1 + \frac{-1}{2} \cdot {x}^{2}\right)\right)\right)} \]

       7. *-lowering-*.f64 N/A

          \[\leadsto y \cdot \color{blue}{\left(y \cdot \left(\frac{1}{6} \cdot \left(1 + \frac{-1}{2} \cdot {x}^{2}\right)\right)\right)} \]

       8. +-commutative N/A

          \[\leadsto y \cdot \left(y \cdot \left(\frac{1}{6} \cdot \color{blue}{\left(\frac{-1}{2} \cdot {x}^{2} + 1\right)}\right)\right) \]

       9. distribute-rgt-in N/A

          \[\leadsto y \cdot \left(y \cdot \color{blue}{\left(\left(\frac{-1}{2} \cdot {x}^{2}\right) \cdot \frac{1}{6} + 1 \cdot \frac{1}{6}\right)}\right) \]

       10. *-commutative N/A

           \[\leadsto y \cdot \left(y \cdot \left(\color{blue}{\left({x}^{2} \cdot \frac{-1}{2}\right)} \cdot \frac{1}{6} + 1 \cdot \frac{1}{6}\right)\right) \]

       11. associate-*l* N/A

           \[\leadsto y \cdot \left(y \cdot \left(\color{blue}{{x}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{1}{6}\right)} + 1 \cdot \frac{1}{6}\right)\right) \]

       12. metadata-eval N/A

           \[\leadsto y \cdot \left(y \cdot \left({x}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{1}{6}\right) + \color{blue}{\frac{1}{6}}\right)\right) \]

       13. accelerator-lowering-fma.f64 N/A

           \[\leadsto y \cdot \left(y \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{-1}{2} \cdot \frac{1}{6}, \frac{1}{6}\right)}\right) \]

       14. unpow2 N/A

           \[\leadsto y \cdot \left(y \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{2} \cdot \frac{1}{6}, \frac{1}{6}\right)\right) \]

       15. *-lowering-*.f64 N/A

           \[\leadsto y \cdot \left(y \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{2} \cdot \frac{1}{6}, \frac{1}{6}\right)\right) \]

       16. metadata-eval 39.0

           \[\leadsto y \cdot \left(y \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{-0.08333333333333333}, 0.16666666666666666\right)\right) \]

   11. Simplified 39.0%

       \[\leadsto \color{blue}{y \cdot \left(y \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.16666666666666666\right)\right)} \]

   if -0.050000000000000003 < (*.f64 (cos.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in 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. *-lowering-*.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. cos-lowering-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. accelerator-lowering-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 87.6%

      \[\leadsto \color{blue}{\cos x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) + 1} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{1}{6} + \frac{1}{120} \cdot {y}^{2}, 1\right)} \]

      3. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{6} + \frac{1}{120} \cdot {y}^{2}, 1\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{6} + \frac{1}{120} \cdot {y}^{2}, 1\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(y \cdot y, \color{blue}{\frac{1}{120} \cdot {y}^{2} + \frac{1}{6}}, 1\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y \cdot y, \color{blue}{{y}^{2} \cdot \frac{1}{120}} + \frac{1}{6}, 1\right) \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(y \cdot y, \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{1}{120}, \frac{1}{6}\right)}, 1\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      9. *-lowering-*.f64 73.9

         \[\leadsto \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, 0.008333333333333333, 0.16666666666666666\right), 1\right) \]

   8. Simplified 73.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), 1\right)} \]
   9. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(y \cdot y, \color{blue}{y \cdot \left(y \cdot \frac{1}{120}\right)} + \frac{1}{6}, 1\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y \cdot y, \color{blue}{\left(y \cdot \frac{1}{120}\right) \cdot y} + \frac{1}{6}, 1\right) \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(y \cdot y, \color{blue}{\mathsf{fma}\left(y \cdot \frac{1}{120}, y, \frac{1}{6}\right)}, 1\right) \]

      4. *-lowering-*.f64 73.9

         \[\leadsto \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot 0.008333333333333333}, y, 0.16666666666666666\right), 1\right) \]

   10. Applied egg-rr 73.9%

       \[\leadsto \mathsf{fma}\left(y \cdot y, \color{blue}{\mathsf{fma}\left(y \cdot 0.008333333333333333, y, 0.16666666666666666\right)}, 1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 24: 67.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos x \cdot \frac{\sinh y}{y} \leq -0.05:\\ \;\;\;\;y \cdot \left(y \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, \left(y \cdot y\right) \cdot 0.008333333333333333, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cos x) (/ (sinh y) y)) -0.05)
   (* y (* y (fma (* x x) -0.08333333333333333 0.16666666666666666)))
   (fma (* y y) (* (* y y) 0.008333333333333333) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((cos(x) * (sinh(y) / y)) <= -0.05) {
		tmp = y * (y * fma((x * x), -0.08333333333333333, 0.16666666666666666));
	} else {
		tmp = fma((y * y), ((y * y) * 0.008333333333333333), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(cos(x) * Float64(sinh(y) / y)) <= -0.05)
		tmp = Float64(y * Float64(y * fma(Float64(x * x), -0.08333333333333333, 0.16666666666666666)));
	else
		tmp = fma(Float64(y * y), Float64(Float64(y * y) * 0.008333333333333333), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Cos[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], -0.05], N[(y * N[(y * N[(N[(x * x), $MachinePrecision] * -0.08333333333333333 + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.008333333333333333), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cos.f64 x) (/.f64 (sinh.f64 y) y)) < -0.050000000000000003

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + \frac{-1}{2} \cdot {x}^{2}\right)} \cdot \frac{\sinh y}{y} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{2} \cdot {x}^{2} + 1\right)} \cdot \frac{\sinh y}{y} \]

      2. unpow2 N/A

         \[\leadsto \left(\frac{-1}{2} \cdot \color{blue}{\left(x \cdot x\right)} + 1\right) \cdot \frac{\sinh y}{y} \]

      3. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(\frac{-1}{2} \cdot x\right) \cdot x} + 1\right) \cdot \frac{\sinh y}{y} \]

      4. *-commutative N/A

         \[\leadsto \left(\color{blue}{x \cdot \left(\frac{-1}{2} \cdot x\right)} + 1\right) \cdot \frac{\sinh y}{y} \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{-1}{2} \cdot x, 1\right)} \cdot \frac{\sinh y}{y} \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{-1}{2}}, 1\right) \cdot \frac{\sinh y}{y} \]

      7. *-lowering-*.f64 45.3

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot -0.5}, 1\right) \cdot \frac{\sinh y}{y} \]

   5. Simplified 45.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot -0.5, 1\right)} \cdot \frac{\sinh y}{y} \]

   6. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{-1}{2}, 1\right) \cdot \color{blue}{\left(1 + \frac{1}{6} \cdot {y}^{2}\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{-1}{2}, 1\right) \cdot \color{blue}{\left(\frac{1}{6} \cdot {y}^{2} + 1\right)} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{-1}{2}, 1\right) \cdot \left(\color{blue}{{y}^{2} \cdot \frac{1}{6}} + 1\right) \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{-1}{2}, 1\right) \cdot \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{1}{6}, 1\right)} \]

      4. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{-1}{2}, 1\right) \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{6}, 1\right) \]

      5. *-lowering-*.f64 38.9

         \[\leadsto \mathsf{fma}\left(x, x \cdot -0.5, 1\right) \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, 0.16666666666666666, 1\right) \]

   8. Simplified 38.9%

      \[\leadsto \mathsf{fma}\left(x, x \cdot -0.5, 1\right) \cdot \color{blue}{\mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right)} \]

   9. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{1}{6} \cdot \left({y}^{2} \cdot \left(1 + \frac{-1}{2} \cdot {x}^{2}\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \left(1 + \frac{-1}{2} \cdot {x}^{2}\right)} \]

       2. *-commutative N/A

          \[\leadsto \color{blue}{\left({y}^{2} \cdot \frac{1}{6}\right)} \cdot \left(1 + \frac{-1}{2} \cdot {x}^{2}\right) \]

       3. associate-*l* N/A

          \[\leadsto \color{blue}{{y}^{2} \cdot \left(\frac{1}{6} \cdot \left(1 + \frac{-1}{2} \cdot {x}^{2}\right)\right)} \]

       4. unpow2 N/A

          \[\leadsto \color{blue}{\left(y \cdot y\right)} \cdot \left(\frac{1}{6} \cdot \left(1 + \frac{-1}{2} \cdot {x}^{2}\right)\right) \]

       5. associate-*l* N/A

          \[\leadsto \color{blue}{y \cdot \left(y \cdot \left(\frac{1}{6} \cdot \left(1 + \frac{-1}{2} \cdot {x}^{2}\right)\right)\right)} \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{y \cdot \left(y \cdot \left(\frac{1}{6} \cdot \left(1 + \frac{-1}{2} \cdot {x}^{2}\right)\right)\right)} \]

       7. *-lowering-*.f64 N/A

          \[\leadsto y \cdot \color{blue}{\left(y \cdot \left(\frac{1}{6} \cdot \left(1 + \frac{-1}{2} \cdot {x}^{2}\right)\right)\right)} \]

       8. +-commutative N/A

          \[\leadsto y \cdot \left(y \cdot \left(\frac{1}{6} \cdot \color{blue}{\left(\frac{-1}{2} \cdot {x}^{2} + 1\right)}\right)\right) \]

       9. distribute-rgt-in N/A

          \[\leadsto y \cdot \left(y \cdot \color{blue}{\left(\left(\frac{-1}{2} \cdot {x}^{2}\right) \cdot \frac{1}{6} + 1 \cdot \frac{1}{6}\right)}\right) \]

       10. *-commutative N/A

           \[\leadsto y \cdot \left(y \cdot \left(\color{blue}{\left({x}^{2} \cdot \frac{-1}{2}\right)} \cdot \frac{1}{6} + 1 \cdot \frac{1}{6}\right)\right) \]

       11. associate-*l* N/A

           \[\leadsto y \cdot \left(y \cdot \left(\color{blue}{{x}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{1}{6}\right)} + 1 \cdot \frac{1}{6}\right)\right) \]

       12. metadata-eval N/A

           \[\leadsto y \cdot \left(y \cdot \left({x}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{1}{6}\right) + \color{blue}{\frac{1}{6}}\right)\right) \]

       13. accelerator-lowering-fma.f64 N/A

           \[\leadsto y \cdot \left(y \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{-1}{2} \cdot \frac{1}{6}, \frac{1}{6}\right)}\right) \]

       14. unpow2 N/A

           \[\leadsto y \cdot \left(y \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{2} \cdot \frac{1}{6}, \frac{1}{6}\right)\right) \]

       15. *-lowering-*.f64 N/A

           \[\leadsto y \cdot \left(y \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{2} \cdot \frac{1}{6}, \frac{1}{6}\right)\right) \]

       16. metadata-eval 39.0

           \[\leadsto y \cdot \left(y \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{-0.08333333333333333}, 0.16666666666666666\right)\right) \]

   11. Simplified 39.0%

       \[\leadsto \color{blue}{y \cdot \left(y \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.16666666666666666\right)\right)} \]

   if -0.050000000000000003 < (*.f64 (cos.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in 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. *-lowering-*.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. cos-lowering-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. accelerator-lowering-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 87.6%

      \[\leadsto \color{blue}{\cos x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) + 1} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{1}{6} + \frac{1}{120} \cdot {y}^{2}, 1\right)} \]

      3. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{6} + \frac{1}{120} \cdot {y}^{2}, 1\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{6} + \frac{1}{120} \cdot {y}^{2}, 1\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(y \cdot y, \color{blue}{\frac{1}{120} \cdot {y}^{2} + \frac{1}{6}}, 1\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y \cdot y, \color{blue}{{y}^{2} \cdot \frac{1}{120}} + \frac{1}{6}, 1\right) \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(y \cdot y, \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{1}{120}, \frac{1}{6}\right)}, 1\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      9. *-lowering-*.f64 73.9

         \[\leadsto \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, 0.008333333333333333, 0.16666666666666666\right), 1\right) \]

   8. Simplified 73.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), 1\right)} \]

   9. Taylor expanded in y around inf

      \[\leadsto \mathsf{fma}\left(y \cdot y, \color{blue}{\frac{1}{120} \cdot {y}^{2}}, 1\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(y \cdot y, \color{blue}{{y}^{2} \cdot \frac{1}{120}}, 1\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(y \cdot y, \color{blue}{{y}^{2} \cdot \frac{1}{120}}, 1\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(y \cdot y, \color{blue}{\left(y \cdot y\right)} \cdot \frac{1}{120}, 1\right) \]

       4. *-lowering-*.f64 72.7

          \[\leadsto \mathsf{fma}\left(y \cdot y, \color{blue}{\left(y \cdot y\right)} \cdot 0.008333333333333333, 1\right) \]

   11. Simplified 72.7%

       \[\leadsto \mathsf{fma}\left(y \cdot y, \color{blue}{\left(y \cdot y\right) \cdot 0.008333333333333333}, 1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 25: 54.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos x \cdot \frac{\sinh y}{y} \leq -0.05:\\ \;\;\;\;-0.5 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cos x) (/ (sinh y) y)) -0.05)
   (* -0.5 (* x x))
   (fma (* y y) 0.16666666666666666 1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((cos(x) * (sinh(y) / y)) <= -0.05) {
		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 (Float64(cos(x) * Float64(sinh(y) / y)) <= -0.05)
		tmp = Float64(-0.5 * Float64(x * x));
	else
		tmp = fma(Float64(y * y), 0.16666666666666666, 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Cos[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], -0.05], N[(-0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(y * y), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cos.f64 x) (/.f64 (sinh.f64 y) y)) < -0.050000000000000003

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\cos x} \]
   4. Step-by-step derivation

      1. cos-lowering-cos.f64 56.5

         \[\leadsto \color{blue}{\cos x} \]

   5. Simplified 56.5%

      \[\leadsto \color{blue}{\cos x} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + \frac{-1}{2} \cdot {x}^{2}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{-1}{2} \cdot {x}^{2} + 1} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1}{2}, {x}^{2}, 1\right)} \]

      3. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{x \cdot x}, 1\right) \]

      4. *-lowering-*.f64 22.6

         \[\leadsto \mathsf{fma}\left(-0.5, \color{blue}{x \cdot x}, 1\right) \]

   8. Simplified 22.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, x \cdot x, 1\right)} \]

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot {x}^{2}} \]
   10. Step-by-step derivation

       1. *-lowering-*.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. *-lowering-*.f64 22.6

          \[\leadsto -0.5 \cdot \color{blue}{\left(x \cdot x\right)} \]

   11. Simplified 22.6%

       \[\leadsto \color{blue}{-0.5 \cdot \left(x \cdot x\right)} \]

   if -0.050000000000000003 < (*.f64 (cos.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \cdot \frac{\sinh y}{y} \]
   4. Step-by-step derivation

   5. Simplified 86.3%

      \[\leadsto \color{blue}{1} \cdot \frac{\sinh y}{y} \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1 + \frac{1}{6} \cdot {y}^{2}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{6} \cdot {y}^{2} + 1} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{{y}^{2} \cdot \frac{1}{6}} + 1 \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{1}{6}, 1\right)} \]

      4. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{6}, 1\right) \]

      5. *-lowering-*.f64 63.5

         \[\leadsto \mathsf{fma}\left(\color{blue}{y \cdot y}, 0.16666666666666666, 1\right) \]

   8. Simplified 63.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 26: 72.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos x \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right) \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (cos x) -0.05)
   (*
    (fma 0.16666666666666666 (* y y) 1.0)
    (fma
     x
     (*
      x
      (fma
       (* x x)
       (fma (* x x) -0.001388888888888889 0.041666666666666664)
       -0.5))
     1.0))
   (/
    (fma
     (fma
      y
      (* y (fma y (* y 0.0001984126984126984) 0.008333333333333333))
      0.16666666666666666)
     (* y (* y y))
     y)
    y)))

C

double code(double x, double y) {
	double tmp;
	if (cos(x) <= -0.05) {
		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 {
		tmp = fma(fma(y, (y * fma(y, (y * 0.0001984126984126984), 0.008333333333333333)), 0.16666666666666666), (y * (y * y)), y) / y;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (cos(x) <= -0.05)
		tmp = Float64(fma(0.16666666666666666, Float64(y * y), 1.0) * fma(x, Float64(x * fma(Float64(x * x), fma(Float64(x * x), -0.001388888888888889, 0.041666666666666664), -0.5)), 1.0));
	else
		tmp = Float64(fma(fma(y, Float64(y * fma(y, Float64(y * 0.0001984126984126984), 0.008333333333333333)), 0.16666666666666666), Float64(y * Float64(y * y)), y) / y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[Cos[x], $MachinePrecision], -0.05], N[(N[(0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.001388888888888889 + 0.041666666666666664), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y * N[(y * N[(y * N[(y * 0.0001984126984126984), $MachinePrecision] + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 x) < -0.050000000000000003

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\cos x + \frac{1}{6} \cdot \left({y}^{2} \cdot \cos x\right)} \]
   4. Step-by-step derivation

      1. *-lft-identity N/A

         \[\leadsto \color{blue}{1 \cdot \cos x} + \frac{1}{6} \cdot \left({y}^{2} \cdot \cos x\right) \]

      2. associate-*r* N/A

         \[\leadsto 1 \cdot \cos x + \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \cos x} \]

      3. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\cos x \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\cos x \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)} \]

      5. cos-lowering-cos.f64 N/A

         \[\leadsto \color{blue}{\cos x} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right) \]

      6. +-commutative N/A

         \[\leadsto \cos x \cdot \color{blue}{\left(\frac{1}{6} \cdot {y}^{2} + 1\right)} \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \cos x \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{6}, {y}^{2}, 1\right)} \]

      8. unpow2 N/A

         \[\leadsto \cos x \cdot \mathsf{fma}\left(\frac{1}{6}, \color{blue}{y \cdot y}, 1\right) \]

      9. *-lowering-*.f64 80.3

         \[\leadsto \cos x \cdot \mathsf{fma}\left(0.16666666666666666, \color{blue}{y \cdot y}, 1\right) \]

   5. Simplified 80.3%

      \[\leadsto \color{blue}{\cos x \cdot \mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right)} \]

   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. accelerator-lowering-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. *-lowering-*.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. accelerator-lowering-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. *-lowering-*.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. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{-1}{720}} + \frac{1}{24}, \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(\frac{1}{6}, y \cdot y, 1\right) \]

      13. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{-1}{720}, \frac{1}{24}\right)}, \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(\frac{1}{6}, y \cdot y, 1\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{720}, \frac{1}{24}\right), \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(\frac{1}{6}, y \cdot y, 1\right) \]

      15. *-lowering-*.f64 43.4

          \[\leadsto \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right) \cdot \mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right) \]

   8. Simplified 43.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right)} \cdot \mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right) \]

   if -0.050000000000000003 < (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 86.3%

      \[\leadsto \color{blue}{1} \cdot \frac{\sinh y}{y} \]

   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

         \[\leadsto 1 \cdot \frac{\color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right) \cdot y}}{y} \]

      2. +-commutative N/A

         \[\leadsto 1 \cdot \frac{\color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) + 1\right)} \cdot y}{y} \]

      3. distribute-lft1-in N/A

         \[\leadsto 1 \cdot \frac{\color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right) \cdot y + y}}{y} \]

      4. *-commutative N/A

         \[\leadsto 1 \cdot \frac{\color{blue}{\left(\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot {y}^{2}\right)} \cdot y + y}{y} \]

      5. associate-*l* N/A

         \[\leadsto 1 \cdot \frac{\color{blue}{\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot \left({y}^{2} \cdot y\right)} + y}{y} \]

      6. *-rgt-identity N/A

         \[\leadsto 1 \cdot \frac{\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot \left({y}^{2} \cdot \color{blue}{\left(y \cdot 1\right)}\right) + y}{y} \]

      7. *-inverses N/A

         \[\leadsto 1 \cdot \frac{\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot \left({y}^{2} \cdot \left(y \cdot \color{blue}{\frac{y}{y}}\right)\right) + y}{y} \]

      8. associate-/l* N/A

         \[\leadsto 1 \cdot \frac{\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot \left({y}^{2} \cdot \color{blue}{\frac{y \cdot y}{y}}\right) + y}{y} \]

      9. unpow2 N/A

         \[\leadsto 1 \cdot \frac{\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot \left({y}^{2} \cdot \frac{\color{blue}{{y}^{2}}}{y}\right) + y}{y} \]

      10. associate-/l* N/A

          \[\leadsto 1 \cdot \frac{\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot \color{blue}{\frac{{y}^{2} \cdot {y}^{2}}{y}} + y}{y} \]

      11. pow-sqr N/A

          \[\leadsto 1 \cdot \frac{\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot \frac{\color{blue}{{y}^{\left(2 \cdot 2\right)}}}{y} + y}{y} \]

      12. metadata-eval N/A

          \[\leadsto 1 \cdot \frac{\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot \frac{{y}^{\color{blue}{4}}}{y} + y}{y} \]

      13. accelerator-lowering-fma.f64 N/A

          \[\leadsto 1 \cdot \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right), \frac{{y}^{4}}{y}, y\right)}}{y} \]

   8. Simplified 78.7%

      \[\leadsto 1 \cdot \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)}}{y} \]
   9. Step-by-step derivation

      1. *-lft-identity N/A

         \[\leadsto \color{blue}{\frac{\left(\left(y \cdot y\right) \cdot \left(y \cdot \left(y \cdot \frac{1}{5040}\right) + \frac{1}{120}\right) + \frac{1}{6}\right) \cdot \left(y \cdot \left(y \cdot y\right)\right) + y}{y}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\left(y \cdot y\right) \cdot \left(y \cdot \left(y \cdot \frac{1}{5040}\right) + \frac{1}{120}\right) + \frac{1}{6}\right) \cdot \left(y \cdot \left(y \cdot y\right)\right) + y}{y}} \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(y \cdot y\right) \cdot \left(y \cdot \left(y \cdot \frac{1}{5040}\right) + \frac{1}{120}\right) + \frac{1}{6}, y \cdot \left(y \cdot y\right), y\right)}}{y} \]

      4. associate-*l* N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{y \cdot \left(y \cdot \left(y \cdot \left(y \cdot \frac{1}{5040}\right) + \frac{1}{120}\right)\right)} + \frac{1}{6}, y \cdot \left(y \cdot y\right), y\right)}{y} \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(y, y \cdot \left(y \cdot \left(y \cdot \frac{1}{5040}\right) + \frac{1}{120}\right), \frac{1}{6}\right)}, y \cdot \left(y \cdot y\right), y\right)}{y} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(y, \color{blue}{y \cdot \left(y \cdot \left(y \cdot \frac{1}{5040}\right) + \frac{1}{120}\right)}, \frac{1}{6}\right), y \cdot \left(y \cdot y\right), y\right)}{y} \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \color{blue}{\mathsf{fma}\left(y, y \cdot \frac{1}{5040}, \frac{1}{120}\right)}, \frac{1}{6}\right), y \cdot \left(y \cdot y\right), y\right)}{y} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, \color{blue}{y \cdot \frac{1}{5040}}, \frac{1}{120}\right), \frac{1}{6}\right), y \cdot \left(y \cdot y\right), y\right)}{y} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), \color{blue}{y \cdot \left(y \cdot y\right)}, y\right)}{y} \]

      10. *-lowering-*.f64 78.7

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y \cdot \color{blue}{\left(y \cdot y\right)}, y\right)}{y} \]

   10. Applied egg-rr 78.7%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos x \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right) \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 27: 35.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos x \leq -0.05:\\ \;\;\;\;-0.5 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= (cos x) -0.05) (* -0.5 (* x x)) 1.0))

C

double code(double x, double y) {
	double tmp;
	if (cos(x) <= -0.05) {
		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) <= (-0.05d0)) 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) <= -0.05) {
		tmp = -0.5 * (x * x);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if math.cos(x) <= -0.05:
		tmp = -0.5 * (x * x)
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (cos(x) <= -0.05)
		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) <= -0.05)
		tmp = -0.5 * (x * x);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[Cos[x], $MachinePrecision], -0.05], N[(-0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 x) < -0.050000000000000003

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\cos x} \]
   4. Step-by-step derivation

      1. cos-lowering-cos.f64 56.5

         \[\leadsto \color{blue}{\cos x} \]

   5. Simplified 56.5%

      \[\leadsto \color{blue}{\cos x} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + \frac{-1}{2} \cdot {x}^{2}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{-1}{2} \cdot {x}^{2} + 1} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1}{2}, {x}^{2}, 1\right)} \]

      3. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{x \cdot x}, 1\right) \]

      4. *-lowering-*.f64 22.6

         \[\leadsto \mathsf{fma}\left(-0.5, \color{blue}{x \cdot x}, 1\right) \]

   8. Simplified 22.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, x \cdot x, 1\right)} \]

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot {x}^{2}} \]
   10. Step-by-step derivation

       1. *-lowering-*.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. *-lowering-*.f64 22.6

          \[\leadsto -0.5 \cdot \color{blue}{\left(x \cdot x\right)} \]

   11. Simplified 22.6%

       \[\leadsto \color{blue}{-0.5 \cdot \left(x \cdot x\right)} \]

   if -0.050000000000000003 < (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. cos-lowering-cos.f64 55.6

         \[\leadsto \color{blue}{\cos x} \]

   5. Simplified 55.6%

      \[\leadsto \color{blue}{\cos x} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \]
   7. Step-by-step derivation

   8. Simplified 42.2%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 28: 28.3% accurate, 217.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 1.0)

C

double code(double x, double y) {
	return 1.0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0
end function

Java

public static double code(double x, double y) {
	return 1.0;
}

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0;
end

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 100.0%

   \[\cos x \cdot \frac{\sinh y}{y} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\cos x} \]
4. Step-by-step derivation

   1. cos-lowering-cos.f64 55.8

      \[\leadsto \color{blue}{\cos x} \]

5. Simplified 55.8%

   \[\leadsto \color{blue}{\cos x} \]

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1} \]
7. Step-by-step derivation

8. Simplified 32.9%

   \[\leadsto \color{blue}{1} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024198 
(FPCore (x y)
  :name "Linear.Quaternion:$csin from linear-1.19.1.3"
  :precision binary64
  (* (cos x) (/ (sinh y) y)))