Linear.Quaternion:$csin from linear-1.19.1.3

Percentage Accurate: 100.0% → 100.0%

Time: 12.6s

Alternatives: 18

Speedup: 1.0×

• 49.5% 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: 99.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y}{y}\\ t_1 := \cos x \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), 1\right)\\ \mathbf{elif}\;t\_1 \leq 0.9999999999999889:\\ \;\;\;\;\cos x \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sinh y) y)) (t_1 (* (cos x) t_0)))
   (if (<= t_1 (- INFINITY))
     (*
      (fma
       x
       (*
        x
        (fma
         x
         (* x (fma x (* x -0.001388888888888889) 0.041666666666666664))
         -0.5))
       1.0)
      (fma y (* y (fma (* y y) 0.008333333333333333 0.16666666666666666)) 1.0))
     (if (<= t_1 0.9999999999999889)
       (*
        (cos x)
        (fma
         y
         (*
          y
          (fma
           y
           (* y (fma (* y y) 0.0001984126984126984 0.008333333333333333))
           0.16666666666666666))
         1.0))
       t_0))))

C

double code(double x, double y) {
	double t_0 = sinh(y) / y;
	double t_1 = cos(x) * t_0;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma(x, (x * fma(x, (x * fma(x, (x * -0.001388888888888889), 0.041666666666666664)), -0.5)), 1.0) * fma(y, (y * fma((y * y), 0.008333333333333333, 0.16666666666666666)), 1.0);
	} else if (t_1 <= 0.9999999999999889) {
		tmp = cos(x) * fma(y, (y * fma(y, (y * fma((y * y), 0.0001984126984126984, 0.008333333333333333)), 0.16666666666666666)), 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sinh(y) / y)
	t_1 = Float64(cos(x) * t_0)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(fma(x, Float64(x * fma(x, Float64(x * fma(x, Float64(x * -0.001388888888888889), 0.041666666666666664)), -0.5)), 1.0) * fma(y, Float64(y * fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666)), 1.0));
	elseif (t_1 <= 0.9999999999999889)
		tmp = Float64(cos(x) * fma(y, Float64(y * fma(y, Float64(y * fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333)), 0.16666666666666666)), 1.0));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(x * N[(x * N[(x * N[(x * N[(x * N[(x * -0.001388888888888889), $MachinePrecision] + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.9999999999999889], N[(N[Cos[x], $MachinePrecision] * N[(y * N[(y * N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

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

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

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

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

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

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 100.0

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

   8. Simplified 100.0%

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

   if -inf.0 < (*.f64 (cos.f64 x) (/.f64 (sinh.f64 y) y)) < 0.9999999999999889

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

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

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

   5. Simplified 98.6%

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

   if 0.9999999999999889 < (*.f64 (cos.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 100.0%

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

      1. *-lft-identity N/A

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

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

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

      3. sinh-lowering-sinh.f64 100.0

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

   7. Applied egg-rr 100.0%

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

Alternative 3: 99.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y}{y}\\ t_1 := \cos x \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), 1\right)\\ \mathbf{elif}\;t\_1 \leq 0.9999999999999889:\\ \;\;\;\;\cos x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sinh y) y)) (t_1 (* (cos x) t_0)))
   (if (<= t_1 (- INFINITY))
     (*
      (fma
       x
       (*
        x
        (fma
         x
         (* x (fma x (* x -0.001388888888888889) 0.041666666666666664))
         -0.5))
       1.0)
      (fma y (* y (fma (* y y) 0.008333333333333333 0.16666666666666666)) 1.0))
     (if (<= t_1 0.9999999999999889)
       (*
        (cos x)
        (fma
         (* y y)
         (fma y (* y 0.008333333333333333) 0.16666666666666666)
         1.0))
       t_0))))

C

double code(double x, double y) {
	double t_0 = sinh(y) / y;
	double t_1 = cos(x) * t_0;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma(x, (x * fma(x, (x * fma(x, (x * -0.001388888888888889), 0.041666666666666664)), -0.5)), 1.0) * fma(y, (y * fma((y * y), 0.008333333333333333, 0.16666666666666666)), 1.0);
	} else if (t_1 <= 0.9999999999999889) {
		tmp = cos(x) * fma((y * y), fma(y, (y * 0.008333333333333333), 0.16666666666666666), 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sinh(y) / y)
	t_1 = Float64(cos(x) * t_0)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(fma(x, Float64(x * fma(x, Float64(x * fma(x, Float64(x * -0.001388888888888889), 0.041666666666666664)), -0.5)), 1.0) * fma(y, Float64(y * fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666)), 1.0));
	elseif (t_1 <= 0.9999999999999889)
		tmp = Float64(cos(x) * fma(Float64(y * y), fma(y, Float64(y * 0.008333333333333333), 0.16666666666666666), 1.0));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(x * N[(x * N[(x * N[(x * N[(x * N[(x * -0.001388888888888889), $MachinePrecision] + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.9999999999999889], N[(N[Cos[x], $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

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

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

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

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

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

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 100.0

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

   8. Simplified 100.0%

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

   if -inf.0 < (*.f64 (cos.f64 x) (/.f64 (sinh.f64 y) y)) < 0.9999999999999889

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

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

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

      3. unpow2 N/A

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

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

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

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

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

      10. *-commutative N/A

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

      11. *-lowering-*.f64 98.3

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

   5. Simplified 98.3%

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

   if 0.9999999999999889 < (*.f64 (cos.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 100.0%

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

      1. *-lft-identity N/A

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

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

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

      3. sinh-lowering-sinh.f64 100.0

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

   7. Applied egg-rr 100.0%

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

Alternative 4: 99.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y}{y}\\ t_1 := \cos x \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), 1\right)\\ \mathbf{elif}\;t\_1 \leq 0.9999999999999889:\\ \;\;\;\;\cos x \cdot \mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sinh y) y)) (t_1 (* (cos x) t_0)))
   (if (<= t_1 (- INFINITY))
     (*
      (fma
       x
       (*
        x
        (fma
         x
         (* x (fma x (* x -0.001388888888888889) 0.041666666666666664))
         -0.5))
       1.0)
      (fma y (* y (fma (* y y) 0.008333333333333333 0.16666666666666666)) 1.0))
     (if (<= t_1 0.9999999999999889)
       (* (cos x) (fma 0.16666666666666666 (* y y) 1.0))
       t_0))))

C

double code(double x, double y) {
	double t_0 = sinh(y) / y;
	double t_1 = cos(x) * t_0;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma(x, (x * fma(x, (x * fma(x, (x * -0.001388888888888889), 0.041666666666666664)), -0.5)), 1.0) * fma(y, (y * fma((y * y), 0.008333333333333333, 0.16666666666666666)), 1.0);
	} else if (t_1 <= 0.9999999999999889) {
		tmp = cos(x) * fma(0.16666666666666666, (y * y), 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sinh(y) / y)
	t_1 = Float64(cos(x) * t_0)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(fma(x, Float64(x * fma(x, Float64(x * fma(x, Float64(x * -0.001388888888888889), 0.041666666666666664)), -0.5)), 1.0) * fma(y, Float64(y * fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666)), 1.0));
	elseif (t_1 <= 0.9999999999999889)
		tmp = Float64(cos(x) * fma(0.16666666666666666, Float64(y * y), 1.0));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(x * N[(x * N[(x * N[(x * N[(x * N[(x * -0.001388888888888889), $MachinePrecision] + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.9999999999999889], N[(N[Cos[x], $MachinePrecision] * N[(0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

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

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

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

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

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

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 100.0

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

   8. Simplified 100.0%

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

   if -inf.0 < (*.f64 (cos.f64 x) (/.f64 (sinh.f64 y) y)) < 0.9999999999999889

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 98.0

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

   5. Simplified 98.0%

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

   if 0.9999999999999889 < (*.f64 (cos.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 100.0%

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

      1. *-lft-identity N/A

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

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

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

      3. sinh-lowering-sinh.f64 100.0

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

   7. Applied egg-rr 100.0%

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

Alternative 5: 99.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y}{y}\\ t_1 := \cos x \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), 1\right)\\ \mathbf{elif}\;t\_1 \leq 0.9999999999999889:\\ \;\;\;\;\cos x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sinh y) y)) (t_1 (* (cos x) t_0)))
   (if (<= t_1 (- INFINITY))
     (*
      (fma
       x
       (*
        x
        (fma
         x
         (* x (fma x (* x -0.001388888888888889) 0.041666666666666664))
         -0.5))
       1.0)
      (fma y (* y (fma (* y y) 0.008333333333333333 0.16666666666666666)) 1.0))
     (if (<= t_1 0.9999999999999889) (cos x) t_0))))

C

double code(double x, double y) {
	double t_0 = sinh(y) / y;
	double t_1 = cos(x) * t_0;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma(x, (x * fma(x, (x * fma(x, (x * -0.001388888888888889), 0.041666666666666664)), -0.5)), 1.0) * fma(y, (y * fma((y * y), 0.008333333333333333, 0.16666666666666666)), 1.0);
	} else if (t_1 <= 0.9999999999999889) {
		tmp = cos(x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sinh(y) / y)
	t_1 = Float64(cos(x) * t_0)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(fma(x, Float64(x * fma(x, Float64(x * fma(x, Float64(x * -0.001388888888888889), 0.041666666666666664)), -0.5)), 1.0) * fma(y, Float64(y * fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666)), 1.0));
	elseif (t_1 <= 0.9999999999999889)
		tmp = cos(x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(x * N[(x * N[(x * N[(x * N[(x * N[(x * -0.001388888888888889), $MachinePrecision] + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.9999999999999889], N[Cos[x], $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

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

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

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

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

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

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 100.0

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

   8. Simplified 100.0%

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

   if -inf.0 < (*.f64 (cos.f64 x) (/.f64 (sinh.f64 y) y)) < 0.9999999999999889

   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 97.6

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

   5. Simplified 97.6%

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

   if 0.9999999999999889 < (*.f64 (cos.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 100.0%

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

      1. *-lft-identity N/A

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

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

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

      3. sinh-lowering-sinh.f64 100.0

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

   7. Applied egg-rr 100.0%

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

Alternative 6: 94.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x \cdot \frac{\sinh y}{y}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), 1\right)\\ \mathbf{elif}\;t\_0 \leq 0.9999999999999889:\\ \;\;\;\;\cos x\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (cos x) (/ (sinh y) y))))
   (if (<= t_0 (- INFINITY))
     (*
      (fma
       x
       (*
        x
        (fma
         x
         (* x (fma x (* x -0.001388888888888889) 0.041666666666666664))
         -0.5))
       1.0)
      (fma y (* y (fma (* y y) 0.008333333333333333 0.16666666666666666)) 1.0))
     (if (<= t_0 0.9999999999999889)
       (cos x)
       (/
        (*
         y
         (fma
          (* y y)
          (fma
           y
           (* y (fma (* y y) 0.0001984126984126984 0.008333333333333333))
           0.16666666666666666)
          1.0))
        y)))))

C

double code(double x, double y) {
	double t_0 = cos(x) * (sinh(y) / y);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma(x, (x * fma(x, (x * fma(x, (x * -0.001388888888888889), 0.041666666666666664)), -0.5)), 1.0) * fma(y, (y * fma((y * y), 0.008333333333333333, 0.16666666666666666)), 1.0);
	} else if (t_0 <= 0.9999999999999889) {
		tmp = cos(x);
	} else {
		tmp = (y * fma((y * y), fma(y, (y * fma((y * y), 0.0001984126984126984, 0.008333333333333333)), 0.16666666666666666), 1.0)) / y;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) * Float64(sinh(y) / y))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(fma(x, Float64(x * fma(x, Float64(x * fma(x, Float64(x * -0.001388888888888889), 0.041666666666666664)), -0.5)), 1.0) * fma(y, Float64(y * fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666)), 1.0));
	elseif (t_0 <= 0.9999999999999889)
		tmp = cos(x);
	else
		tmp = Float64(Float64(y * fma(Float64(y * y), fma(y, Float64(y * fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333)), 0.16666666666666666), 1.0)) / y);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(x * N[(x * N[(x * N[(x * N[(x * N[(x * -0.001388888888888889), $MachinePrecision] + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.9999999999999889], N[Cos[x], $MachinePrecision], N[(N[(y * N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

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

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

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

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

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

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 100.0

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

   8. Simplified 100.0%

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

   if -inf.0 < (*.f64 (cos.f64 x) (/.f64 (sinh.f64 y) y)) < 0.9999999999999889

   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 97.6

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

   5. Simplified 97.6%

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

   if 0.9999999999999889 < (*.f64 (cos.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 100.0%

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

      1. *-lft-identity N/A

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

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

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

      3. sinh-lowering-sinh.f64 100.0

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

   7. Applied egg-rr 100.0%

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

   8. Taylor expanded in y around 0

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

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

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

      2. +-commutative N/A

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

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

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

      4. unpow2 N/A

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

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

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

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

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

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

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

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

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 93.3

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

   10. Simplified 93.3%

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

Alternative 7: 53.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos x \cdot \frac{\sinh y}{y} \leq -0.005:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot -0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cos x) (/ (sinh y) y)) -0.005)
   (fma x (* x -0.5) 1.0)
   (fma 0.16666666666666666 (* y y) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((cos(x) * (sinh(y) / y)) <= -0.005) {
		tmp = fma(x, (x * -0.5), 1.0);
	} else {
		tmp = fma(0.16666666666666666, (y * y), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(cos(x) * Float64(sinh(y) / y)) <= -0.005)
		tmp = fma(x, Float64(x * -0.5), 1.0);
	else
		tmp = fma(0.16666666666666666, Float64(y * y), 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], -0.005], N[(x * N[(x * -0.5), $MachinePrecision] + 1.0), $MachinePrecision], N[(0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   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 47.2

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

   5. Simplified 47.2%

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

   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. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 29.6

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

   8. Simplified 29.6%

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

   if -0.0050000000000000001 < (*.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

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

      1. +-commutative N/A

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 75.0

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

   5. Simplified 75.0%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 59.6

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

   8. Simplified 59.6%

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

Alternative 8: 46.2% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. cos-lowering-cos.f64 79.8

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

   5. Simplified 79.8%

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

   6. Taylor expanded in x around 0

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

   8. Simplified 45.4%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 44.9

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

   5. Simplified 44.9%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 44.9

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

   8. Simplified 44.9%

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

   9. Taylor expanded in y around inf

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

       1. *-commutative N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

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

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

       6. *-commutative N/A

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

       7. *-lowering-*.f64 44.9

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

   11. Simplified 44.9%

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

Alternative 9: 72.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos x \leq -0.005:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (cos x) -0.005)
   (*
    (fma
     x
     (*
      x
      (fma
       x
       (* x (fma x (* x -0.001388888888888889) 0.041666666666666664))
       -0.5))
     1.0)
    (fma y (* y (fma (* y y) 0.008333333333333333 0.16666666666666666)) 1.0))
   (/
    (*
     y
     (fma
      (* y y)
      (fma
       y
       (* y (fma (* y y) 0.0001984126984126984 0.008333333333333333))
       0.16666666666666666)
      1.0))
    y)))

C

double code(double x, double y) {
	double tmp;
	if (cos(x) <= -0.005) {
		tmp = fma(x, (x * fma(x, (x * fma(x, (x * -0.001388888888888889), 0.041666666666666664)), -0.5)), 1.0) * fma(y, (y * fma((y * y), 0.008333333333333333, 0.16666666666666666)), 1.0);
	} else {
		tmp = (y * fma((y * y), fma(y, (y * fma((y * y), 0.0001984126984126984, 0.008333333333333333)), 0.16666666666666666), 1.0)) / y;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (cos(x) <= -0.005)
		tmp = Float64(fma(x, Float64(x * fma(x, Float64(x * fma(x, Float64(x * -0.001388888888888889), 0.041666666666666664)), -0.5)), 1.0) * fma(y, Float64(y * fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666)), 1.0));
	else
		tmp = Float64(Float64(y * fma(Float64(y * y), fma(y, Float64(y * fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333)), 0.16666666666666666), 1.0)) / y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[Cos[x], $MachinePrecision], -0.005], N[(N[(x * N[(x * N[(x * N[(x * N[(x * N[(x * -0.001388888888888889), $MachinePrecision] + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(y * N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

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

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

   5. Simplified 54.9%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

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

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

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

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 54.9

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

   8. Simplified 54.9%

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

   if -0.0050000000000000001 < (cos.f64 x)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 84.1%

      \[\leadsto \color{blue}{1} \cdot \frac{\sinh y}{y} \]
   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 84.1

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

   7. Applied egg-rr 84.1%

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

   8. Taylor expanded in y around 0

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

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

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

      2. +-commutative N/A

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

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

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

      4. unpow2 N/A

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

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

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

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

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

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

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

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

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 78.8

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

   10. Simplified 78.8%

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

Alternative 10: 71.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right)\\ \mathbf{if}\;\cos x \leq -0.005:\\ \;\;\;\;\mathsf{fma}\left(-0.5, x \cdot x, 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, t\_0, 0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot t\_0, 0.16666666666666666\right), 1\right)}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma (* y y) 0.0001984126984126984 0.008333333333333333)))
   (if (<= (cos x) -0.005)
     (*
      (fma -0.5 (* x x) 1.0)
      (fma y (* y (fma (* y y) t_0 0.16666666666666666)) 1.0))
     (/ (* y (fma (* y y) (fma y (* y t_0) 0.16666666666666666) 1.0)) y))))

C

double code(double x, double y) {
	double t_0 = fma((y * y), 0.0001984126984126984, 0.008333333333333333);
	double tmp;
	if (cos(x) <= -0.005) {
		tmp = fma(-0.5, (x * x), 1.0) * fma(y, (y * fma((y * y), t_0, 0.16666666666666666)), 1.0);
	} else {
		tmp = (y * fma((y * y), fma(y, (y * t_0), 0.16666666666666666), 1.0)) / y;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333)
	tmp = 0.0
	if (cos(x) <= -0.005)
		tmp = Float64(fma(-0.5, Float64(x * x), 1.0) * fma(y, Float64(y * fma(Float64(y * y), t_0, 0.16666666666666666)), 1.0));
	else
		tmp = Float64(Float64(y * fma(Float64(y * y), fma(y, Float64(y * t_0), 0.16666666666666666), 1.0)) / y);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision]}, If[LessEqual[N[Cos[x], $MachinePrecision], -0.005], N[(N[(-0.5 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * t$95$0 + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(y * N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * t$95$0), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   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. accelerator-lowering-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 55.4

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

   5. Simplified 55.4%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

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

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

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

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

      6. +-commutative N/A

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

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

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

      8. unpow2 N/A

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

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

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 53.9

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

   8. Simplified 53.9%

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

   if -0.0050000000000000001 < (cos.f64 x)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 84.1%

      \[\leadsto \color{blue}{1} \cdot \frac{\sinh y}{y} \]
   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 84.1

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

   7. Applied egg-rr 84.1%

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

   8. Taylor expanded in y around 0

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

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

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

      2. +-commutative N/A

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

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

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

      4. unpow2 N/A

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

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

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

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

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

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

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

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

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 78.8

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

   10. Simplified 78.8%

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

Alternative 11: 71.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)\\ \mathbf{if}\;\cos x \leq -0.005:\\ \;\;\;\;\mathsf{fma}\left(-0.5, x \cdot x, 1\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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) -0.005) (* (fma -0.5 (* x x) 1.0) t_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) <= -0.005) {
		tmp = fma(-0.5, (x * x), 1.0) * t_0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(y, Float64(y * fma(Float64(y * y), fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666)), 1.0)
	tmp = 0.0
	if (cos(x) <= -0.005)
		tmp = Float64(fma(-0.5, Float64(x * x), 1.0) * t_0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[N[Cos[x], $MachinePrecision], -0.005], N[(N[(-0.5 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

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

   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. accelerator-lowering-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 55.4

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

   5. Simplified 55.4%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

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

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

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

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

      6. +-commutative N/A

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

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

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

      8. unpow2 N/A

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

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

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 53.9

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

   8. Simplified 53.9%

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

   if -0.0050000000000000001 < (cos.f64 x)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 84.1%

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

   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. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

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

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

      8. +-commutative N/A

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

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

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

      10. unpow2 N/A

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

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

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

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

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

      15. unpow2 N/A

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

      16. *-lowering-*.f64 76.1

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

   8. Simplified 76.1%

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

Alternative 12: 70.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos x \leq -0.005:\\ \;\;\;\;\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), 1\right) \cdot \mathsf{fma}\left(-0.5, x \cdot x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (cos x) -0.005)
   (*
    (fma y (* y (fma (* y y) 0.008333333333333333 0.16666666666666666)) 1.0)
    (fma -0.5 (* x x) 1.0))
   (fma
    y
    (*
     y
     (fma
      (* y y)
      (fma (* y y) 0.0001984126984126984 0.008333333333333333)
      0.16666666666666666))
    1.0)))

C

double code(double x, double y) {
	double tmp;
	if (cos(x) <= -0.005) {
		tmp = fma(y, (y * fma((y * y), 0.008333333333333333, 0.16666666666666666)), 1.0) * fma(-0.5, (x * x), 1.0);
	} else {
		tmp = fma(y, (y * fma((y * y), fma((y * y), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666)), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (cos(x) <= -0.005)
		tmp = Float64(fma(y, Float64(y * fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666)), 1.0) * fma(-0.5, Float64(x * x), 1.0));
	else
		tmp = fma(y, Float64(y * fma(Float64(y * y), fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666)), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[Cos[x], $MachinePrecision], -0.005], N[(N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(-0.5 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   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. accelerator-lowering-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 55.4

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

   5. Simplified 55.4%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

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

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

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

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 53.8

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

   8. Simplified 53.8%

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

   if -0.0050000000000000001 < (cos.f64 x)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 84.1%

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

   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. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

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

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

      8. +-commutative N/A

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

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

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

      10. unpow2 N/A

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

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

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

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

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

      15. unpow2 N/A

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

      16. *-lowering-*.f64 76.1

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

   8. Simplified 76.1%

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

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos x \leq -0.005:\\ \;\;\;\;\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), 1\right) \cdot \mathsf{fma}\left(-0.5, x \cdot x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 69.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos x \leq -0.005:\\ \;\;\;\;\mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right) \cdot \mathsf{fma}\left(-0.5, x \cdot x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (cos x) -0.005)
   (* (fma 0.16666666666666666 (* y y) 1.0) (fma -0.5 (* x x) 1.0))
   (fma
    y
    (*
     y
     (fma
      (* y y)
      (fma (* y y) 0.0001984126984126984 0.008333333333333333)
      0.16666666666666666))
    1.0)))

C

double code(double x, double y) {
	double tmp;
	if (cos(x) <= -0.005) {
		tmp = fma(0.16666666666666666, (y * y), 1.0) * fma(-0.5, (x * x), 1.0);
	} else {
		tmp = fma(y, (y * fma((y * y), fma((y * y), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666)), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (cos(x) <= -0.005)
		tmp = Float64(fma(0.16666666666666666, Float64(y * y), 1.0) * fma(-0.5, Float64(x * x), 1.0));
	else
		tmp = fma(y, Float64(y * fma(Float64(y * y), fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666)), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[Cos[x], $MachinePrecision], -0.005], N[(N[(0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * N[(-0.5 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   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. accelerator-lowering-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 55.4

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

   5. Simplified 55.4%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 50.8

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

   8. Simplified 50.8%

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

   if -0.0050000000000000001 < (cos.f64 x)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 84.1%

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

   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. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

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

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

      8. +-commutative N/A

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

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

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

      10. unpow2 N/A

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

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

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

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

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

      15. unpow2 N/A

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

      16. *-lowering-*.f64 76.1

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

   8. Simplified 76.1%

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

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos x \leq -0.005:\\ \;\;\;\;\mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right) \cdot \mathsf{fma}\left(-0.5, x \cdot x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 67.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos x \leq -0.005:\\ \;\;\;\;\mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right) \cdot \mathsf{fma}\left(-0.5, x \cdot x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (cos x) -0.005)
   (* (fma 0.16666666666666666 (* y y) 1.0) (fma -0.5 (* x x) 1.0))
   (fma y (* y (fma y (* y 0.008333333333333333) 0.16666666666666666)) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (cos(x) <= -0.005) {
		tmp = fma(0.16666666666666666, (y * y), 1.0) * fma(-0.5, (x * x), 1.0);
	} else {
		tmp = fma(y, (y * fma(y, (y * 0.008333333333333333), 0.16666666666666666)), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (cos(x) <= -0.005)
		tmp = Float64(fma(0.16666666666666666, Float64(y * y), 1.0) * fma(-0.5, Float64(x * x), 1.0));
	else
		tmp = fma(y, Float64(y * fma(y, Float64(y * 0.008333333333333333), 0.16666666666666666)), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[Cos[x], $MachinePrecision], -0.005], N[(N[(0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * N[(-0.5 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(y * N[(y * N[(y * N[(y * 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   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. accelerator-lowering-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 55.4

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

   5. Simplified 55.4%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 50.8

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

   8. Simplified 50.8%

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

   if -0.0050000000000000001 < (cos.f64 x)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 84.1%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

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

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

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

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

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

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

      11. *-commutative N/A

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

      12. *-lowering-*.f64 70.7

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

   8. Simplified 70.7%

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

4. Final simplification 65.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos x \leq -0.005:\\ \;\;\;\;\mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right) \cdot \mathsf{fma}\left(-0.5, x \cdot x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 67.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos x \leq -0.005:\\ \;\;\;\;x \cdot \left(x \cdot \mathsf{fma}\left(y \cdot y, -0.08333333333333333, -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (cos x) -0.005)
   (* x (* x (fma (* y y) -0.08333333333333333 -0.5)))
   (fma y (* y (fma y (* y 0.008333333333333333) 0.16666666666666666)) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (cos(x) <= -0.005) {
		tmp = x * (x * fma((y * y), -0.08333333333333333, -0.5));
	} else {
		tmp = fma(y, (y * fma(y, (y * 0.008333333333333333), 0.16666666666666666)), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (cos(x) <= -0.005)
		tmp = Float64(x * Float64(x * fma(Float64(y * y), -0.08333333333333333, -0.5)));
	else
		tmp = fma(y, Float64(y * fma(y, Float64(y * 0.008333333333333333), 0.16666666666666666)), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[Cos[x], $MachinePrecision], -0.005], N[(x * N[(x * N[(N[(y * y), $MachinePrecision] * -0.08333333333333333 + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(y * N[(y * N[(y * 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   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. accelerator-lowering-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 55.4

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

   5. Simplified 55.4%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 50.8

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

   8. Simplified 50.8%

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

   9. Taylor expanded in x around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. unpow2 N/A

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

       5. associate-*l* N/A

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

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

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

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

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

       8. distribute-rgt-in N/A

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

       9. metadata-eval N/A

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

       10. +-commutative N/A

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

       11. *-commutative N/A

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

       12. associate-*l* N/A

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

       13. metadata-eval N/A

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

       14. metadata-eval N/A

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

       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 50.8

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

   11. Simplified 50.8%

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

   if -0.0050000000000000001 < (cos.f64 x)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 84.1%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

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

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

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

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

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

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

      11. *-commutative N/A

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

      12. *-lowering-*.f64 70.7

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

   8. Simplified 70.7%

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

Alternative 16: 58.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos x \leq -0.005:\\ \;\;\;\;x \cdot \left(x \cdot \mathsf{fma}\left(y \cdot y, -0.08333333333333333, -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (cos x) -0.005)
   (* x (* x (fma (* y y) -0.08333333333333333 -0.5)))
   (fma 0.16666666666666666 (* y y) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (cos(x) <= -0.005) {
		tmp = x * (x * fma((y * y), -0.08333333333333333, -0.5));
	} else {
		tmp = fma(0.16666666666666666, (y * y), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (cos(x) <= -0.005)
		tmp = Float64(x * Float64(x * fma(Float64(y * y), -0.08333333333333333, -0.5)));
	else
		tmp = fma(0.16666666666666666, Float64(y * y), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[Cos[x], $MachinePrecision], -0.005], N[(x * N[(x * N[(N[(y * y), $MachinePrecision] * -0.08333333333333333 + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   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. accelerator-lowering-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 55.4

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

   5. Simplified 55.4%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 50.8

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

   8. Simplified 50.8%

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

   9. Taylor expanded in x around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. unpow2 N/A

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

       5. associate-*l* N/A

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

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

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

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

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

       8. distribute-rgt-in N/A

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

       9. metadata-eval N/A

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

       10. +-commutative N/A

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

       11. *-commutative N/A

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

       12. associate-*l* N/A

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

       13. metadata-eval N/A

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

       14. metadata-eval N/A

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

       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 50.8

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

   11. Simplified 50.8%

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

   if -0.0050000000000000001 < (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

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

      1. +-commutative N/A

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 75.0

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

   5. Simplified 75.0%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 59.6

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

   8. Simplified 59.6%

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

Alternative 17: 46.2% accurate, 18.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (fma 0.16666666666666666 (* y y) 1.0))

C

double code(double x, double y) {
	return fma(0.16666666666666666, (y * y), 1.0);
}

Julia

function code(x, y)
	return fma(0.16666666666666666, Float64(y * y), 1.0)
end

Wolfram

code[x_, y_] := N[(0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

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

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

   3. unpow2 N/A

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

   4. *-lowering-*.f64 74.9

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

5. Simplified 74.9%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

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

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

   3. unpow2 N/A

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

   4. *-lowering-*.f64 45.3

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

8. Simplified 45.3%

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

Alternative 18: 28.2% accurate, 217.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 1.0)

C

double code(double x, double y) {
	return 1.0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0
end function

Java

public static double code(double x, double y) {
	return 1.0;
}

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0;
end

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in y around 0

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

   1. cos-lowering-cos.f64 54.1

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

5. Simplified 54.1%

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

6. Taylor expanded in x around 0

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

8. Simplified 31.3%

   \[\leadsto \color{blue}{1} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024196 
(FPCore (x y)
  :name "Linear.Quaternion:$csin from linear-1.19.1.3"
  :precision binary64
  (* (cos x) (/ (sinh y) y)))