Linear.Quaternion:$csin from linear-1.19.1.3

Percentage Accurate: 100.0% → 100.0%

Time: 12.2s

Alternatives: 27

Speedup: 1.0×

• 49.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

Alternative 2: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y}{y}\\ t_1 := \cos x \cdot t\_0\\ t_2 := y \cdot \left(y \cdot \left(y \cdot y\right)\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right) \cdot \left(\left(0.0001984126984126984 \cdot \left(y \cdot y\right)\right) \cdot t\_2\right)\\ \mathbf{elif}\;t\_1 \leq 0.9999999999999779:\\ \;\;\;\;\cos x \cdot \left(y \cdot \left(y \cdot 0.16666666666666666\right) + \mathsf{fma}\left(t\_2, \mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right), 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sinh y) y)) (t_1 (* (cos x) t_0)) (t_2 (* y (* y (* y y)))))
   (if (<= t_1 (- INFINITY))
     (*
      (fma
       (* x x)
       (fma
        x
        (* x (fma x (* x -0.001388888888888889) 0.041666666666666664))
        -0.5)
       1.0)
      (* (* 0.0001984126984126984 (* y y)) t_2))
     (if (<= t_1 0.9999999999999779)
       (*
        (cos x)
        (+
         (* y (* y 0.16666666666666666))
         (fma
          t_2
          (fma y (* y 0.0001984126984126984) 0.008333333333333333)
          1.0)))
       (* t_0 1.0)))))

C

double code(double x, double y) {
	double t_0 = sinh(y) / y;
	double t_1 = cos(x) * t_0;
	double t_2 = y * (y * (y * y));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma((x * x), fma(x, (x * fma(x, (x * -0.001388888888888889), 0.041666666666666664)), -0.5), 1.0) * ((0.0001984126984126984 * (y * y)) * t_2);
	} else if (t_1 <= 0.9999999999999779) {
		tmp = cos(x) * ((y * (y * 0.16666666666666666)) + fma(t_2, fma(y, (y * 0.0001984126984126984), 0.008333333333333333), 1.0));
	} else {
		tmp = t_0 * 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sinh(y) / y)
	t_1 = Float64(cos(x) * t_0)
	t_2 = Float64(y * Float64(y * Float64(y * y)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(fma(Float64(x * x), fma(x, Float64(x * fma(x, Float64(x * -0.001388888888888889), 0.041666666666666664)), -0.5), 1.0) * Float64(Float64(0.0001984126984126984 * Float64(y * y)) * t_2));
	elseif (t_1 <= 0.9999999999999779)
		tmp = Float64(cos(x) * Float64(Float64(y * Float64(y * 0.16666666666666666)) + fma(t_2, fma(y, Float64(y * 0.0001984126984126984), 0.008333333333333333), 1.0)));
	else
		tmp = Float64(t_0 * 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(y * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(x * N[(x * -0.001388888888888889), $MachinePrecision] + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(0.0001984126984126984 * N[(y * y), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.9999999999999779], N[(N[Cos[x], $MachinePrecision] * N[(N[(y * N[(y * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 * N[(y * N[(y * 0.0001984126984126984), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * 1.0), $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 y around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 74.6%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)\right)} \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right) + 1\right)} \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{2}, {x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}, 1\right)} \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, {x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}, 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, {x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}, 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(x \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right)\right)} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, x \cdot \color{blue}{\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) \cdot x\right)} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, x \cdot \left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) \cdot x\right) + \color{blue}{\frac{-1}{2}}, 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) \cdot x, \frac{-1}{2}\right)}, 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right)}, \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right)}, \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\left(\frac{-1}{720} \cdot {x}^{2} + \frac{1}{24}\right)}, \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{-1}{720}} + \frac{1}{24}\right), \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{-1}{720} + \frac{1}{24}\right), \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      16. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{x \cdot \left(x \cdot \frac{-1}{720}\right)} + \frac{1}{24}\right), \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      17. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{-1}{720}, \frac{1}{24}\right)}, \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      18. lower-*.f64 100.0

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot -0.001388888888888889}, 0.041666666666666664\right), -0.5\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right) \]

   8. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right)} \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right) \]

   9. Taylor expanded in y around inf

      \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \frac{-1}{720}, \frac{1}{24}\right), \frac{-1}{2}\right), 1\right) \cdot \left(\frac{1}{5040} \cdot \color{blue}{{y}^{6}}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 100.0%

       \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right) \cdot \left(\left(\left(y \cdot y\right) \cdot 0.0001984126984126984\right) \cdot \color{blue}{\left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)}\right) \]

   if -inf.0 < (*.f64 (cos.f64 x) (/.f64 (sinh.f64 y) y)) < 0.999999999999977907

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 97.5%

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

   7. Applied rewrites 97.5%

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

   if 0.999999999999977907 < (*.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 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos x \cdot \frac{\sinh y}{y} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right) \cdot \left(\left(0.0001984126984126984 \cdot \left(y \cdot y\right)\right) \cdot \left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right)\\ \mathbf{elif}\;\cos x \cdot \frac{\sinh y}{y} \leq 0.9999999999999779:\\ \;\;\;\;\cos x \cdot \left(y \cdot \left(y \cdot 0.16666666666666666\right) + \mathsf{fma}\left(y \cdot \left(y \cdot \left(y \cdot y\right)\right), \mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right), 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sinh y}{y} \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y}{y}\\ t_1 := \cos x \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right) \cdot \left(\left(0.0001984126984126984 \cdot \left(y \cdot y\right)\right) \cdot \left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right)\\ \mathbf{elif}\;t\_1 \leq 0.9999999999999779:\\ \;\;\;\;\cos x \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sinh y) y)) (t_1 (* (cos x) t_0)))
   (if (<= t_1 (- INFINITY))
     (*
      (fma
       (* x x)
       (fma
        x
        (* x (fma x (* x -0.001388888888888889) 0.041666666666666664))
        -0.5)
       1.0)
      (* (* 0.0001984126984126984 (* y y)) (* y (* y (* y y)))))
     (if (<= t_1 0.9999999999999779)
       (*
        (cos x)
        (fma
         y
         (*
          y
          (fma
           y
           (* y (fma (* y y) 0.0001984126984126984 0.008333333333333333))
           0.16666666666666666))
         1.0))
       (* t_0 1.0)))))

C

double code(double x, double y) {
	double t_0 = sinh(y) / y;
	double t_1 = cos(x) * t_0;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma((x * x), fma(x, (x * fma(x, (x * -0.001388888888888889), 0.041666666666666664)), -0.5), 1.0) * ((0.0001984126984126984 * (y * y)) * (y * (y * (y * y))));
	} else if (t_1 <= 0.9999999999999779) {
		tmp = cos(x) * fma(y, (y * fma(y, (y * fma((y * y), 0.0001984126984126984, 0.008333333333333333)), 0.16666666666666666)), 1.0);
	} else {
		tmp = t_0 * 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sinh(y) / y)
	t_1 = Float64(cos(x) * t_0)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(fma(Float64(x * x), fma(x, Float64(x * fma(x, Float64(x * -0.001388888888888889), 0.041666666666666664)), -0.5), 1.0) * Float64(Float64(0.0001984126984126984 * Float64(y * y)) * Float64(y * Float64(y * Float64(y * y)))));
	elseif (t_1 <= 0.9999999999999779)
		tmp = Float64(cos(x) * fma(y, Float64(y * fma(y, Float64(y * fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333)), 0.16666666666666666)), 1.0));
	else
		tmp = Float64(t_0 * 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(x * N[(x * -0.001388888888888889), $MachinePrecision] + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(0.0001984126984126984 * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(y * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.9999999999999779], N[(N[Cos[x], $MachinePrecision] * N[(y * N[(y * N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * 1.0), $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 y around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 74.6%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)\right)} \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right) + 1\right)} \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{2}, {x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}, 1\right)} \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, {x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}, 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, {x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}, 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(x \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right)\right)} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, x \cdot \color{blue}{\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) \cdot x\right)} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, x \cdot \left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) \cdot x\right) + \color{blue}{\frac{-1}{2}}, 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) \cdot x, \frac{-1}{2}\right)}, 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right)}, \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right)}, \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\left(\frac{-1}{720} \cdot {x}^{2} + \frac{1}{24}\right)}, \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{-1}{720}} + \frac{1}{24}\right), \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{-1}{720} + \frac{1}{24}\right), \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      16. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{x \cdot \left(x \cdot \frac{-1}{720}\right)} + \frac{1}{24}\right), \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      17. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{-1}{720}, \frac{1}{24}\right)}, \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      18. lower-*.f64 100.0

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot -0.001388888888888889}, 0.041666666666666664\right), -0.5\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right) \]

   8. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right)} \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right) \]

   9. Taylor expanded in y around inf

      \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \frac{-1}{720}, \frac{1}{24}\right), \frac{-1}{2}\right), 1\right) \cdot \left(\frac{1}{5040} \cdot \color{blue}{{y}^{6}}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 100.0%

       \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right) \cdot \left(\left(\left(y \cdot y\right) \cdot 0.0001984126984126984\right) \cdot \color{blue}{\left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)}\right) \]

   if -inf.0 < (*.f64 (cos.f64 x) (/.f64 (sinh.f64 y) y)) < 0.999999999999977907

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 97.5%

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

   if 0.999999999999977907 < (*.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 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos x \cdot \frac{\sinh y}{y} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right) \cdot \left(\left(0.0001984126984126984 \cdot \left(y \cdot y\right)\right) \cdot \left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right)\\ \mathbf{elif}\;\cos x \cdot \frac{\sinh y}{y} \leq 0.9999999999999779:\\ \;\;\;\;\cos x \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sinh y}{y} \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y}{y}\\ t_1 := \cos x \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right) \cdot \left(\left(0.0001984126984126984 \cdot \left(y \cdot y\right)\right) \cdot \left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right)\\ \mathbf{elif}\;t\_1 \leq 0.9999999999999779:\\ \;\;\;\;\cos x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sinh y) y)) (t_1 (* (cos x) t_0)))
   (if (<= t_1 (- INFINITY))
     (*
      (fma
       (* x x)
       (fma
        x
        (* x (fma x (* x -0.001388888888888889) 0.041666666666666664))
        -0.5)
       1.0)
      (* (* 0.0001984126984126984 (* y y)) (* y (* y (* y y)))))
     (if (<= t_1 0.9999999999999779)
       (*
        (cos x)
        (fma
         (* y y)
         (fma y (* y 0.008333333333333333) 0.16666666666666666)
         1.0))
       (* t_0 1.0)))))

C

double code(double x, double y) {
	double t_0 = sinh(y) / y;
	double t_1 = cos(x) * t_0;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma((x * x), fma(x, (x * fma(x, (x * -0.001388888888888889), 0.041666666666666664)), -0.5), 1.0) * ((0.0001984126984126984 * (y * y)) * (y * (y * (y * y))));
	} else if (t_1 <= 0.9999999999999779) {
		tmp = cos(x) * fma((y * y), fma(y, (y * 0.008333333333333333), 0.16666666666666666), 1.0);
	} else {
		tmp = t_0 * 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sinh(y) / y)
	t_1 = Float64(cos(x) * t_0)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(fma(Float64(x * x), fma(x, Float64(x * fma(x, Float64(x * -0.001388888888888889), 0.041666666666666664)), -0.5), 1.0) * Float64(Float64(0.0001984126984126984 * Float64(y * y)) * Float64(y * Float64(y * Float64(y * y)))));
	elseif (t_1 <= 0.9999999999999779)
		tmp = Float64(cos(x) * fma(Float64(y * y), fma(y, Float64(y * 0.008333333333333333), 0.16666666666666666), 1.0));
	else
		tmp = Float64(t_0 * 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(x * N[(x * -0.001388888888888889), $MachinePrecision] + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(0.0001984126984126984 * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(y * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.9999999999999779], N[(N[Cos[x], $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * 1.0), $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 y around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 74.6%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)\right)} \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right) + 1\right)} \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{2}, {x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}, 1\right)} \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, {x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}, 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, {x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}, 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(x \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right)\right)} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, x \cdot \color{blue}{\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) \cdot x\right)} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, x \cdot \left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) \cdot x\right) + \color{blue}{\frac{-1}{2}}, 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) \cdot x, \frac{-1}{2}\right)}, 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right)}, \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right)}, \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\left(\frac{-1}{720} \cdot {x}^{2} + \frac{1}{24}\right)}, \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{-1}{720}} + \frac{1}{24}\right), \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{-1}{720} + \frac{1}{24}\right), \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      16. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{x \cdot \left(x \cdot \frac{-1}{720}\right)} + \frac{1}{24}\right), \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      17. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{-1}{720}, \frac{1}{24}\right)}, \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      18. lower-*.f64 100.0

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot -0.001388888888888889}, 0.041666666666666664\right), -0.5\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right) \]

   8. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right)} \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right) \]

   9. Taylor expanded in y around inf

      \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \frac{-1}{720}, \frac{1}{24}\right), \frac{-1}{2}\right), 1\right) \cdot \left(\frac{1}{5040} \cdot \color{blue}{{y}^{6}}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 100.0%

       \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right) \cdot \left(\left(\left(y \cdot y\right) \cdot 0.0001984126984126984\right) \cdot \color{blue}{\left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)}\right) \]

   if -inf.0 < (*.f64 (cos.f64 x) (/.f64 (sinh.f64 y) y)) < 0.999999999999977907

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 97.4

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

   5. Applied rewrites 97.4%

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

   if 0.999999999999977907 < (*.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 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos x \cdot \frac{\sinh y}{y} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right) \cdot \left(\left(0.0001984126984126984 \cdot \left(y \cdot y\right)\right) \cdot \left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right)\\ \mathbf{elif}\;\cos x \cdot \frac{\sinh y}{y} \leq 0.9999999999999779:\\ \;\;\;\;\cos x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sinh y}{y} \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y}{y}\\ t_1 := \cos x \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right) \cdot \left(\left(0.0001984126984126984 \cdot \left(y \cdot y\right)\right) \cdot \left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right)\\ \mathbf{elif}\;t\_1 \leq 0.9999999999999779:\\ \;\;\;\;\cos x \cdot \mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sinh y) y)) (t_1 (* (cos x) t_0)))
   (if (<= t_1 (- INFINITY))
     (*
      (fma
       (* x x)
       (fma
        x
        (* x (fma x (* x -0.001388888888888889) 0.041666666666666664))
        -0.5)
       1.0)
      (* (* 0.0001984126984126984 (* y y)) (* y (* y (* y y)))))
     (if (<= t_1 0.9999999999999779)
       (* (cos x) (fma 0.16666666666666666 (* y y) 1.0))
       (* t_0 1.0)))))

C

double code(double x, double y) {
	double t_0 = sinh(y) / y;
	double t_1 = cos(x) * t_0;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma((x * x), fma(x, (x * fma(x, (x * -0.001388888888888889), 0.041666666666666664)), -0.5), 1.0) * ((0.0001984126984126984 * (y * y)) * (y * (y * (y * y))));
	} else if (t_1 <= 0.9999999999999779) {
		tmp = cos(x) * fma(0.16666666666666666, (y * y), 1.0);
	} else {
		tmp = t_0 * 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sinh(y) / y)
	t_1 = Float64(cos(x) * t_0)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(fma(Float64(x * x), fma(x, Float64(x * fma(x, Float64(x * -0.001388888888888889), 0.041666666666666664)), -0.5), 1.0) * Float64(Float64(0.0001984126984126984 * Float64(y * y)) * Float64(y * Float64(y * Float64(y * y)))));
	elseif (t_1 <= 0.9999999999999779)
		tmp = Float64(cos(x) * fma(0.16666666666666666, Float64(y * y), 1.0));
	else
		tmp = Float64(t_0 * 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(x * N[(x * -0.001388888888888889), $MachinePrecision] + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(0.0001984126984126984 * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(y * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.9999999999999779], N[(N[Cos[x], $MachinePrecision] * N[(0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * 1.0), $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 y around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 74.6%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)\right)} \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right) + 1\right)} \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{2}, {x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}, 1\right)} \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, {x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}, 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, {x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}, 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(x \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right)\right)} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, x \cdot \color{blue}{\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) \cdot x\right)} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, x \cdot \left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) \cdot x\right) + \color{blue}{\frac{-1}{2}}, 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) \cdot x, \frac{-1}{2}\right)}, 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right)}, \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right)}, \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\left(\frac{-1}{720} \cdot {x}^{2} + \frac{1}{24}\right)}, \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{-1}{720}} + \frac{1}{24}\right), \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{-1}{720} + \frac{1}{24}\right), \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      16. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{x \cdot \left(x \cdot \frac{-1}{720}\right)} + \frac{1}{24}\right), \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      17. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{-1}{720}, \frac{1}{24}\right)}, \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      18. lower-*.f64 100.0

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot -0.001388888888888889}, 0.041666666666666664\right), -0.5\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right) \]

   8. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right)} \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right) \]

   9. Taylor expanded in y around inf

      \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \frac{-1}{720}, \frac{1}{24}\right), \frac{-1}{2}\right), 1\right) \cdot \left(\frac{1}{5040} \cdot \color{blue}{{y}^{6}}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 100.0%

       \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right) \cdot \left(\left(\left(y \cdot y\right) \cdot 0.0001984126984126984\right) \cdot \color{blue}{\left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)}\right) \]

   if -inf.0 < (*.f64 (cos.f64 x) (/.f64 (sinh.f64 y) y)) < 0.999999999999977907

   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. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 97.3

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

   5. Applied rewrites 97.3%

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

   if 0.999999999999977907 < (*.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 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos x \cdot \frac{\sinh y}{y} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right) \cdot \left(\left(0.0001984126984126984 \cdot \left(y \cdot y\right)\right) \cdot \left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right)\\ \mathbf{elif}\;\cos x \cdot \frac{\sinh y}{y} \leq 0.9999999999999779:\\ \;\;\;\;\cos x \cdot \mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sinh y}{y} \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y}{y}\\ t_1 := \cos x \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right) \cdot \left(\left(0.0001984126984126984 \cdot \left(y \cdot y\right)\right) \cdot \left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right)\\ \mathbf{elif}\;t\_1 \leq 0.9999999999999779:\\ \;\;\;\;\cos x\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sinh y) y)) (t_1 (* (cos x) t_0)))
   (if (<= t_1 (- INFINITY))
     (*
      (fma
       (* x x)
       (fma
        x
        (* x (fma x (* x -0.001388888888888889) 0.041666666666666664))
        -0.5)
       1.0)
      (* (* 0.0001984126984126984 (* y y)) (* y (* y (* y y)))))
     (if (<= t_1 0.9999999999999779) (cos x) (* t_0 1.0)))))

C

double code(double x, double y) {
	double t_0 = sinh(y) / y;
	double t_1 = cos(x) * t_0;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma((x * x), fma(x, (x * fma(x, (x * -0.001388888888888889), 0.041666666666666664)), -0.5), 1.0) * ((0.0001984126984126984 * (y * y)) * (y * (y * (y * y))));
	} else if (t_1 <= 0.9999999999999779) {
		tmp = cos(x);
	} else {
		tmp = t_0 * 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sinh(y) / y)
	t_1 = Float64(cos(x) * t_0)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(fma(Float64(x * x), fma(x, Float64(x * fma(x, Float64(x * -0.001388888888888889), 0.041666666666666664)), -0.5), 1.0) * Float64(Float64(0.0001984126984126984 * Float64(y * y)) * Float64(y * Float64(y * Float64(y * y)))));
	elseif (t_1 <= 0.9999999999999779)
		tmp = cos(x);
	else
		tmp = Float64(t_0 * 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(x * N[(x * -0.001388888888888889), $MachinePrecision] + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(0.0001984126984126984 * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(y * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.9999999999999779], N[Cos[x], $MachinePrecision], N[(t$95$0 * 1.0), $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 y around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 74.6%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)\right)} \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right) + 1\right)} \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{2}, {x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}, 1\right)} \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, {x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}, 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, {x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}, 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(x \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right)\right)} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, x \cdot \color{blue}{\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) \cdot x\right)} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, x \cdot \left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) \cdot x\right) + \color{blue}{\frac{-1}{2}}, 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) \cdot x, \frac{-1}{2}\right)}, 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right)}, \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right)}, \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\left(\frac{-1}{720} \cdot {x}^{2} + \frac{1}{24}\right)}, \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{-1}{720}} + \frac{1}{24}\right), \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{-1}{720} + \frac{1}{24}\right), \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      16. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{x \cdot \left(x \cdot \frac{-1}{720}\right)} + \frac{1}{24}\right), \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      17. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{-1}{720}, \frac{1}{24}\right)}, \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      18. lower-*.f64 100.0

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot -0.001388888888888889}, 0.041666666666666664\right), -0.5\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right) \]

   8. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right)} \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right) \]

   9. Taylor expanded in y around inf

      \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \frac{-1}{720}, \frac{1}{24}\right), \frac{-1}{2}\right), 1\right) \cdot \left(\frac{1}{5040} \cdot \color{blue}{{y}^{6}}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 100.0%

       \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right) \cdot \left(\left(\left(y \cdot y\right) \cdot 0.0001984126984126984\right) \cdot \color{blue}{\left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)}\right) \]

   if -inf.0 < (*.f64 (cos.f64 x) (/.f64 (sinh.f64 y) y)) < 0.999999999999977907

   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 97.2

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

   5. Applied rewrites 97.2%

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

   if 0.999999999999977907 < (*.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 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos x \cdot \frac{\sinh y}{y} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right) \cdot \left(\left(0.0001984126984126984 \cdot \left(y \cdot y\right)\right) \cdot \left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right)\\ \mathbf{elif}\;\cos x \cdot \frac{\sinh y}{y} \leq 0.9999999999999779:\\ \;\;\;\;\cos x\\ \mathbf{else}:\\ \;\;\;\;\frac{\sinh y}{y} \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 95.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x \cdot \frac{\sinh y}{y}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right) \cdot \left(\left(0.0001984126984126984 \cdot \left(y \cdot y\right)\right) \cdot \left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right)\\ \mathbf{elif}\;t\_0 \leq 0.9999999999999779:\\ \;\;\;\;\cos x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right) \cdot \mathsf{fma}\left(x, x \cdot \left(\left(x \cdot x\right) \cdot 0.041666666666666664\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (cos x) (/ (sinh y) y))))
   (if (<= t_0 (- INFINITY))
     (*
      (fma
       (* x x)
       (fma
        x
        (* x (fma x (* x -0.001388888888888889) 0.041666666666666664))
        -0.5)
       1.0)
      (* (* 0.0001984126984126984 (* y y)) (* y (* y (* y y)))))
     (if (<= t_0 0.9999999999999779)
       (cos x)
       (*
        (fma
         y
         (*
          y
          (fma
           y
           (* y (fma (* y y) 0.0001984126984126984 0.008333333333333333))
           0.16666666666666666))
         1.0)
        (fma x (* x (* (* x x) 0.041666666666666664)) 1.0))))))

C

double code(double x, double y) {
	double t_0 = cos(x) * (sinh(y) / y);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma((x * x), fma(x, (x * fma(x, (x * -0.001388888888888889), 0.041666666666666664)), -0.5), 1.0) * ((0.0001984126984126984 * (y * y)) * (y * (y * (y * y))));
	} else if (t_0 <= 0.9999999999999779) {
		tmp = cos(x);
	} else {
		tmp = fma(y, (y * fma(y, (y * fma((y * y), 0.0001984126984126984, 0.008333333333333333)), 0.16666666666666666)), 1.0) * fma(x, (x * ((x * x) * 0.041666666666666664)), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) * Float64(sinh(y) / y))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(fma(Float64(x * x), fma(x, Float64(x * fma(x, Float64(x * -0.001388888888888889), 0.041666666666666664)), -0.5), 1.0) * Float64(Float64(0.0001984126984126984 * Float64(y * y)) * Float64(y * Float64(y * Float64(y * y)))));
	elseif (t_0 <= 0.9999999999999779)
		tmp = cos(x);
	else
		tmp = Float64(fma(y, Float64(y * fma(y, Float64(y * fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333)), 0.16666666666666666)), 1.0) * fma(x, Float64(x * Float64(Float64(x * x) * 0.041666666666666664)), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(x * N[(x * -0.001388888888888889), $MachinePrecision] + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(0.0001984126984126984 * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(y * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.9999999999999779], N[Cos[x], $MachinePrecision], N[(N[(y * N[(y * N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 74.6%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)\right)} \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right) + 1\right)} \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{2}, {x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}, 1\right)} \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, {x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}, 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, {x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}, 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(x \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right)\right)} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, x \cdot \color{blue}{\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) \cdot x\right)} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, x \cdot \left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) \cdot x\right) + \color{blue}{\frac{-1}{2}}, 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) \cdot x, \frac{-1}{2}\right)}, 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right)}, \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right)}, \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\left(\frac{-1}{720} \cdot {x}^{2} + \frac{1}{24}\right)}, \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{-1}{720}} + \frac{1}{24}\right), \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{-1}{720} + \frac{1}{24}\right), \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      16. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{x \cdot \left(x \cdot \frac{-1}{720}\right)} + \frac{1}{24}\right), \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      17. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{-1}{720}, \frac{1}{24}\right)}, \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      18. lower-*.f64 100.0

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot -0.001388888888888889}, 0.041666666666666664\right), -0.5\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right) \]

   8. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right)} \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right) \]

   9. Taylor expanded in y around inf

      \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \frac{-1}{720}, \frac{1}{24}\right), \frac{-1}{2}\right), 1\right) \cdot \left(\frac{1}{5040} \cdot \color{blue}{{y}^{6}}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 100.0%

       \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right) \cdot \left(\left(\left(y \cdot y\right) \cdot 0.0001984126984126984\right) \cdot \color{blue}{\left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)}\right) \]

   if -inf.0 < (*.f64 (cos.f64 x) (/.f64 (sinh.f64 y) y)) < 0.999999999999977907

   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 97.2

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

   5. Applied rewrites 97.2%

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

   if 0.999999999999977907 < (*.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

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 89.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 89.6%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. lower-fma.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. sub-neg N/A

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

       7. *-commutative N/A

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

       8. unpow2 N/A

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

       9. associate-*l* N/A

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

       10. metadata-eval N/A

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

       11. lower-fma.f64 N/A

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

       12. lower-*.f64 92.5

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

   11. Applied rewrites 92.5%

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

   12. Taylor expanded in x around inf

       \[\leadsto \mathsf{fma}\left(x, \frac{1}{24} \cdot \color{blue}{{x}^{3}}, 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]
   13. Step-by-step derivation

   14. Applied rewrites 92.5%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos x \cdot \frac{\sinh y}{y} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right) \cdot \left(\left(0.0001984126984126984 \cdot \left(y \cdot y\right)\right) \cdot \left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right)\\ \mathbf{elif}\;\cos x \cdot \frac{\sinh y}{y} \leq 0.9999999999999779:\\ \;\;\;\;\cos x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right) \cdot \mathsf{fma}\left(x, x \cdot \left(\left(x \cdot x\right) \cdot 0.041666666666666664\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 73.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x \cdot \frac{\sinh y}{y}\\ t_1 := \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)\\ \mathbf{if}\;t\_0 \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right) \cdot t\_1\\ \mathbf{elif}\;t\_0 \leq 0.998:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(y, y \cdot \left(y \cdot \left(y \cdot \left(0.0001984126984126984 \cdot \left(y \cdot y\right)\right)\right)\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.041666666666666664, -0.5\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (cos x) (/ (sinh y) y)))
        (t_1
         (fma
          y
          (*
           y
           (fma
            y
            (* y (fma (* y y) 0.0001984126984126984 0.008333333333333333))
            0.16666666666666666))
          1.0)))
   (if (<= t_0 -0.05)
     (*
      (fma
       (* x x)
       (fma
        x
        (* x (fma x (* x -0.001388888888888889) 0.041666666666666664))
        -0.5)
       1.0)
      t_1)
     (if (<= t_0 0.998)
       (* 1.0 (fma y (* y (* y (* y (* 0.0001984126984126984 (* y y))))) 1.0))
       (* t_1 (fma x (* x (fma x (* x 0.041666666666666664) -0.5)) 1.0))))))

C

double code(double x, double y) {
	double t_0 = cos(x) * (sinh(y) / y);
	double t_1 = fma(y, (y * fma(y, (y * fma((y * y), 0.0001984126984126984, 0.008333333333333333)), 0.16666666666666666)), 1.0);
	double tmp;
	if (t_0 <= -0.05) {
		tmp = fma((x * x), fma(x, (x * fma(x, (x * -0.001388888888888889), 0.041666666666666664)), -0.5), 1.0) * t_1;
	} else if (t_0 <= 0.998) {
		tmp = 1.0 * fma(y, (y * (y * (y * (0.0001984126984126984 * (y * y))))), 1.0);
	} else {
		tmp = t_1 * fma(x, (x * fma(x, (x * 0.041666666666666664), -0.5)), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) * Float64(sinh(y) / y))
	t_1 = fma(y, Float64(y * fma(y, Float64(y * fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333)), 0.16666666666666666)), 1.0)
	tmp = 0.0
	if (t_0 <= -0.05)
		tmp = Float64(fma(Float64(x * x), fma(x, Float64(x * fma(x, Float64(x * -0.001388888888888889), 0.041666666666666664)), -0.5), 1.0) * t_1);
	elseif (t_0 <= 0.998)
		tmp = Float64(1.0 * fma(y, Float64(y * Float64(y * Float64(y * Float64(0.0001984126984126984 * Float64(y * y))))), 1.0));
	else
		tmp = Float64(t_1 * fma(x, Float64(x * fma(x, Float64(x * 0.041666666666666664), -0.5)), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(y * N[(y * N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[t$95$0, -0.05], N[(N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(x * N[(x * -0.001388888888888889), $MachinePrecision] + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t$95$0, 0.998], N[(1.0 * N[(y * N[(y * N[(y * N[(y * N[(0.0001984126984126984 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(x * N[(x * N[(x * N[(x * 0.041666666666666664), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 85.5%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)\right)} \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right) + 1\right)} \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{2}, {x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}, 1\right)} \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, {x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}, 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, {x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}, 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(x \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right)\right)} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, x \cdot \color{blue}{\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) \cdot x\right)} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, x \cdot \left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) \cdot x\right) + \color{blue}{\frac{-1}{2}}, 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) \cdot x, \frac{-1}{2}\right)}, 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right)}, \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right)}, \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\left(\frac{-1}{720} \cdot {x}^{2} + \frac{1}{24}\right)}, \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{-1}{720}} + \frac{1}{24}\right), \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{-1}{720} + \frac{1}{24}\right), \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      16. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{x \cdot \left(x \cdot \frac{-1}{720}\right)} + \frac{1}{24}\right), \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      17. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{-1}{720}, \frac{1}{24}\right)}, \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      18. lower-*.f64 49.8

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot -0.001388888888888889}, 0.041666666666666664\right), -0.5\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right) \]

   8. Applied rewrites 49.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right)} \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right) \]

   if -0.050000000000000003 < (*.f64 (cos.f64 x) (/.f64 (sinh.f64 y) y)) < 0.998

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 21.5%

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

   9. Taylor expanded in y around inf

      \[\leadsto 1 \cdot \mathsf{fma}\left(y, \frac{1}{5040} \cdot \color{blue}{{y}^{5}}, 1\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 21.5%

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

   if 0.998 < (*.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

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 89.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 89.3%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. lower-fma.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. sub-neg N/A

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

       7. *-commutative N/A

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

       8. unpow2 N/A

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

       9. associate-*l* N/A

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

       10. metadata-eval N/A

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

       11. lower-fma.f64 N/A

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

       12. lower-*.f64 92.7

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

   11. Applied rewrites 92.7%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos x \cdot \frac{\sinh y}{y} \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)\\ \mathbf{elif}\;\cos x \cdot \frac{\sinh y}{y} \leq 0.998:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(y, y \cdot \left(y \cdot \left(y \cdot \left(0.0001984126984126984 \cdot \left(y \cdot y\right)\right)\right)\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right) \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.041666666666666664, -0.5\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 73.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x \cdot \frac{\sinh y}{y}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+45}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right) \cdot \left(\left(0.0001984126984126984 \cdot \left(y \cdot y\right)\right) \cdot \left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right)\\ \mathbf{elif}\;t\_0 \leq 0.998:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right) \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.041666666666666664, -0.5\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (cos x) (/ (sinh y) y))))
   (if (<= t_0 -5e+45)
     (*
      (fma
       (* x x)
       (fma
        x
        (* x (fma x (* x -0.001388888888888889) 0.041666666666666664))
        -0.5)
       1.0)
      (* (* 0.0001984126984126984 (* y y)) (* y (* y (* y y)))))
     (if (<= t_0 0.998)
       1.0
       (*
        (fma
         y
         (*
          y
          (fma
           y
           (* y (fma (* y y) 0.0001984126984126984 0.008333333333333333))
           0.16666666666666666))
         1.0)
        (fma x (* x (fma x (* x 0.041666666666666664) -0.5)) 1.0))))))

C

double code(double x, double y) {
	double t_0 = cos(x) * (sinh(y) / y);
	double tmp;
	if (t_0 <= -5e+45) {
		tmp = fma((x * x), fma(x, (x * fma(x, (x * -0.001388888888888889), 0.041666666666666664)), -0.5), 1.0) * ((0.0001984126984126984 * (y * y)) * (y * (y * (y * y))));
	} else if (t_0 <= 0.998) {
		tmp = 1.0;
	} else {
		tmp = fma(y, (y * fma(y, (y * fma((y * y), 0.0001984126984126984, 0.008333333333333333)), 0.16666666666666666)), 1.0) * fma(x, (x * fma(x, (x * 0.041666666666666664), -0.5)), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) * Float64(sinh(y) / y))
	tmp = 0.0
	if (t_0 <= -5e+45)
		tmp = Float64(fma(Float64(x * x), fma(x, Float64(x * fma(x, Float64(x * -0.001388888888888889), 0.041666666666666664)), -0.5), 1.0) * Float64(Float64(0.0001984126984126984 * Float64(y * y)) * Float64(y * Float64(y * Float64(y * y)))));
	elseif (t_0 <= 0.998)
		tmp = 1.0;
	else
		tmp = Float64(fma(y, Float64(y * fma(y, Float64(y * fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333)), 0.16666666666666666)), 1.0) * fma(x, Float64(x * fma(x, Float64(x * 0.041666666666666664), -0.5)), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e+45], N[(N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(x * N[(x * -0.001388888888888889), $MachinePrecision] + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(0.0001984126984126984 * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(y * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.998], 1.0, N[(N[(y * N[(y * N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x * N[(x * N[(x * N[(x * 0.041666666666666664), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (cos.f64 x) (/.f64 (sinh.f64 y) y)) < -5e45

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 72.4%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)\right)} \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right) + 1\right)} \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{2}, {x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}, 1\right)} \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, {x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}, 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, {x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}, 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(x \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right)\right)} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, x \cdot \color{blue}{\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) \cdot x\right)} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, x \cdot \left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) \cdot x\right) + \color{blue}{\frac{-1}{2}}, 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) \cdot x, \frac{-1}{2}\right)}, 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right)}, \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right)}, \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\left(\frac{-1}{720} \cdot {x}^{2} + \frac{1}{24}\right)}, \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{-1}{720}} + \frac{1}{24}\right), \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{-1}{720} + \frac{1}{24}\right), \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      16. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{x \cdot \left(x \cdot \frac{-1}{720}\right)} + \frac{1}{24}\right), \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      17. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{-1}{720}, \frac{1}{24}\right)}, \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      18. lower-*.f64 96.9

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot -0.001388888888888889}, 0.041666666666666664\right), -0.5\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right) \]

   8. Applied rewrites 96.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right)} \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right) \]

   9. Taylor expanded in y around inf

      \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \frac{-1}{720}, \frac{1}{24}\right), \frac{-1}{2}\right), 1\right) \cdot \left(\frac{1}{5040} \cdot \color{blue}{{y}^{6}}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 96.9%

       \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right) \cdot \left(\left(\left(y \cdot y\right) \cdot 0.0001984126984126984\right) \cdot \color{blue}{\left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)}\right) \]

   if -5e45 < (*.f64 (cos.f64 x) (/.f64 (sinh.f64 y) y)) < 0.998

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-cos.f64 98.6

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

   5. Applied rewrites 98.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 10.3%

      \[\leadsto 1 \]

   if 0.998 < (*.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

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 89.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 89.3%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. lower-fma.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. sub-neg N/A

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

       7. *-commutative N/A

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

       8. unpow2 N/A

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

       9. associate-*l* N/A

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

       10. metadata-eval N/A

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

       11. lower-fma.f64 N/A

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

       12. lower-*.f64 92.7

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

   11. Applied rewrites 92.7%

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

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos x \cdot \frac{\sinh y}{y} \leq -5 \cdot 10^{+45}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right) \cdot \left(\left(0.0001984126984126984 \cdot \left(y \cdot y\right)\right) \cdot \left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right)\\ \mathbf{elif}\;\cos x \cdot \frac{\sinh y}{y} \leq 0.998:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right) \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.041666666666666664, -0.5\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 73.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x \cdot \frac{\sinh y}{y}\\ \mathbf{if}\;t\_0 \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)\\ \mathbf{elif}\;t\_0 \leq 0.998:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(y, y \cdot \left(y \cdot \left(y \cdot \left(0.0001984126984126984 \cdot \left(y \cdot y\right)\right)\right)\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right) \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.041666666666666664, -0.5\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (cos x) (/ (sinh y) y))))
   (if (<= t_0 -0.05)
     (*
      (fma
       (* x x)
       (fma
        x
        (* x (fma x (* x -0.001388888888888889) 0.041666666666666664))
        -0.5)
       1.0)
      (fma (* y y) (fma y (* y 0.008333333333333333) 0.16666666666666666) 1.0))
     (if (<= t_0 0.998)
       (* 1.0 (fma y (* y (* y (* y (* 0.0001984126984126984 (* y y))))) 1.0))
       (*
        (fma
         y
         (*
          y
          (fma
           y
           (* y (fma (* y y) 0.0001984126984126984 0.008333333333333333))
           0.16666666666666666))
         1.0)
        (fma x (* x (fma x (* x 0.041666666666666664) -0.5)) 1.0))))))

C

double code(double x, double y) {
	double t_0 = cos(x) * (sinh(y) / y);
	double tmp;
	if (t_0 <= -0.05) {
		tmp = fma((x * x), fma(x, (x * fma(x, (x * -0.001388888888888889), 0.041666666666666664)), -0.5), 1.0) * fma((y * y), fma(y, (y * 0.008333333333333333), 0.16666666666666666), 1.0);
	} else if (t_0 <= 0.998) {
		tmp = 1.0 * fma(y, (y * (y * (y * (0.0001984126984126984 * (y * y))))), 1.0);
	} else {
		tmp = fma(y, (y * fma(y, (y * fma((y * y), 0.0001984126984126984, 0.008333333333333333)), 0.16666666666666666)), 1.0) * fma(x, (x * fma(x, (x * 0.041666666666666664), -0.5)), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) * Float64(sinh(y) / y))
	tmp = 0.0
	if (t_0 <= -0.05)
		tmp = Float64(fma(Float64(x * x), fma(x, Float64(x * fma(x, Float64(x * -0.001388888888888889), 0.041666666666666664)), -0.5), 1.0) * fma(Float64(y * y), fma(y, Float64(y * 0.008333333333333333), 0.16666666666666666), 1.0));
	elseif (t_0 <= 0.998)
		tmp = Float64(1.0 * fma(y, Float64(y * Float64(y * Float64(y * Float64(0.0001984126984126984 * Float64(y * y))))), 1.0));
	else
		tmp = Float64(fma(y, Float64(y * fma(y, Float64(y * fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333)), 0.16666666666666666)), 1.0) * fma(x, Float64(x * fma(x, Float64(x * 0.041666666666666664), -0.5)), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.05], N[(N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(x * N[(x * -0.001388888888888889), $MachinePrecision] + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.998], N[(1.0 * N[(y * N[(y * N[(y * N[(y * N[(0.0001984126984126984 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(y * N[(y * N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x * N[(x * N[(x * N[(x * 0.041666666666666664), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 85.5%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)\right)} \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right) + 1\right)} \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{2}, {x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}, 1\right)} \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, {x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}, 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, {x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}, 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(x \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right)\right)} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, x \cdot \color{blue}{\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) \cdot x\right)} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, x \cdot \left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) \cdot x\right) + \color{blue}{\frac{-1}{2}}, 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) \cdot x, \frac{-1}{2}\right)}, 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right)}, \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right)}, \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\left(\frac{-1}{720} \cdot {x}^{2} + \frac{1}{24}\right)}, \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{-1}{720}} + \frac{1}{24}\right), \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{-1}{720} + \frac{1}{24}\right), \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      16. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{x \cdot \left(x \cdot \frac{-1}{720}\right)} + \frac{1}{24}\right), \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      17. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{-1}{720}, \frac{1}{24}\right)}, \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      18. lower-*.f64 49.8

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot -0.001388888888888889}, 0.041666666666666664\right), -0.5\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right) \]

   8. Applied rewrites 49.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right)} \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right) \]

   9. Taylor expanded in y around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 N/A

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

       5. +-commutative N/A

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

       6. unpow2 N/A

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

       7. associate-*r* N/A

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

       8. *-commutative N/A

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

       9. lower-fma.f64 N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 48.3

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

   11. Applied rewrites 48.3%

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

   if -0.050000000000000003 < (*.f64 (cos.f64 x) (/.f64 (sinh.f64 y) y)) < 0.998

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 21.5%

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

   9. Taylor expanded in y around inf

      \[\leadsto 1 \cdot \mathsf{fma}\left(y, \frac{1}{5040} \cdot \color{blue}{{y}^{5}}, 1\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 21.5%

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

   if 0.998 < (*.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

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 89.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 89.3%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. lower-fma.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. sub-neg N/A

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

       7. *-commutative N/A

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

       8. unpow2 N/A

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

       9. associate-*l* N/A

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

       10. metadata-eval N/A

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

       11. lower-fma.f64 N/A

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

       12. lower-*.f64 92.7

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

   11. Applied rewrites 92.7%

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

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos x \cdot \frac{\sinh y}{y} \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)\\ \mathbf{elif}\;\cos x \cdot \frac{\sinh y}{y} \leq 0.998:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(y, y \cdot \left(y \cdot \left(y \cdot \left(0.0001984126984126984 \cdot \left(y \cdot y\right)\right)\right)\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right) \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.041666666666666664, -0.5\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 73.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x \cdot \frac{\sinh y}{y}\\ \mathbf{if}\;t\_0 \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right) \cdot \mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{elif}\;t\_0 \leq 0.998:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(y, y \cdot \left(y \cdot \left(y \cdot \left(0.0001984126984126984 \cdot \left(y \cdot y\right)\right)\right)\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right) \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.041666666666666664, -0.5\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (cos x) (/ (sinh y) y))))
   (if (<= t_0 -0.05)
     (*
      (fma
       (* x x)
       (fma
        x
        (* x (fma x (* x -0.001388888888888889) 0.041666666666666664))
        -0.5)
       1.0)
      (fma 0.16666666666666666 (* y y) 1.0))
     (if (<= t_0 0.998)
       (* 1.0 (fma y (* y (* y (* y (* 0.0001984126984126984 (* y y))))) 1.0))
       (*
        (fma
         y
         (*
          y
          (fma
           y
           (* y (fma (* y y) 0.0001984126984126984 0.008333333333333333))
           0.16666666666666666))
         1.0)
        (fma x (* x (fma x (* x 0.041666666666666664) -0.5)) 1.0))))))

C

double code(double x, double y) {
	double t_0 = cos(x) * (sinh(y) / y);
	double tmp;
	if (t_0 <= -0.05) {
		tmp = fma((x * x), fma(x, (x * fma(x, (x * -0.001388888888888889), 0.041666666666666664)), -0.5), 1.0) * fma(0.16666666666666666, (y * y), 1.0);
	} else if (t_0 <= 0.998) {
		tmp = 1.0 * fma(y, (y * (y * (y * (0.0001984126984126984 * (y * y))))), 1.0);
	} else {
		tmp = fma(y, (y * fma(y, (y * fma((y * y), 0.0001984126984126984, 0.008333333333333333)), 0.16666666666666666)), 1.0) * fma(x, (x * fma(x, (x * 0.041666666666666664), -0.5)), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) * Float64(sinh(y) / y))
	tmp = 0.0
	if (t_0 <= -0.05)
		tmp = Float64(fma(Float64(x * x), fma(x, Float64(x * fma(x, Float64(x * -0.001388888888888889), 0.041666666666666664)), -0.5), 1.0) * fma(0.16666666666666666, Float64(y * y), 1.0));
	elseif (t_0 <= 0.998)
		tmp = Float64(1.0 * fma(y, Float64(y * Float64(y * Float64(y * Float64(0.0001984126984126984 * Float64(y * y))))), 1.0));
	else
		tmp = Float64(fma(y, Float64(y * fma(y, Float64(y * fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333)), 0.16666666666666666)), 1.0) * fma(x, Float64(x * fma(x, Float64(x * 0.041666666666666664), -0.5)), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.05], N[(N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(x * N[(x * -0.001388888888888889), $MachinePrecision] + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * N[(0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.998], N[(1.0 * N[(y * N[(y * N[(y * N[(y * N[(0.0001984126984126984 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(y * N[(y * N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x * N[(x * N[(x * N[(x * 0.041666666666666664), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   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. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 73.1

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

   5. Applied rewrites 73.1%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. unpow2 N/A

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

      16. associate-*l* N/A

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

      17. lower-fma.f64 N/A

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

      18. lower-*.f64 48.3

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

   8. Applied rewrites 48.3%

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

   if -0.050000000000000003 < (*.f64 (cos.f64 x) (/.f64 (sinh.f64 y) y)) < 0.998

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 21.5%

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

   9. Taylor expanded in y around inf

      \[\leadsto 1 \cdot \mathsf{fma}\left(y, \frac{1}{5040} \cdot \color{blue}{{y}^{5}}, 1\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 21.5%

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

   if 0.998 < (*.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

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 89.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 89.3%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. lower-fma.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. sub-neg N/A

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

       7. *-commutative N/A

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

       8. unpow2 N/A

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

       9. associate-*l* N/A

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

       10. metadata-eval N/A

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

       11. lower-fma.f64 N/A

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

       12. lower-*.f64 92.7

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

   11. Applied rewrites 92.7%

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

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos x \cdot \frac{\sinh y}{y} \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right) \cdot \mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{elif}\;\cos x \cdot \frac{\sinh y}{y} \leq 0.998:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(y, y \cdot \left(y \cdot \left(y \cdot \left(0.0001984126984126984 \cdot \left(y \cdot y\right)\right)\right)\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right) \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.041666666666666664, -0.5\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 70.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x \cdot \frac{\sinh y}{y}\\ \mathbf{if}\;t\_0 \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right) \cdot \mathsf{fma}\left(x, x \cdot -0.5, 1\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(0.0001984126984126984 \cdot \left(y \cdot y\right)\right) \cdot \left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (cos x) (/ (sinh y) y))))
   (if (<= t_0 -0.05)
     (* (fma 0.16666666666666666 (* y y) 1.0) (fma x (* x -0.5) 1.0))
     (if (<= t_0 2.0)
       (*
        1.0
        (fma
         y
         (* y (fma y (* y 0.008333333333333333) 0.16666666666666666))
         1.0))
       (* (* (* 0.0001984126984126984 (* y y)) (* y (* y (* y y)))) 1.0)))))

C

double code(double x, double y) {
	double t_0 = cos(x) * (sinh(y) / y);
	double tmp;
	if (t_0 <= -0.05) {
		tmp = fma(0.16666666666666666, (y * y), 1.0) * fma(x, (x * -0.5), 1.0);
	} else if (t_0 <= 2.0) {
		tmp = 1.0 * fma(y, (y * fma(y, (y * 0.008333333333333333), 0.16666666666666666)), 1.0);
	} else {
		tmp = ((0.0001984126984126984 * (y * y)) * (y * (y * (y * y)))) * 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) * Float64(sinh(y) / y))
	tmp = 0.0
	if (t_0 <= -0.05)
		tmp = Float64(fma(0.16666666666666666, Float64(y * y), 1.0) * fma(x, Float64(x * -0.5), 1.0));
	elseif (t_0 <= 2.0)
		tmp = Float64(1.0 * fma(y, Float64(y * fma(y, Float64(y * 0.008333333333333333), 0.16666666666666666)), 1.0));
	else
		tmp = Float64(Float64(Float64(0.0001984126984126984 * Float64(y * y)) * Float64(y * Float64(y * Float64(y * y)))) * 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.05], N[(N[(0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x * N[(x * -0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(1.0 * N[(y * N[(y * N[(y * N[(y * 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.0001984126984126984 * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(y * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \cos x \cdot \color{blue}{\left(1 + \frac{1}{6} \cdot {y}^{2}\right)} \]
   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. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 73.1

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

   5. Applied rewrites 73.1%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 42.4

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

   8. Applied rewrites 42.4%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 77.4%

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

   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. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 77.4

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

   8. Applied rewrites 77.4%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 83.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 83.2%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 83.2%

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

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos x \cdot \frac{\sinh y}{y} \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right) \cdot \mathsf{fma}\left(x, x \cdot -0.5, 1\right)\\ \mathbf{elif}\;\cos x \cdot \frac{\sinh y}{y} \leq 2:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(0.0001984126984126984 \cdot \left(y \cdot y\right)\right) \cdot \left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right) \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 67.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 49.9

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

   5. Applied rewrites 49.9%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 42.4

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

   8. Applied rewrites 42.4%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 41.8%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \cos x \cdot \color{blue}{\left(1 + \frac{1}{6} \cdot {y}^{2}\right)} \]
   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. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 77.4%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. 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. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 71.3

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

   8. Applied rewrites 71.3%

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

   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 71.3%

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

Alternative 14: 63.5% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-cos.f64 51.1

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

   5. Applied rewrites 51.1%

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

   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 24.3%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \cos x \cdot \color{blue}{\left(1 + \frac{1}{6} \cdot {y}^{2}\right)} \]
   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. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 77.4%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. 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. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 71.3

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

   8. Applied rewrites 71.3%

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

   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 71.3%

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

Alternative 15: 63.5% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-cos.f64 51.1

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

   5. Applied rewrites 51.1%

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

   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 24.3%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \cos x \cdot \color{blue}{\left(1 + \frac{1}{6} \cdot {y}^{2}\right)} \]
   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. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 77.4%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. 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. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 71.3

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

   8. Applied rewrites 71.3%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 71.3%

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

4. Final simplification 61.9%

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

Alternative 16: 72.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos x \cdot \frac{\sinh y}{y} \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right) \cdot \mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cos x) (/ (sinh y) y)) -0.05)
   (*
    (fma
     (* x x)
     (fma
      x
      (* x (fma x (* x -0.001388888888888889) 0.041666666666666664))
      -0.5)
     1.0)
    (fma 0.16666666666666666 (* y y) 1.0))
   (*
    1.0
    (/
     (fma
      (fma
       y
       (* y (fma y (* y 0.0001984126984126984) 0.008333333333333333))
       0.16666666666666666)
      (* y (* y y))
      y)
     y))))

C

double code(double x, double y) {
	double tmp;
	if ((cos(x) * (sinh(y) / y)) <= -0.05) {
		tmp = fma((x * x), fma(x, (x * fma(x, (x * -0.001388888888888889), 0.041666666666666664)), -0.5), 1.0) * fma(0.16666666666666666, (y * y), 1.0);
	} else {
		tmp = 1.0 * (fma(fma(y, (y * fma(y, (y * 0.0001984126984126984), 0.008333333333333333)), 0.16666666666666666), (y * (y * y)), y) / y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(cos(x) * Float64(sinh(y) / y)) <= -0.05)
		tmp = Float64(fma(Float64(x * x), fma(x, Float64(x * fma(x, Float64(x * -0.001388888888888889), 0.041666666666666664)), -0.5), 1.0) * fma(0.16666666666666666, Float64(y * y), 1.0));
	else
		tmp = Float64(1.0 * Float64(fma(fma(y, Float64(y * fma(y, Float64(y * 0.0001984126984126984), 0.008333333333333333)), 0.16666666666666666), Float64(y * Float64(y * y)), y) / y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Cos[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], -0.05], N[(N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(x * N[(x * -0.001388888888888889), $MachinePrecision] + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * N[(0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 * N[(N[(N[(y * N[(y * N[(y * N[(y * 0.0001984126984126984), $MachinePrecision] + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \cos x \cdot \color{blue}{\left(1 + \frac{1}{6} \cdot {y}^{2}\right)} \]
   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. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 73.1

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

   5. Applied rewrites 73.1%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. unpow2 N/A

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

      16. associate-*l* N/A

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

      17. lower-fma.f64 N/A

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

      18. lower-*.f64 48.3

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

   8. Applied rewrites 48.3%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 89.3%

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

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. unpow2 N/A

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

      7. unpow3 N/A

         \[\leadsto 1 \cdot \frac{\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot \color{blue}{{y}^{3}} + y}{y} \]

      8. lower-fma.f64 N/A

         \[\leadsto 1 \cdot \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right), {y}^{3}, y\right)}}{y} \]

   8. Applied rewrites 80.9%

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

Alternative 17: 71.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos x \cdot \frac{\sinh y}{y} \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right) \cdot \mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cos x) (/ (sinh y) y)) -0.05)
   (*
    (fma
     (* x x)
     (fma
      x
      (* x (fma x (* x -0.001388888888888889) 0.041666666666666664))
      -0.5)
     1.0)
    (fma 0.16666666666666666 (* y y) 1.0))
   (*
    1.0
    (fma
     y
     (*
      y
      (fma
       y
       (* y (fma y (* y 0.0001984126984126984) 0.008333333333333333))
       0.16666666666666666))
     1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((cos(x) * (sinh(y) / y)) <= -0.05) {
		tmp = fma((x * x), fma(x, (x * fma(x, (x * -0.001388888888888889), 0.041666666666666664)), -0.5), 1.0) * fma(0.16666666666666666, (y * y), 1.0);
	} else {
		tmp = 1.0 * fma(y, (y * fma(y, (y * fma(y, (y * 0.0001984126984126984), 0.008333333333333333)), 0.16666666666666666)), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(cos(x) * Float64(sinh(y) / y)) <= -0.05)
		tmp = Float64(fma(Float64(x * x), fma(x, Float64(x * fma(x, Float64(x * -0.001388888888888889), 0.041666666666666664)), -0.5), 1.0) * fma(0.16666666666666666, Float64(y * y), 1.0));
	else
		tmp = Float64(1.0 * fma(y, Float64(y * fma(y, Float64(y * fma(y, Float64(y * 0.0001984126984126984), 0.008333333333333333)), 0.16666666666666666)), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Cos[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], -0.05], N[(N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(x * N[(x * -0.001388888888888889), $MachinePrecision] + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * N[(0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 * N[(y * N[(y * N[(y * N[(y * N[(y * N[(y * 0.0001984126984126984), $MachinePrecision] + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \cos x \cdot \color{blue}{\left(1 + \frac{1}{6} \cdot {y}^{2}\right)} \]
   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. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 73.1

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

   5. Applied rewrites 73.1%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. unpow2 N/A

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

      16. associate-*l* N/A

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

      17. lower-fma.f64 N/A

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

      18. lower-*.f64 48.3

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

   8. Applied rewrites 48.3%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 89.3%

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

   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. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

   8. Applied rewrites 80.5%

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

Alternative 18: 71.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right)\\ \mathbf{if}\;\cos x \cdot \frac{\sinh y}{y} \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot -0.5, 1\right) \cdot \mathsf{fma}\left(y, y \cdot \left(y \cdot t\_0\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, t\_0, 0.16666666666666666\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (fma y (* y 0.0001984126984126984) 0.008333333333333333))))
   (if (<= (* (cos x) (/ (sinh y) y)) -0.05)
     (* (fma x (* x -0.5) 1.0) (fma y (* y (* y t_0)) 1.0))
     (* 1.0 (fma y (* y (fma y t_0 0.16666666666666666)) 1.0)))))

C

double code(double x, double y) {
	double t_0 = y * fma(y, (y * 0.0001984126984126984), 0.008333333333333333);
	double tmp;
	if ((cos(x) * (sinh(y) / y)) <= -0.05) {
		tmp = fma(x, (x * -0.5), 1.0) * fma(y, (y * (y * t_0)), 1.0);
	} else {
		tmp = 1.0 * fma(y, (y * fma(y, t_0, 0.16666666666666666)), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(y * fma(y, Float64(y * 0.0001984126984126984), 0.008333333333333333))
	tmp = 0.0
	if (Float64(cos(x) * Float64(sinh(y) / y)) <= -0.05)
		tmp = Float64(fma(x, Float64(x * -0.5), 1.0) * fma(y, Float64(y * Float64(y * t_0)), 1.0));
	else
		tmp = Float64(1.0 * fma(y, Float64(y * fma(y, t_0, 0.16666666666666666)), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(y * N[(y * 0.0001984126984126984), $MachinePrecision] + 0.008333333333333333), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Cos[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], -0.05], N[(N[(x * N[(x * -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * N[(y * N[(y * N[(y * t$95$0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 * N[(y * N[(y * N[(y * t$95$0 + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 85.5%

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

   6. Taylor expanded in y around inf

      \[\leadsto \cos x \cdot \mathsf{fma}\left(y, {y}^{5} \cdot \color{blue}{\left(\frac{1}{5040} + \frac{1}{120} \cdot \frac{1}{{y}^{2}}\right)}, 1\right) \]

   8. Applied rewrites 85.2%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 46.9

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

   11. Applied rewrites 46.9%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 89.3%

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

   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. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

   8. Applied rewrites 80.5%

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

Alternative 19: 70.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos x \cdot \frac{\sinh y}{y} \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right) \cdot \mathsf{fma}\left(x, x \cdot -0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cos x) (/ (sinh y) y)) -0.05)
   (* (fma 0.16666666666666666 (* y y) 1.0) (fma x (* x -0.5) 1.0))
   (*
    1.0
    (fma
     y
     (*
      y
      (fma
       y
       (* y (fma y (* y 0.0001984126984126984) 0.008333333333333333))
       0.16666666666666666))
     1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((cos(x) * (sinh(y) / y)) <= -0.05) {
		tmp = fma(0.16666666666666666, (y * y), 1.0) * fma(x, (x * -0.5), 1.0);
	} else {
		tmp = 1.0 * fma(y, (y * fma(y, (y * fma(y, (y * 0.0001984126984126984), 0.008333333333333333)), 0.16666666666666666)), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(cos(x) * Float64(sinh(y) / y)) <= -0.05)
		tmp = Float64(fma(0.16666666666666666, Float64(y * y), 1.0) * fma(x, Float64(x * -0.5), 1.0));
	else
		tmp = Float64(1.0 * fma(y, Float64(y * fma(y, Float64(y * fma(y, Float64(y * 0.0001984126984126984), 0.008333333333333333)), 0.16666666666666666)), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Cos[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], -0.05], N[(N[(0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x * N[(x * -0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 * N[(y * N[(y * N[(y * N[(y * N[(y * N[(y * 0.0001984126984126984), $MachinePrecision] + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \cos x \cdot \color{blue}{\left(1 + \frac{1}{6} \cdot {y}^{2}\right)} \]
   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. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 73.1

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

   5. Applied rewrites 73.1%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 42.4

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

   8. Applied rewrites 42.4%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 89.3%

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

   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. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

   8. Applied rewrites 80.5%

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

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos x \cdot \frac{\sinh y}{y} \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right) \cdot \mathsf{fma}\left(x, x \cdot -0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 70.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos x \cdot \frac{\sinh y}{y} \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right) \cdot \mathsf{fma}\left(x, x \cdot -0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(y, y \cdot \left(y \cdot \left(y \cdot \left(0.0001984126984126984 \cdot \left(y \cdot y\right)\right)\right)\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cos x) (/ (sinh y) y)) -0.05)
   (* (fma 0.16666666666666666 (* y y) 1.0) (fma x (* x -0.5) 1.0))
   (* 1.0 (fma y (* y (* y (* y (* 0.0001984126984126984 (* y y))))) 1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((cos(x) * (sinh(y) / y)) <= -0.05) {
		tmp = fma(0.16666666666666666, (y * y), 1.0) * fma(x, (x * -0.5), 1.0);
	} else {
		tmp = 1.0 * fma(y, (y * (y * (y * (0.0001984126984126984 * (y * y))))), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(cos(x) * Float64(sinh(y) / y)) <= -0.05)
		tmp = Float64(fma(0.16666666666666666, Float64(y * y), 1.0) * fma(x, Float64(x * -0.5), 1.0));
	else
		tmp = Float64(1.0 * fma(y, Float64(y * Float64(y * Float64(y * Float64(0.0001984126984126984 * Float64(y * y))))), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Cos[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], -0.05], N[(N[(0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x * N[(x * -0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 * N[(y * N[(y * N[(y * N[(y * N[(0.0001984126984126984 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \cos x \cdot \color{blue}{\left(1 + \frac{1}{6} \cdot {y}^{2}\right)} \]
   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. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 73.1

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

   5. Applied rewrites 73.1%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 42.4

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

   8. Applied rewrites 42.4%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 91.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 80.5%

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

   9. Taylor expanded in y around inf

      \[\leadsto 1 \cdot \mathsf{fma}\left(y, \frac{1}{5040} \cdot \color{blue}{{y}^{5}}, 1\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 80.3%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos x \cdot \frac{\sinh y}{y} \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right) \cdot \mathsf{fma}\left(x, x \cdot -0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(y, y \cdot \left(y \cdot \left(y \cdot \left(0.0001984126984126984 \cdot \left(y \cdot y\right)\right)\right)\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 67.8% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \cos x \cdot \color{blue}{\left(1 + \frac{1}{6} \cdot {y}^{2}\right)} \]
   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. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 73.1

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

   5. Applied rewrites 73.1%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 42.4

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

   8. Applied rewrites 42.4%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 89.3%

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

   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. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 74.2

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

   8. Applied rewrites 74.2%

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

4. Final simplification 66.4%

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

Alternative 22: 54.5% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-cos.f64 51.1

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

   5. Applied rewrites 51.1%

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

   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 24.3%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in 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. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 74.0

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

   5. Applied rewrites 74.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 63.3%

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

Alternative 23: 35.0% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-cos.f64 51.1

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

   5. Applied rewrites 51.1%

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

   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 24.3%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-cos.f64 48.6

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

   5. Applied rewrites 48.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 37.9%

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

Alternative 24: 35.0% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((cos(x) * (sinh(y) / y)) <= (-0.05d0)) then
        tmp = (x * x) * (-0.5d0)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((Math.cos(x) * (Math.sinh(y) / y)) <= -0.05) {
		tmp = (x * x) * -0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (math.cos(x) * (math.sinh(y) / y)) <= -0.05:
		tmp = (x * x) * -0.5
	else:
		tmp = 1.0
	return tmp

Julia

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

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((cos(x) * (sinh(y) / y)) <= -0.05)
		tmp = (x * x) * -0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-cos.f64 51.1

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

   5. Applied rewrites 51.1%

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

   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 24.3%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 24.2%

       \[\leadsto \left(x \cdot x\right) \cdot -0.5 \]

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-cos.f64 48.6

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

   5. Applied rewrites 48.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 37.9%

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

Alternative 25: 70.4% accurate, 1.5× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (cos x) -0.01)
   (* (fma 0.16666666666666666 (* y y) 1.0) (fma x (* x -0.5) 1.0))
   (*
    1.0
    (fma
     y
     (* y (* y (* y (fma y (* y 0.0001984126984126984) 0.008333333333333333))))
     1.0))))

C

double code(double x, double y) {
	double tmp;
	if (cos(x) <= -0.01) {
		tmp = fma(0.16666666666666666, (y * y), 1.0) * fma(x, (x * -0.5), 1.0);
	} else {
		tmp = 1.0 * fma(y, (y * (y * (y * fma(y, (y * 0.0001984126984126984), 0.008333333333333333)))), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (cos(x) <= -0.01)
		tmp = Float64(fma(0.16666666666666666, Float64(y * y), 1.0) * fma(x, Float64(x * -0.5), 1.0));
	else
		tmp = Float64(1.0 * fma(y, Float64(y * Float64(y * Float64(y * fma(y, Float64(y * 0.0001984126984126984), 0.008333333333333333)))), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[Cos[x], $MachinePrecision], -0.01], N[(N[(0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x * N[(x * -0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 * N[(y * N[(y * N[(y * N[(y * N[(y * N[(y * 0.0001984126984126984), $MachinePrecision] + 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 x) < -0.0100000000000000002

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \cos x \cdot \color{blue}{\left(1 + \frac{1}{6} \cdot {y}^{2}\right)} \]
   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. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 73.1

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

   5. Applied rewrites 73.1%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 42.4

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

   8. Applied rewrites 42.4%

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

   if -0.0100000000000000002 < (cos.f64 x)

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 91.1%

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

   6. Taylor expanded in y around inf

      \[\leadsto \cos x \cdot \mathsf{fma}\left(y, {y}^{5} \cdot \color{blue}{\left(\frac{1}{5040} + \frac{1}{120} \cdot \frac{1}{{y}^{2}}\right)}, 1\right) \]

   8. Applied rewrites 90.9%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 80.3%

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

4. Final simplification 71.0%

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

Alternative 26: 67.6% accurate, 1.6× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (cos x) -0.01)
   (* (fma x (* x -0.5) 1.0) (* y (* y 0.16666666666666666)))
   (*
    1.0
    (fma y (* y (fma y (* y 0.008333333333333333) 0.16666666666666666)) 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (cos(x) <= -0.01) {
		tmp = fma(x, (x * -0.5), 1.0) * (y * (y * 0.16666666666666666));
	} else {
		tmp = 1.0 * fma(y, (y * fma(y, (y * 0.008333333333333333), 0.16666666666666666)), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (cos(x) <= -0.01)
		tmp = Float64(fma(x, Float64(x * -0.5), 1.0) * Float64(y * Float64(y * 0.16666666666666666)));
	else
		tmp = Float64(1.0 * fma(y, Float64(y * fma(y, Float64(y * 0.008333333333333333), 0.16666666666666666)), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[Cos[x], $MachinePrecision], -0.01], N[(N[(x * N[(x * -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * N[(y * N[(y * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 * N[(y * N[(y * N[(y * N[(y * 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 x) < -0.0100000000000000002

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 49.9

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

   5. Applied rewrites 49.9%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 42.4

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

   8. Applied rewrites 42.4%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 41.8%

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

   if -0.0100000000000000002 < (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 89.3%

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

   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. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 74.2

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

   8. Applied rewrites 74.2%

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

Alternative 27: 28.4% 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 49.2

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

5. Applied rewrites 49.2%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 28.9%

   \[\leadsto 1 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024233 
(FPCore (x y)
  :name "Linear.Quaternion:$csin from linear-1.19.1.3"
  :precision binary64
  (* (cos x) (/ (sinh y) y)))