Linear.Quaternion:$csin from linear-1.19.1.3

Percentage Accurate: 100.0% → 100.0%

Time: 8.6s

Alternatives: 19

Speedup: 1.0×

• Could not converge on a ground truth

  1. x = 2.638044578863187e+201
  2. y = 2.0714841247865068e+97

• 50.0% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[\cos x \cdot \frac{\sinh y}{y} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. un-div-inv N/A

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

   5. lower-/.f64 N/A

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

   6. lower-/.f64 100.0

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

4. Applied rewrites 100.0%

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

Alternative 2: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y}{y}\\ t_1 := t\_0 \cdot \cos x\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(-0.5, x \cdot x, 1\right) \cdot t\_0\\ \mathbf{elif}\;t\_1 \leq 0.99999999336492:\\ \;\;\;\;\frac{\cos x}{\mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{y}{\sinh y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sinh y) y)) (t_1 (* t_0 (cos x))))
   (if (<= t_1 (- INFINITY))
     (* (fma -0.5 (* x x) 1.0) t_0)
     (if (<= t_1 0.99999999336492)
       (/ (cos x) (fma -0.16666666666666666 (* y y) 1.0))
       (/ 1.0 (/ y (sinh y)))))))

C

double code(double x, double y) {
	double t_0 = sinh(y) / y;
	double t_1 = t_0 * cos(x);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma(-0.5, (x * x), 1.0) * t_0;
	} else if (t_1 <= 0.99999999336492) {
		tmp = cos(x) / fma(-0.16666666666666666, (y * y), 1.0);
	} else {
		tmp = 1.0 / (y / sinh(y));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sinh(y) / y)
	t_1 = Float64(t_0 * cos(x))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(fma(-0.5, Float64(x * x), 1.0) * t_0);
	elseif (t_1 <= 0.99999999336492)
		tmp = Float64(cos(x) / fma(-0.16666666666666666, Float64(y * y), 1.0));
	else
		tmp = Float64(1.0 / Float64(y / sinh(y)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[Cos[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(-0.5 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[t$95$1, 0.99999999336492], N[(N[Cos[x], $MachinePrecision] / N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(y / N[Sinh[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in 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. lower-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. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -inf.0 < (*.f64 (cos.f64 x) (/.f64 (sinh.f64 y) y)) < 0.99999999336492

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 100.0

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

   7. Applied rewrites 100.0%

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

   if 0.99999999336492 < (*.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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 100.0%

      \[\leadsto \frac{\color{blue}{1}}{\frac{y}{\sinh y}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y}{y} \cdot \cos x \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(-0.5, x \cdot x, 1\right) \cdot \frac{\sinh y}{y}\\ \mathbf{elif}\;\frac{\sinh y}{y} \cdot \cos x \leq 0.99999999336492:\\ \;\;\;\;\frac{\cos x}{\mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{y}{\sinh y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y}{y} \cdot \cos x\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984 \cdot \left(y \cdot y\right), y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot \mathsf{fma}\left(-0.5, x \cdot x, 1\right)\\ \mathbf{elif}\;t\_0 \leq 0.99999999336492:\\ \;\;\;\;\frac{\cos x}{\mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{y}{\sinh y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (/ (sinh y) y) (cos x))))
   (if (<= t_0 (- INFINITY))
     (*
      (fma
       (fma (* 0.0001984126984126984 (* y y)) (* y y) 0.16666666666666666)
       (* y y)
       1.0)
      (fma -0.5 (* x x) 1.0))
     (if (<= t_0 0.99999999336492)
       (/ (cos x) (fma -0.16666666666666666 (* y y) 1.0))
       (/ 1.0 (/ y (sinh y)))))))

C

double code(double x, double y) {
	double t_0 = (sinh(y) / y) * cos(x);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma(fma((0.0001984126984126984 * (y * y)), (y * y), 0.16666666666666666), (y * y), 1.0) * fma(-0.5, (x * x), 1.0);
	} else if (t_0 <= 0.99999999336492) {
		tmp = cos(x) / fma(-0.16666666666666666, (y * y), 1.0);
	} else {
		tmp = 1.0 / (y / sinh(y));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(sinh(y) / y) * cos(x))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(fma(fma(Float64(0.0001984126984126984 * Float64(y * y)), Float64(y * y), 0.16666666666666666), Float64(y * y), 1.0) * fma(-0.5, Float64(x * x), 1.0));
	elseif (t_0 <= 0.99999999336492)
		tmp = Float64(cos(x) / fma(-0.16666666666666666, Float64(y * y), 1.0));
	else
		tmp = Float64(1.0 / Float64(y / sinh(y)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[(0.0001984126984126984 * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(y * y), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * N[(-0.5 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.99999999336492], N[(N[Cos[x], $MachinePrecision] / N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(y / N[Sinh[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in 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. lower-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. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 93.2

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

   8. Applied rewrites 93.2%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 93.2%

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

   if -inf.0 < (*.f64 (cos.f64 x) (/.f64 (sinh.f64 y) y)) < 0.99999999336492

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 100.0

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

   7. Applied rewrites 100.0%

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

   if 0.99999999336492 < (*.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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 100.0%

      \[\leadsto \frac{\color{blue}{1}}{\frac{y}{\sinh y}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y}{y} \cdot \cos x \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984 \cdot \left(y \cdot y\right), y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot \mathsf{fma}\left(-0.5, x \cdot x, 1\right)\\ \mathbf{elif}\;\frac{\sinh y}{y} \cdot \cos x \leq 0.99999999336492:\\ \;\;\;\;\frac{\cos x}{\mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{y}{\sinh y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y}{y} \cdot \cos x\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984 \cdot \left(y \cdot y\right), y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot \mathsf{fma}\left(-0.5, x \cdot x, 1\right)\\ \mathbf{elif}\;t\_0 \leq 0.99999999336492:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right) \cdot \cos x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{y}{\sinh y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (/ (sinh y) y) (cos x))))
   (if (<= t_0 (- INFINITY))
     (*
      (fma
       (fma (* 0.0001984126984126984 (* y y)) (* y y) 0.16666666666666666)
       (* y y)
       1.0)
      (fma -0.5 (* x x) 1.0))
     (if (<= t_0 0.99999999336492)
       (* (fma (* y y) 0.16666666666666666 1.0) (cos x))
       (/ 1.0 (/ y (sinh y)))))))

C

double code(double x, double y) {
	double t_0 = (sinh(y) / y) * cos(x);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma(fma((0.0001984126984126984 * (y * y)), (y * y), 0.16666666666666666), (y * y), 1.0) * fma(-0.5, (x * x), 1.0);
	} else if (t_0 <= 0.99999999336492) {
		tmp = fma((y * y), 0.16666666666666666, 1.0) * cos(x);
	} else {
		tmp = 1.0 / (y / sinh(y));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(sinh(y) / y) * cos(x))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(fma(fma(Float64(0.0001984126984126984 * Float64(y * y)), Float64(y * y), 0.16666666666666666), Float64(y * y), 1.0) * fma(-0.5, Float64(x * x), 1.0));
	elseif (t_0 <= 0.99999999336492)
		tmp = Float64(fma(Float64(y * y), 0.16666666666666666, 1.0) * cos(x));
	else
		tmp = Float64(1.0 / Float64(y / sinh(y)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[(0.0001984126984126984 * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(y * y), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * N[(-0.5 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.99999999336492], N[(N[(N[(y * y), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(y / N[Sinh[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in 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. lower-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. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 93.2

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

   8. Applied rewrites 93.2%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 93.2%

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

   if -inf.0 < (*.f64 (cos.f64 x) (/.f64 (sinh.f64 y) y)) < 0.99999999336492

   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if 0.99999999336492 < (*.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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 100.0%

      \[\leadsto \frac{\color{blue}{1}}{\frac{y}{\sinh y}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y}{y} \cdot \cos x \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984 \cdot \left(y \cdot y\right), y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot \mathsf{fma}\left(-0.5, x \cdot x, 1\right)\\ \mathbf{elif}\;\frac{\sinh y}{y} \cdot \cos x \leq 0.99999999336492:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right) \cdot \cos x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{y}{\sinh y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y}{y}\\ t_1 := t\_0 \cdot \cos x\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984 \cdot \left(y \cdot y\right), y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot \mathsf{fma}\left(-0.5, x \cdot x, 1\right)\\ \mathbf{elif}\;t\_1 \leq 0.99999999336492:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right) \cdot \cos x\\ \mathbf{else}:\\ \;\;\;\;1 \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sinh y) y)) (t_1 (* t_0 (cos x))))
   (if (<= t_1 (- INFINITY))
     (*
      (fma
       (fma (* 0.0001984126984126984 (* y y)) (* y y) 0.16666666666666666)
       (* y y)
       1.0)
      (fma -0.5 (* x x) 1.0))
     (if (<= t_1 0.99999999336492)
       (* (fma (* y y) 0.16666666666666666 1.0) (cos x))
       (* 1.0 t_0)))))

C

double code(double x, double y) {
	double t_0 = sinh(y) / y;
	double t_1 = t_0 * cos(x);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma(fma((0.0001984126984126984 * (y * y)), (y * y), 0.16666666666666666), (y * y), 1.0) * fma(-0.5, (x * x), 1.0);
	} else if (t_1 <= 0.99999999336492) {
		tmp = fma((y * y), 0.16666666666666666, 1.0) * cos(x);
	} else {
		tmp = 1.0 * t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sinh(y) / y)
	t_1 = Float64(t_0 * cos(x))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(fma(fma(Float64(0.0001984126984126984 * Float64(y * y)), Float64(y * y), 0.16666666666666666), Float64(y * y), 1.0) * fma(-0.5, Float64(x * x), 1.0));
	elseif (t_1 <= 0.99999999336492)
		tmp = Float64(fma(Float64(y * y), 0.16666666666666666, 1.0) * cos(x));
	else
		tmp = Float64(1.0 * 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[(t$95$0 * N[Cos[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(N[(0.0001984126984126984 * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(y * y), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * N[(-0.5 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.99999999336492], N[(N[(N[(y * y), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision], N[(1.0 * t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-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. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 93.2

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

   8. Applied rewrites 93.2%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 93.2%

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

   if -inf.0 < (*.f64 (cos.f64 x) (/.f64 (sinh.f64 y) y)) < 0.99999999336492

   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if 0.99999999336492 < (*.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. Applied rewrites 100.0%

      \[\leadsto \color{blue}{1} \cdot \frac{\sinh y}{y} \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y}{y} \cdot \cos x \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984 \cdot \left(y \cdot y\right), y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot \mathsf{fma}\left(-0.5, x \cdot x, 1\right)\\ \mathbf{elif}\;\frac{\sinh y}{y} \cdot \cos x \leq 0.99999999336492:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right) \cdot \cos x\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \frac{\sinh y}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y}{y}\\ t_1 := t\_0 \cdot \cos x\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984 \cdot \left(y \cdot y\right), y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot \mathsf{fma}\left(-0.5, x \cdot x, 1\right)\\ \mathbf{elif}\;t\_1 \leq 0.99999999336492:\\ \;\;\;\;\cos x\\ \mathbf{else}:\\ \;\;\;\;1 \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sinh y) y)) (t_1 (* t_0 (cos x))))
   (if (<= t_1 (- INFINITY))
     (*
      (fma
       (fma (* 0.0001984126984126984 (* y y)) (* y y) 0.16666666666666666)
       (* y y)
       1.0)
      (fma -0.5 (* x x) 1.0))
     (if (<= t_1 0.99999999336492) (cos x) (* 1.0 t_0)))))

C

double code(double x, double y) {
	double t_0 = sinh(y) / y;
	double t_1 = t_0 * cos(x);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma(fma((0.0001984126984126984 * (y * y)), (y * y), 0.16666666666666666), (y * y), 1.0) * fma(-0.5, (x * x), 1.0);
	} else if (t_1 <= 0.99999999336492) {
		tmp = cos(x);
	} else {
		tmp = 1.0 * t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sinh(y) / y)
	t_1 = Float64(t_0 * cos(x))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(fma(fma(Float64(0.0001984126984126984 * Float64(y * y)), Float64(y * y), 0.16666666666666666), Float64(y * y), 1.0) * fma(-0.5, Float64(x * x), 1.0));
	elseif (t_1 <= 0.99999999336492)
		tmp = cos(x);
	else
		tmp = Float64(1.0 * 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[(t$95$0 * N[Cos[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(N[(0.0001984126984126984 * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(y * y), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * N[(-0.5 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.99999999336492], N[Cos[x], $MachinePrecision], N[(1.0 * t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-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. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 93.2

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

   8. Applied rewrites 93.2%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 93.2%

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

   if -inf.0 < (*.f64 (cos.f64 x) (/.f64 (sinh.f64 y) y)) < 0.99999999336492

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-cos.f64 99.9

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

   5. Applied rewrites 99.9%

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

   if 0.99999999336492 < (*.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. Applied rewrites 100.0%

      \[\leadsto \color{blue}{1} \cdot \frac{\sinh y}{y} \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y}{y} \cdot \cos x \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984 \cdot \left(y \cdot y\right), y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot \mathsf{fma}\left(-0.5, x \cdot x, 1\right)\\ \mathbf{elif}\;\frac{\sinh y}{y} \cdot \cos x \leq 0.99999999336492:\\ \;\;\;\;\cos x\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \frac{\sinh y}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 95.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y}{y} \cdot \cos x\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984 \cdot \left(y \cdot y\right), y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot \mathsf{fma}\left(-0.5, x \cdot x, 1\right)\\ \mathbf{elif}\;t\_0 \leq 0.99999999336492:\\ \;\;\;\;\cos x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984, y \cdot y, 0.008333333333333333\right), y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot \mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right), x \cdot x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (/ (sinh y) y) (cos x))))
   (if (<= t_0 (- INFINITY))
     (*
      (fma
       (fma (* 0.0001984126984126984 (* y y)) (* y y) 0.16666666666666666)
       (* y y)
       1.0)
      (fma -0.5 (* x x) 1.0))
     (if (<= t_0 0.99999999336492)
       (cos x)
       (*
        (fma
         (fma
          (fma 0.0001984126984126984 (* y y) 0.008333333333333333)
          (* y y)
          0.16666666666666666)
         (* y y)
         1.0)
        (fma (* 0.041666666666666664 (* x x)) (* x x) 1.0))))))

C

double code(double x, double y) {
	double t_0 = (sinh(y) / y) * cos(x);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma(fma((0.0001984126984126984 * (y * y)), (y * y), 0.16666666666666666), (y * y), 1.0) * fma(-0.5, (x * x), 1.0);
	} else if (t_0 <= 0.99999999336492) {
		tmp = cos(x);
	} else {
		tmp = fma(fma(fma(0.0001984126984126984, (y * y), 0.008333333333333333), (y * y), 0.16666666666666666), (y * y), 1.0) * fma((0.041666666666666664 * (x * x)), (x * x), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(sinh(y) / y) * cos(x))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(fma(fma(Float64(0.0001984126984126984 * Float64(y * y)), Float64(y * y), 0.16666666666666666), Float64(y * y), 1.0) * fma(-0.5, Float64(x * x), 1.0));
	elseif (t_0 <= 0.99999999336492)
		tmp = cos(x);
	else
		tmp = Float64(fma(fma(fma(0.0001984126984126984, Float64(y * y), 0.008333333333333333), Float64(y * y), 0.16666666666666666), Float64(y * y), 1.0) * fma(Float64(0.041666666666666664 * Float64(x * x)), Float64(x * x), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[(0.0001984126984126984 * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(y * y), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * N[(-0.5 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.99999999336492], N[Cos[x], $MachinePrecision], N[(N[(N[(N[(0.0001984126984126984 * N[(y * y), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] * N[(y * y), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(0.041666666666666664 * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in 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. lower-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. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 93.2

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

   8. Applied rewrites 93.2%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 93.2%

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

   if -inf.0 < (*.f64 (cos.f64 x) (/.f64 (sinh.f64 y) y)) < 0.99999999336492

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-cos.f64 99.9

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

   5. Applied rewrites 99.9%

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

   if 0.99999999336492 < (*.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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 68.3

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

   5. Applied rewrites 68.3%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 75.8

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

   8. Applied rewrites 75.8%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 75.8%

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

   12. Taylor expanded in y around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. +-commutative N/A

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

       5. *-commutative N/A

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

       6. lower-fma.f64 N/A

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

       7. +-commutative N/A

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

       8. lower-fma.f64 N/A

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

       9. unpow2 N/A

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

       10. lower-*.f64 N/A

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

       11. unpow2 N/A

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

       12. lower-*.f64 N/A

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

       13. unpow2 N/A

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

       14. lower-*.f64 95.0

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

   14. Applied rewrites 95.0%

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

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y}{y} \cdot \cos x \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984 \cdot \left(y \cdot y\right), y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot \mathsf{fma}\left(-0.5, x \cdot x, 1\right)\\ \mathbf{elif}\;\frac{\sinh y}{y} \cdot \cos x \leq 0.99999999336492:\\ \;\;\;\;\cos x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984, y \cdot y, 0.008333333333333333\right), y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot \mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right), x \cdot x, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 62.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y}{y} \cdot \cos x\\ \mathbf{if}\;t\_0 \leq -0.005:\\ \;\;\;\;\mathsf{fma}\left(-0.5, x \cdot x, 1\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right) \cdot y\right) \cdot y\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (/ (sinh y) y) (cos x))))
   (if (<= t_0 -0.005)
     (fma -0.5 (* x x) 1.0)
     (if (<= t_0 2.0)
       1.0
       (*
        (* (* (fma 0.008333333333333333 (* y y) 0.16666666666666666) y) y)
        1.0)))))

C

double code(double x, double y) {
	double t_0 = (sinh(y) / y) * cos(x);
	double tmp;
	if (t_0 <= -0.005) {
		tmp = fma(-0.5, (x * x), 1.0);
	} else if (t_0 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = ((fma(0.008333333333333333, (y * y), 0.16666666666666666) * y) * y) * 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(sinh(y) / y) * cos(x))
	tmp = 0.0
	if (t_0 <= -0.005)
		tmp = fma(-0.5, Float64(x * x), 1.0);
	elseif (t_0 <= 2.0)
		tmp = 1.0;
	else
		tmp = Float64(Float64(Float64(fma(0.008333333333333333, Float64(y * y), 0.16666666666666666) * y) * y) * 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.005], N[(-0.5 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[t$95$0, 2.0], 1.0, N[(N[(N[(N[(0.008333333333333333 * N[(y * y), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * y), $MachinePrecision] * y), $MachinePrecision] * 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 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. lower-cos.f64 45.6

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

   5. Applied rewrites 45.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 29.4%

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

   if -0.0050000000000000001 < (*.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. lower-cos.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

      \[\leadsto 1 \]
   7. Step-by-step derivation

   8. Applied rewrites 73.4%

      \[\leadsto 1 \]

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 80.5

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

   8. Applied rewrites 80.5%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 80.5%

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

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y}{y} \cdot \cos x \leq -0.005:\\ \;\;\;\;\mathsf{fma}\left(-0.5, x \cdot x, 1\right)\\ \mathbf{elif}\;\frac{\sinh y}{y} \cdot \cos x \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right) \cdot y\right) \cdot y\right) \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 53.6% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision], -0.005], N[(-0.5 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(N[(y * y), $MachinePrecision] * 0.16666666666666666 + 1.0), $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. lower-cos.f64 45.6

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

   5. Applied rewrites 45.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 29.4%

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

   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 x around 0

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

   5. Applied rewrites 86.7%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 60.6

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

   8. Applied rewrites 60.6%

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

4. Final simplification 51.7%

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

Alternative 10: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Final simplification 100.0%

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

Alternative 11: 71.0% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_] := If[LessEqual[N[Cos[x], $MachinePrecision], -0.005], N[(N[(N[(N[(0.0001984126984126984 * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(y * y), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * N[(-0.5 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.0001984126984126984 * N[(y * y), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] * N[(y * y), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $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. lower-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. lower-*.f64 58.0

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

   5. Applied rewrites 58.0%

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

   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 54.2

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

   8. Applied rewrites 54.2%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 54.2%

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

   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. Applied rewrites 86.7%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 81.6

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

   8. Applied rewrites 81.6%

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

4. Final simplification 73.8%

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

Alternative 12: 70.8% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_] := If[LessEqual[N[Cos[x], $MachinePrecision], -0.005], N[(N[(N[(0.008333333333333333 * N[(y * y), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * N[(-0.5 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.0001984126984126984 * N[(y * y), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] * N[(y * y), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $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. lower-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. lower-*.f64 58.0

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

   5. Applied rewrites 58.0%

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

   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 52.9

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

   8. Applied rewrites 52.9%

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

   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. Applied rewrites 86.7%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 81.6

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

   8. Applied rewrites 81.6%

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

4. Final simplification 73.4%

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

Alternative 13: 69.2% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_] := If[LessEqual[N[Cos[x], $MachinePrecision], -0.005], N[(N[(N[(N[(-0.001388888888888889 * N[(x * x), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(N[(N[(0.0001984126984126984 * N[(y * y), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] * N[(y * y), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $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 y around 0

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

      1. lower-cos.f64 45.6

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

   5. Applied rewrites 45.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 48.8%

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

   9. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left({x}^{4} \cdot \left(\frac{1}{24} \cdot \frac{1}{{x}^{2}} - \frac{1}{720}\right), x \cdot x, 1\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 48.8%

       \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(-0.001388888888888889, x \cdot x, 0.041666666666666664\right) \cdot x\right) \cdot x, x \cdot x, 1\right) \]

   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. Applied rewrites 86.7%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 81.6

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

   8. Applied rewrites 81.6%

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

4. Final simplification 72.2%

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

Alternative 14: 66.4% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_] := If[LessEqual[N[Cos[x], $MachinePrecision], -0.005], N[(N[(N[(N[(-0.001388888888888889 * N[(x * x), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(N[(0.008333333333333333 * N[(y * y), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $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 y around 0

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

      1. lower-cos.f64 45.6

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

   5. Applied rewrites 45.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 48.8%

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

   9. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left({x}^{4} \cdot \left(\frac{1}{24} \cdot \frac{1}{{x}^{2}} - \frac{1}{720}\right), x \cdot x, 1\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 48.8%

       \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(-0.001388888888888889, x \cdot x, 0.041666666666666664\right) \cdot x\right) \cdot x, x \cdot x, 1\right) \]

   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. Applied rewrites 86.7%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 76.9

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

   8. Applied rewrites 76.9%

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

4. Final simplification 68.9%

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

Alternative 15: 67.1% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_] := If[LessEqual[N[Cos[x], $MachinePrecision], -0.005], N[(N[(N[(y * y), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision] * N[(-0.5 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.008333333333333333 * N[(y * y), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $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. lower-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. lower-*.f64 58.0

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

   5. Applied rewrites 58.0%

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

   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 46.5

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

   8. Applied rewrites 46.5%

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

   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. Applied rewrites 86.7%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 76.9

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

   8. Applied rewrites 76.9%

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

4. Final simplification 68.3%

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

Alternative 16: 62.6% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_] := If[LessEqual[N[Cos[x], $MachinePrecision], -0.005], N[(-0.5 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(N[(0.008333333333333333 * N[(y * y), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $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 y around 0

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

      1. lower-cos.f64 45.6

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

   5. Applied rewrites 45.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 29.4%

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

   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. Applied rewrites 86.7%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 76.9

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

   8. Applied rewrites 76.9%

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

4. Final simplification 63.4%

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

Alternative 17: 62.4% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_] := If[LessEqual[N[Cos[x], $MachinePrecision], -0.005], N[(-0.5 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(N[(0.008333333333333333 * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $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 y around 0

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

      1. lower-cos.f64 45.6

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

   5. Applied rewrites 45.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 29.4%

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

   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. Applied rewrites 86.7%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 76.9

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

   8. Applied rewrites 76.9%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 76.9%

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

4. Final simplification 63.4%

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

Alternative 18: 35.0% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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 y around 0

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

      1. lower-cos.f64 45.6

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

   5. Applied rewrites 45.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 29.4%

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

   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 y around 0

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

      1. lower-cos.f64 51.3

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

   5. Applied rewrites 51.3%

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

   6. Taylor expanded in x around 0

      \[\leadsto 1 \]
   7. Step-by-step derivation

   8. Applied rewrites 38.1%

      \[\leadsto 1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 27.7% accurate, 217.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

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

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in y around 0

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

   1. lower-cos.f64 49.7

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

5. Applied rewrites 49.7%

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

6. Taylor expanded in x around 0

   \[\leadsto 1 \]
7. Step-by-step derivation

8. Applied rewrites 27.5%

   \[\leadsto 1 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024243 
(FPCore (x y)
  :name "Linear.Quaternion:$csin from linear-1.19.1.3"
  :precision binary64
  (* (cos x) (/ (sinh y) y)))