Linear.Quaternion:$csin from linear-1.19.1.3

Percentage Accurate: 100.0% → 99.2%

Time: 11.4s

Alternatives: 21

Speedup: 1.0×

• Could not converge on a ground truth

  1. x = 1.7778518688450889e-193
  2. y = 2.0115826969203702e+160

• 48.9% 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: 99.2% accurate, 0.4× speedup

Math

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

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 -0.5) 1.0)
      (fma
       (* y y)
       (fma
        y
        (* y (fma (* y y) 0.0001984126984126984 0.008333333333333333))
        0.16666666666666666)
       1.0))
     (if (<= t_1 2.0)
       (*
        (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 * -0.5), 1.0) * fma((y * y), fma(y, (y * fma((y * y), 0.0001984126984126984, 0.008333333333333333)), 0.16666666666666666), 1.0);
	} else if (t_1 <= 2.0) {
		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(x, Float64(x * -0.5), 1.0) * fma(Float64(y * y), fma(y, Float64(y * fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333)), 0.16666666666666666), 1.0));
	elseif (t_1 <= 2.0)
		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[(x * N[(x * -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2.0], 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 x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 100.0

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

   8. Applied rewrites 100.0%

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

   if -inf.0 < (*.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

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

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

   5. Applied rewrites 100.0%

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

   if 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%

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

4. Final simplification 100.0%

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

Alternative 2: 99.1% accurate, 0.4× speedup

Math

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

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 -0.5) 1.0)
      (fma
       (* y y)
       (fma
        y
        (* y (fma (* y y) 0.0001984126984126984 0.008333333333333333))
        0.16666666666666666)
       1.0))
     (if (<= t_1 2.0)
       (* (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 * -0.5), 1.0) * fma((y * y), fma(y, (y * fma((y * y), 0.0001984126984126984, 0.008333333333333333)), 0.16666666666666666), 1.0);
	} else if (t_1 <= 2.0) {
		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(x, Float64(x * -0.5), 1.0) * fma(Float64(y * y), fma(y, Float64(y * fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333)), 0.16666666666666666), 1.0));
	elseif (t_1 <= 2.0)
		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[(x * N[(x * -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2.0], 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 x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 100.0

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

   8. Applied rewrites 100.0%

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

   if -inf.0 < (*.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 99.8

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

   5. Applied rewrites 99.8%

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

   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%

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

4. Final simplification 99.9%

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

Alternative 3: 99.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y}{y}\\ t_1 := \cos x \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot -0.5, 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)\\ \mathbf{elif}\;t\_1 \leq 0.9999999999999689:\\ \;\;\;\;\cos x\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot 1\\ \end{array} \end{array} \]

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 -0.5) 1.0)
      (fma
       (* y y)
       (fma
        y
        (* y (fma (* y y) 0.0001984126984126984 0.008333333333333333))
        0.16666666666666666)
       1.0))
     (if (<= t_1 0.9999999999999689) (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 * -0.5), 1.0) * fma((y * y), fma(y, (y * fma((y * y), 0.0001984126984126984, 0.008333333333333333)), 0.16666666666666666), 1.0);
	} else if (t_1 <= 0.9999999999999689) {
		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(x, Float64(x * -0.5), 1.0) * fma(Float64(y * y), fma(y, Float64(y * fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333)), 0.16666666666666666), 1.0));
	elseif (t_1 <= 0.9999999999999689)
		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[(x * N[(x * -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.9999999999999689], 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 x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 100.0

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

   8. Applied rewrites 100.0%

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

   if -inf.0 < (*.f64 (cos.f64 x) (/.f64 (sinh.f64 y) y)) < 0.99999999999996891

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-cos.f64 99.0

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

   5. Applied rewrites 99.0%

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

   if 0.99999999999996891 < (*.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 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos x \cdot \frac{\sinh y}{y} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot -0.5, 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)\\ \mathbf{elif}\;\cos x \cdot \frac{\sinh y}{y} \leq 0.9999999999999689:\\ \;\;\;\;\cos x\\ \mathbf{else}:\\ \;\;\;\;\frac{\sinh y}{y} \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x \cdot \frac{\sinh y}{y}\\ t_1 := \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot -0.5, 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot t\_1, 0.16666666666666666\right), 1\right)\\ \mathbf{elif}\;t\_0 \leq 0.99999:\\ \;\;\;\;\cos x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.041666666666666664, -0.5\right), 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, t\_1, 0.16666666666666666\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (cos x) (/ (sinh y) y)))
        (t_1 (fma (* y y) 0.0001984126984126984 0.008333333333333333)))
   (if (<= t_0 (- INFINITY))
     (*
      (fma x (* x -0.5) 1.0)
      (fma (* y y) (fma y (* y t_1) 0.16666666666666666) 1.0))
     (if (<= t_0 0.99999)
       (cos x)
       (*
        (fma x (* x (fma x (* x 0.041666666666666664) -0.5)) 1.0)
        (fma (* y y) (fma (* y y) t_1 0.16666666666666666) 1.0))))))

C

double code(double x, double y) {
	double t_0 = cos(x) * (sinh(y) / y);
	double t_1 = fma((y * y), 0.0001984126984126984, 0.008333333333333333);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma(x, (x * -0.5), 1.0) * fma((y * y), fma(y, (y * t_1), 0.16666666666666666), 1.0);
	} else if (t_0 <= 0.99999) {
		tmp = cos(x);
	} else {
		tmp = fma(x, (x * fma(x, (x * 0.041666666666666664), -0.5)), 1.0) * fma((y * y), fma((y * y), t_1, 0.16666666666666666), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) * Float64(sinh(y) / y))
	t_1 = fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(fma(x, Float64(x * -0.5), 1.0) * fma(Float64(y * y), fma(y, Float64(y * t_1), 0.16666666666666666), 1.0));
	elseif (t_0 <= 0.99999)
		tmp = cos(x);
	else
		tmp = Float64(fma(x, Float64(x * fma(x, Float64(x * 0.041666666666666664), -0.5)), 1.0) * fma(Float64(y * y), fma(Float64(y * y), t_1, 0.16666666666666666), 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[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(x * N[(x * -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * t$95$1), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.99999], N[Cos[x], $MachinePrecision], N[(N[(x * N[(x * N[(x * N[(x * 0.041666666666666664), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * t$95$1 + 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 100.0

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

   8. Applied rewrites 100.0%

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

   if -inf.0 < (*.f64 (cos.f64 x) (/.f64 (sinh.f64 y) y)) < 0.999990000000000046

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-cos.f64 99.0

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

   5. Applied rewrites 99.0%

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

   if 0.999990000000000046 < (*.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 99.9%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 69.3

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

   8. Applied rewrites 69.3%

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

   9. Taylor expanded in y around 0

      \[\leadsto 1 \cdot \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)} \]
   10. 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. lower-fma.f64 N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 N/A

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

       5. +-commutative N/A

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

       6. lower-fma.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 N/A

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

       9. +-commutative N/A

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

       10. *-commutative N/A

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

       11. lower-fma.f64 N/A

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

       12. unpow2 N/A

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

       13. lower-*.f64 90.4

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

   11. Applied rewrites 90.4%

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

   12. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. lower-fma.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. sub-neg N/A

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

       7. *-commutative N/A

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

       8. unpow2 N/A

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

       9. associate-*l* N/A

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

       10. metadata-eval N/A

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

       11. lower-fma.f64 N/A

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

       12. lower-*.f64 91.7

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

   14. Applied rewrites 91.7%

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

Alternative 5: 71.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x \cdot \frac{\sinh y}{y}\\ \mathbf{if}\;t\_0 \leq -0.01:\\ \;\;\;\;\left(x \cdot \left(x \cdot -0.5\right)\right) \cdot \mathsf{fma}\left(y, y \cdot 0.16666666666666666, 1\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\left(y \cdot y\right) \cdot \left(y \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (cos x) (/ (sinh y) y))))
   (if (<= t_0 -0.01)
     (* (* x (* x -0.5)) (fma y (* y 0.16666666666666666) 1.0))
     (if (<= t_0 2.0)
       (*
        1.0
        (fma
         y
         (* y (fma y (* y 0.008333333333333333) 0.16666666666666666))
         1.0))
       (*
        1.0
        (*
         (* y y)
         (*
          y
          (* y (fma (* y y) 0.0001984126984126984 0.008333333333333333)))))))))

C

double code(double x, double y) {
	double t_0 = cos(x) * (sinh(y) / y);
	double tmp;
	if (t_0 <= -0.01) {
		tmp = (x * (x * -0.5)) * fma(y, (y * 0.16666666666666666), 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 = 1.0 * ((y * y) * (y * (y * fma((y * y), 0.0001984126984126984, 0.008333333333333333))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) * Float64(sinh(y) / y))
	tmp = 0.0
	if (t_0 <= -0.01)
		tmp = Float64(Float64(x * Float64(x * -0.5)) * fma(y, Float64(y * 0.16666666666666666), 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(1.0 * Float64(Float64(y * y) * Float64(y * Float64(y * fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.01], N[(N[(x * N[(x * -0.5), $MachinePrecision]), $MachinePrecision] * N[(y * N[(y * 0.16666666666666666), $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[(1.0 * N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (cos.f64 x) (/.f64 (sinh.f64 y) y)) < -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 60.3

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

   5. Applied rewrites 60.3%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. 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 53.6

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

   8. Applied rewrites 53.6%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 53.6%

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

   if -0.0100000000000000002 < (*.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 77.1%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 77.1

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

   8. Applied rewrites 77.1%

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

   9. Taylor expanded in y around 0

      \[\leadsto 1 \cdot \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)} \]
   10. 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. lower-fma.f64 N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 N/A

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

       5. +-commutative N/A

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

       6. lower-fma.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 N/A

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

       9. +-commutative N/A

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

       10. *-commutative N/A

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

       11. lower-fma.f64 N/A

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

       12. unpow2 N/A

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

       13. lower-*.f64 77.1

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

   11. Applied rewrites 77.1%

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

   12. Taylor expanded in y around 0

       \[\leadsto 1 \cdot \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} \]
   13. 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. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 N/A

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

       8. +-commutative N/A

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

       9. *-commutative N/A

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

       10. unpow2 N/A

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

       11. associate-*l* N/A

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

       12. *-commutative N/A

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

       13. lower-fma.f64 N/A

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

       14. *-commutative N/A

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

       15. lower-*.f64 77.1

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

   14. Applied rewrites 77.1%

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

   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 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

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 48.3

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

   8. Applied rewrites 48.3%

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

   9. Taylor expanded in y around 0

      \[\leadsto 1 \cdot \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)} \]
   10. 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. lower-fma.f64 N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 N/A

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

       5. +-commutative N/A

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

       6. lower-fma.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 N/A

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

       9. +-commutative N/A

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

       10. *-commutative N/A

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

       11. lower-fma.f64 N/A

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

       12. unpow2 N/A

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

       13. lower-*.f64 84.0

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

   11. Applied rewrites 84.0%

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

   12. Taylor expanded in y around inf

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

   14. Applied rewrites 84.0%

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

Alternative 6: 71.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x \cdot \frac{\sinh y}{y}\\ \mathbf{if}\;t\_0 \leq -0.01:\\ \;\;\;\;\left(x \cdot \left(x \cdot -0.5\right)\right) \cdot \mathsf{fma}\left(y, y \cdot 0.16666666666666666, 1\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\left(y \cdot y\right) \cdot \left(y \cdot \left(0.0001984126984126984 \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (cos x) (/ (sinh y) y))))
   (if (<= t_0 -0.01)
     (* (* x (* x -0.5)) (fma y (* y 0.16666666666666666) 1.0))
     (if (<= t_0 2.0)
       (*
        1.0
        (fma
         y
         (* y (fma y (* y 0.008333333333333333) 0.16666666666666666))
         1.0))
       (* 1.0 (* (* y y) (* y (* 0.0001984126984126984 (* y (* y y))))))))))

C

double code(double x, double y) {
	double t_0 = cos(x) * (sinh(y) / y);
	double tmp;
	if (t_0 <= -0.01) {
		tmp = (x * (x * -0.5)) * fma(y, (y * 0.16666666666666666), 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 = 1.0 * ((y * y) * (y * (0.0001984126984126984 * (y * (y * y)))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) * Float64(sinh(y) / y))
	tmp = 0.0
	if (t_0 <= -0.01)
		tmp = Float64(Float64(x * Float64(x * -0.5)) * fma(y, Float64(y * 0.16666666666666666), 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(1.0 * Float64(Float64(y * y) * Float64(y * Float64(0.0001984126984126984 * Float64(y * Float64(y * y))))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.01], N[(N[(x * N[(x * -0.5), $MachinePrecision]), $MachinePrecision] * N[(y * N[(y * 0.16666666666666666), $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[(1.0 * N[(N[(y * y), $MachinePrecision] * N[(y * N[(0.0001984126984126984 * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (cos.f64 x) (/.f64 (sinh.f64 y) y)) < -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 60.3

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

   5. Applied rewrites 60.3%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. 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 53.6

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

   8. Applied rewrites 53.6%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 53.6%

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

   if -0.0100000000000000002 < (*.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 77.1%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 77.1

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

   8. Applied rewrites 77.1%

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

   9. Taylor expanded in y around 0

      \[\leadsto 1 \cdot \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)} \]
   10. 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. lower-fma.f64 N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 N/A

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

       5. +-commutative N/A

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

       6. lower-fma.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 N/A

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

       9. +-commutative N/A

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

       10. *-commutative N/A

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

       11. lower-fma.f64 N/A

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

       12. unpow2 N/A

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

       13. lower-*.f64 77.1

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

   11. Applied rewrites 77.1%

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

   12. Taylor expanded in y around 0

       \[\leadsto 1 \cdot \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} \]
   13. 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. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 N/A

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

       8. +-commutative N/A

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

       9. *-commutative N/A

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

       10. unpow2 N/A

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

       11. associate-*l* N/A

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

       12. *-commutative N/A

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

       13. lower-fma.f64 N/A

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

       14. *-commutative N/A

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

       15. lower-*.f64 77.1

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

   14. Applied rewrites 77.1%

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

   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 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

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 48.3

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

   8. Applied rewrites 48.3%

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

   9. Taylor expanded in y around 0

      \[\leadsto 1 \cdot \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)} \]
   10. 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. lower-fma.f64 N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 N/A

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

       5. +-commutative N/A

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

       6. lower-fma.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 N/A

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

       9. +-commutative N/A

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

       10. *-commutative N/A

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

       11. lower-fma.f64 N/A

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

       12. unpow2 N/A

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

       13. lower-*.f64 84.0

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

   11. Applied rewrites 84.0%

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

   12. Taylor expanded in y around inf

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

   14. Applied rewrites 84.0%

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

Alternative 7: 54.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x \cdot \frac{\sinh y}{y}\\ \mathbf{if}\;t\_0 \leq -0.01:\\ \;\;\;\;x \cdot \left(x \cdot -0.5\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(y \cdot \left(y \cdot 0.16666666666666666\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (cos x) (/ (sinh y) y))))
   (if (<= t_0 -0.01)
     (* x (* x -0.5))
     (if (<= t_0 2.0) 1.0 (* 1.0 (* y (* y 0.16666666666666666)))))))

C

double code(double x, double y) {
	double t_0 = cos(x) * (sinh(y) / y);
	double tmp;
	if (t_0 <= -0.01) {
		tmp = x * (x * -0.5);
	} else if (t_0 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = 1.0 * (y * (y * 0.16666666666666666));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos(x) * (sinh(y) / y)
    if (t_0 <= (-0.01d0)) then
        tmp = x * (x * (-0.5d0))
    else if (t_0 <= 2.0d0) then
        tmp = 1.0d0
    else
        tmp = 1.0d0 * (y * (y * 0.16666666666666666d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.cos(x) * (Math.sinh(y) / y);
	double tmp;
	if (t_0 <= -0.01) {
		tmp = x * (x * -0.5);
	} else if (t_0 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = 1.0 * (y * (y * 0.16666666666666666));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.cos(x) * (math.sinh(y) / y)
	tmp = 0
	if t_0 <= -0.01:
		tmp = x * (x * -0.5)
	elif t_0 <= 2.0:
		tmp = 1.0
	else:
		tmp = 1.0 * (y * (y * 0.16666666666666666))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(cos(x) * Float64(sinh(y) / y))
	tmp = 0.0
	if (t_0 <= -0.01)
		tmp = Float64(x * Float64(x * -0.5));
	elseif (t_0 <= 2.0)
		tmp = 1.0;
	else
		tmp = Float64(1.0 * Float64(y * Float64(y * 0.16666666666666666)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = cos(x) * (sinh(y) / y);
	tmp = 0.0;
	if (t_0 <= -0.01)
		tmp = x * (x * -0.5);
	elseif (t_0 <= 2.0)
		tmp = 1.0;
	else
		tmp = 1.0 * (y * (y * 0.16666666666666666));
	end
	tmp_2 = 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, -0.01], N[(x * N[(x * -0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2.0], 1.0, N[(1.0 * N[(y * N[(y * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (cos.f64 x) (/.f64 (sinh.f64 y) y)) < -0.0100000000000000002

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-cos.f64 42.5

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

   5. Applied rewrites 42.5%

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

   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 34.4%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 34.4%

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

   if -0.0100000000000000002 < (*.f64 (cos.f64 x) (/.f64 (sinh.f64 y) y)) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-cos.f64 99.7

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 76.8%

      \[\leadsto 1 \]

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 48.3

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

   8. Applied rewrites 48.3%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 48.3%

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

Alternative 8: 72.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)\\ \mathbf{if}\;\cos x \cdot \frac{\sinh y}{y} \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot -0.5, 1\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \frac{y \cdot t\_0}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (fma
          (* y y)
          (fma
           y
           (* y (fma (* y y) 0.0001984126984126984 0.008333333333333333))
           0.16666666666666666)
          1.0)))
   (if (<= (* (cos x) (/ (sinh y) y)) -0.01)
     (* (fma x (* x -0.5) 1.0) t_0)
     (* 1.0 (/ (* y t_0) y)))))

C

double code(double x, double y) {
	double t_0 = fma((y * y), fma(y, (y * fma((y * y), 0.0001984126984126984, 0.008333333333333333)), 0.16666666666666666), 1.0);
	double tmp;
	if ((cos(x) * (sinh(y) / y)) <= -0.01) {
		tmp = fma(x, (x * -0.5), 1.0) * t_0;
	} else {
		tmp = 1.0 * ((y * t_0) / y);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(Float64(y * y), fma(y, Float64(y * fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333)), 0.16666666666666666), 1.0)
	tmp = 0.0
	if (Float64(cos(x) * Float64(sinh(y) / y)) <= -0.01)
		tmp = Float64(fma(x, Float64(x * -0.5), 1.0) * t_0);
	else
		tmp = Float64(1.0 * Float64(Float64(y * t_0) / y));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[N[(N[Cos[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], -0.01], N[(N[(x * N[(x * -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision], N[(1.0 * N[(N[(y * t$95$0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cos.f64 x) (/.f64 (sinh.f64 y) y)) < -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 60.3

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

   5. Applied rewrites 60.3%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 60.3

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

   8. Applied rewrites 60.3%

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

   if -0.0100000000000000002 < (*.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 88.8%

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

   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. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 81.1

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

   8. Applied rewrites 81.1%

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

Alternative 9: 70.9% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cos x) (/ (sinh y) y)) -0.01)
   (* (* x (* x -0.5)) (fma y (* y 0.16666666666666666) 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) * (sinh(y) / y)) <= -0.01) {
		tmp = (x * (x * -0.5)) * fma(y, (y * 0.16666666666666666), 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 (Float64(cos(x) * Float64(sinh(y) / y)) <= -0.01)
		tmp = Float64(Float64(x * Float64(x * -0.5)) * fma(y, Float64(y * 0.16666666666666666), 1.0));
	else
		tmp = Float64(1.0 * fma(Float64(y * y), Float64(y * Float64(y * fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333))), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Cos[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], -0.01], N[(N[(x * N[(x * -0.5), $MachinePrecision]), $MachinePrecision] * N[(y * N[(y * 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 * N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $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)) < -0.0100000000000000002

   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 60.3

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

   5. Applied rewrites 60.3%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. 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 53.6

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

   8. Applied rewrites 53.6%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 53.6%

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

   if -0.0100000000000000002 < (*.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 88.8%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 62.4

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

   8. Applied rewrites 62.4%

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

   9. Taylor expanded in y around 0

      \[\leadsto 1 \cdot \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)} \]
   10. 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. lower-fma.f64 N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 N/A

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

       5. +-commutative N/A

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

       6. lower-fma.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 N/A

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

       9. +-commutative N/A

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

       10. *-commutative N/A

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

       11. lower-fma.f64 N/A

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

       12. unpow2 N/A

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

       13. lower-*.f64 80.7

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

   11. Applied rewrites 80.7%

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

   12. Taylor expanded in y around inf

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

   14. Applied rewrites 80.5%

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

Alternative 10: 68.4% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cos x) (/ (sinh y) y)) -0.01)
   (* (* x (* x -0.5)) (fma y (* y 0.16666666666666666) 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)) <= -0.01) {
		tmp = (x * (x * -0.5)) * fma(y, (y * 0.16666666666666666), 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)) <= -0.01)
		tmp = Float64(Float64(x * Float64(x * -0.5)) * fma(y, Float64(y * 0.16666666666666666), 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], -0.01], N[(N[(x * N[(x * -0.5), $MachinePrecision]), $MachinePrecision] * N[(y * N[(y * 0.16666666666666666), $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)) < -0.0100000000000000002

   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 60.3

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

   5. Applied rewrites 60.3%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. 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 53.6

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

   8. Applied rewrites 53.6%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 53.6%

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

   if -0.0100000000000000002 < (*.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 88.8%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 62.4

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

   8. Applied rewrites 62.4%

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

   9. Taylor expanded in y around 0

      \[\leadsto 1 \cdot \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)} \]
   10. 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. lower-fma.f64 N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 N/A

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

       5. +-commutative N/A

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

       6. lower-fma.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 N/A

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

       9. +-commutative N/A

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

       10. *-commutative N/A

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

       11. lower-fma.f64 N/A

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

       12. unpow2 N/A

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

       13. lower-*.f64 80.7

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

   11. Applied rewrites 80.7%

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

   12. Taylor expanded in y around 0

       \[\leadsto 1 \cdot \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} \]
   13. 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. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 N/A

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

       8. +-commutative N/A

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

       9. *-commutative N/A

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

       10. unpow2 N/A

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

       11. associate-*l* N/A

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

       12. *-commutative N/A

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

       13. lower-fma.f64 N/A

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

       14. *-commutative N/A

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

       15. lower-*.f64 74.1

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

   14. Applied rewrites 74.1%

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

Alternative 11: 60.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos x \leq -0.01:\\ \;\;\;\;\left(x \cdot \left(x \cdot -0.5\right)\right) \cdot \mathsf{fma}\left(y, y \cdot 0.16666666666666666, 1\right)\\ \mathbf{elif}\;\cos x \leq 0.995:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.041666666666666664, -0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (cos x) -0.01)
   (* (* x (* x -0.5)) (fma y (* y 0.16666666666666666) 1.0))
   (if (<= (cos x) 0.995)
     (fma (* x x) (fma x (* x 0.041666666666666664) -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) <= -0.01) {
		tmp = (x * (x * -0.5)) * fma(y, (y * 0.16666666666666666), 1.0);
	} else if (cos(x) <= 0.995) {
		tmp = fma((x * x), fma(x, (x * 0.041666666666666664), -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 (cos(x) <= -0.01)
		tmp = Float64(Float64(x * Float64(x * -0.5)) * fma(y, Float64(y * 0.16666666666666666), 1.0));
	elseif (cos(x) <= 0.995)
		tmp = fma(Float64(x * x), fma(x, Float64(x * 0.041666666666666664), -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[Cos[x], $MachinePrecision], -0.01], N[(N[(x * N[(x * -0.5), $MachinePrecision]), $MachinePrecision] * N[(y * N[(y * 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Cos[x], $MachinePrecision], 0.995], N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * 0.041666666666666664), $MachinePrecision] + -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 3 regimes

2. if (cos.f64 x) < -0.0100000000000000002

   1. Initial program 100.0%

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

   3. Taylor expanded in 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 60.3

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

   5. Applied rewrites 60.3%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. 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 53.6

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

   8. Applied rewrites 53.6%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 53.6%

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

   if -0.0100000000000000002 < (cos.f64 x) < 0.994999999999999996

   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

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

   8. Applied rewrites 39.0%

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

   if 0.994999999999999996 < (cos.f64 x)

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. 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 75.3

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

   5. Applied rewrites 75.3%

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

   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 73.8%

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

Alternative 12: 60.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos x \leq 0.005:\\ \;\;\;\;\left(x \cdot \left(x \cdot -0.5\right)\right) \cdot \left(y \cdot \left(y \cdot 0.16666666666666666\right)\right)\\ \mathbf{elif}\;\cos x \leq 0.995:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.041666666666666664, -0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (cos x) 0.005)
   (* (* x (* x -0.5)) (* y (* y 0.16666666666666666)))
   (if (<= (cos x) 0.995)
     (fma (* x x) (fma x (* x 0.041666666666666664) -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) <= 0.005) {
		tmp = (x * (x * -0.5)) * (y * (y * 0.16666666666666666));
	} else if (cos(x) <= 0.995) {
		tmp = fma((x * x), fma(x, (x * 0.041666666666666664), -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 (cos(x) <= 0.005)
		tmp = Float64(Float64(x * Float64(x * -0.5)) * Float64(y * Float64(y * 0.16666666666666666)));
	elseif (cos(x) <= 0.995)
		tmp = fma(Float64(x * x), fma(x, Float64(x * 0.041666666666666664), -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[Cos[x], $MachinePrecision], 0.005], N[(N[(x * N[(x * -0.5), $MachinePrecision]), $MachinePrecision] * N[(y * N[(y * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Cos[x], $MachinePrecision], 0.995], N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * 0.041666666666666664), $MachinePrecision] + -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 3 regimes

2. if (cos.f64 x) < 0.0050000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. 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 59.4

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

   5. Applied rewrites 59.4%

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

   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 52.9

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

   8. Applied rewrites 52.9%

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

   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 52.3%

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

   12. Taylor expanded in x around inf

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

   14. Applied rewrites 52.3%

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

   if 0.0050000000000000001 < (cos.f64 x) < 0.994999999999999996

   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 47.5

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

   5. Applied rewrites 47.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 39.8%

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

   if 0.994999999999999996 < (cos.f64 x)

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. 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 75.3

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

   5. Applied rewrites 75.3%

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

   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 73.8%

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

Alternative 13: 55.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos x \leq -0.01:\\ \;\;\;\;x \cdot \left(x \cdot -0.5\right)\\ \mathbf{elif}\;\cos x \leq 0.995:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.041666666666666664, -0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (cos x) -0.01)
   (* x (* x -0.5))
   (if (<= (cos x) 0.995)
     (fma (* x x) (fma x (* x 0.041666666666666664) -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) <= -0.01) {
		tmp = x * (x * -0.5);
	} else if (cos(x) <= 0.995) {
		tmp = fma((x * x), fma(x, (x * 0.041666666666666664), -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 (cos(x) <= -0.01)
		tmp = Float64(x * Float64(x * -0.5));
	elseif (cos(x) <= 0.995)
		tmp = fma(Float64(x * x), fma(x, Float64(x * 0.041666666666666664), -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[Cos[x], $MachinePrecision], -0.01], N[(x * N[(x * -0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Cos[x], $MachinePrecision], 0.995], N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * 0.041666666666666664), $MachinePrecision] + -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 3 regimes

2. if (cos.f64 x) < -0.0100000000000000002

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-cos.f64 42.5

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

   5. Applied rewrites 42.5%

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

   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 34.4%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 34.4%

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

   if -0.0100000000000000002 < (cos.f64 x) < 0.994999999999999996

   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

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

   8. Applied rewrites 39.0%

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

   if 0.994999999999999996 < (cos.f64 x)

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. 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 75.3

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

   5. Applied rewrites 75.3%

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

   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 73.8%

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

Alternative 14: 36.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos x \cdot \frac{\sinh y}{y} \leq -0.01:\\ \;\;\;\;x \cdot \left(x \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cos x) (/ (sinh y) y)) -0.01) (* x (* x -0.5)) 1.0))

C

double code(double x, double y) {
	double tmp;
	if ((cos(x) * (sinh(y) / y)) <= -0.01) {
		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)) <= (-0.01d0)) 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)) <= -0.01) {
		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)) <= -0.01:
		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)) <= -0.01)
		tmp = Float64(x * Float64(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)) <= -0.01)
		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], -0.01], N[(x * N[(x * -0.5), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cos.f64 x) (/.f64 (sinh.f64 y) y)) < -0.0100000000000000002

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-cos.f64 42.5

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

   5. Applied rewrites 42.5%

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

   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 34.4%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 34.4%

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

   if -0.0100000000000000002 < (*.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 50.4

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

   5. Applied rewrites 50.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 39.2%

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

Alternative 15: 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 16: 71.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)\\ \mathbf{if}\;\cos x \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot -0.5, 1\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (fma
          (* y y)
          (fma
           y
           (* y (fma (* y y) 0.0001984126984126984 0.008333333333333333))
           0.16666666666666666)
          1.0)))
   (if (<= (cos x) -0.01) (* (fma x (* x -0.5) 1.0) t_0) (* t_0 1.0))))

C

double code(double x, double y) {
	double t_0 = fma((y * y), fma(y, (y * fma((y * y), 0.0001984126984126984, 0.008333333333333333)), 0.16666666666666666), 1.0);
	double tmp;
	if (cos(x) <= -0.01) {
		tmp = fma(x, (x * -0.5), 1.0) * t_0;
	} else {
		tmp = t_0 * 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(Float64(y * y), fma(y, Float64(y * fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333)), 0.16666666666666666), 1.0)
	tmp = 0.0
	if (cos(x) <= -0.01)
		tmp = Float64(fma(x, Float64(x * -0.5), 1.0) * t_0);
	else
		tmp = Float64(t_0 * 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[N[Cos[x], $MachinePrecision], -0.01], N[(N[(x * N[(x * -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision], N[(t$95$0 * 1.0), $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 60.3

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

   5. Applied rewrites 60.3%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 60.3

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

   8. Applied rewrites 60.3%

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

   if -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 88.8%

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

   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. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 80.7

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

   8. Applied rewrites 80.7%

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

4. Final simplification 75.1%

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

Alternative 17: 71.7% accurate, 1.5× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (cos x) -0.01)
   (*
    (fma x (* x -0.5) 1.0)
    (fma (* y y) (* (* y y) 0.008333333333333333) 1.0))
   (*
    (fma
     (* y y)
     (fma
      y
      (* y (fma (* y y) 0.0001984126984126984 0.008333333333333333))
      0.16666666666666666)
     1.0)
    1.0)))

C

double code(double x, double y) {
	double tmp;
	if (cos(x) <= -0.01) {
		tmp = fma(x, (x * -0.5), 1.0) * fma((y * y), ((y * y) * 0.008333333333333333), 1.0);
	} else {
		tmp = fma((y * y), fma(y, (y * fma((y * y), 0.0001984126984126984, 0.008333333333333333)), 0.16666666666666666), 1.0) * 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) * fma(Float64(y * y), Float64(Float64(y * y) * 0.008333333333333333), 1.0));
	else
		tmp = Float64(fma(Float64(y * y), fma(y, Float64(y * fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333)), 0.16666666666666666), 1.0) * 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[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.008333333333333333), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * 1.0), $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 60.3

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

   5. Applied rewrites 60.3%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 57.7

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

   8. Applied rewrites 57.7%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 57.7%

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

   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 x around 0

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

   5. Applied rewrites 88.8%

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

   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. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 80.7

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

   8. Applied rewrites 80.7%

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

4. Final simplification 74.4%

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

Alternative 18: 71.6% accurate, 1.5× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (cos x) -0.01)
   (*
    (fma x (* x -0.5) 1.0)
    (fma (* y y) (* (* y y) 0.008333333333333333) 1.0))
   (*
    1.0
    (fma
     (* y y)
     (fma (* y y) (* (* y y) 0.0001984126984126984) 0.16666666666666666)
     1.0))))

C

double code(double x, double y) {
	double tmp;
	if (cos(x) <= -0.01) {
		tmp = fma(x, (x * -0.5), 1.0) * fma((y * y), ((y * y) * 0.008333333333333333), 1.0);
	} else {
		tmp = 1.0 * fma((y * y), fma((y * y), ((y * y) * 0.0001984126984126984), 0.16666666666666666), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (cos(x) <= -0.01)
		tmp = Float64(fma(x, Float64(x * -0.5), 1.0) * fma(Float64(y * y), Float64(Float64(y * y) * 0.008333333333333333), 1.0));
	else
		tmp = Float64(1.0 * fma(Float64(y * y), fma(Float64(y * y), Float64(Float64(y * y) * 0.0001984126984126984), 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[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.008333333333333333), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984), $MachinePrecision] + 0.16666666666666666), $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 60.3

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

   5. Applied rewrites 60.3%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 57.7

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

   8. Applied rewrites 57.7%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 57.7%

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

   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 x around 0

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

   5. Applied rewrites 88.8%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 62.4

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

   8. Applied rewrites 62.4%

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

   9. Taylor expanded in y around 0

      \[\leadsto 1 \cdot \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)} \]
   10. 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. lower-fma.f64 N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 N/A

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

       5. +-commutative N/A

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

       6. lower-fma.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 N/A

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

       9. +-commutative N/A

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

       10. *-commutative N/A

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

       11. lower-fma.f64 N/A

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

       12. unpow2 N/A

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

       13. lower-*.f64 80.7

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

   11. Applied rewrites 80.7%

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

   12. Taylor expanded in y around inf

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

   14. Applied rewrites 80.7%

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

Alternative 19: 71.0% accurate, 1.5× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (cos x) -0.01)
   (* (* x (* x -0.5)) (fma y (* y 0.16666666666666666) 1.0))
   (*
    1.0
    (fma
     (* y y)
     (fma (* y y) (* (* y y) 0.0001984126984126984) 0.16666666666666666)
     1.0))))

C

double code(double x, double y) {
	double tmp;
	if (cos(x) <= -0.01) {
		tmp = (x * (x * -0.5)) * fma(y, (y * 0.16666666666666666), 1.0);
	} else {
		tmp = 1.0 * fma((y * y), fma((y * y), ((y * y) * 0.0001984126984126984), 0.16666666666666666), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (cos(x) <= -0.01)
		tmp = Float64(Float64(x * Float64(x * -0.5)) * fma(y, Float64(y * 0.16666666666666666), 1.0));
	else
		tmp = Float64(1.0 * fma(Float64(y * y), fma(Float64(y * y), Float64(Float64(y * y) * 0.0001984126984126984), 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]), $MachinePrecision] * N[(y * N[(y * 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984), $MachinePrecision] + 0.16666666666666666), $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 60.3

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

   5. Applied rewrites 60.3%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. 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 53.6

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

   8. Applied rewrites 53.6%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 53.6%

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

   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 x around 0

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

   5. Applied rewrites 88.8%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 62.4

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

   8. Applied rewrites 62.4%

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

   9. Taylor expanded in y around 0

      \[\leadsto 1 \cdot \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)} \]
   10. 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. lower-fma.f64 N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 N/A

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

       5. +-commutative N/A

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

       6. lower-fma.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 N/A

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

       9. +-commutative N/A

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

       10. *-commutative N/A

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

       11. lower-fma.f64 N/A

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

       12. unpow2 N/A

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

       13. lower-*.f64 80.7

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

   11. Applied rewrites 80.7%

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

   12. Taylor expanded in y around inf

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

   14. Applied rewrites 80.7%

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

Alternative 20: 54.6% accurate, 1.8× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (cos x) -0.01)
   (* x (* x -0.5))
   (* 1.0 (fma 0.16666666666666666 (* y y) 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (cos(x) <= -0.01) {
		tmp = x * (x * -0.5);
	} else {
		tmp = 1.0 * fma(0.16666666666666666, (y * y), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (cos(x) <= -0.01)
		tmp = Float64(x * Float64(x * -0.5));
	else
		tmp = Float64(1.0 * fma(0.16666666666666666, Float64(y * y), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[Cos[x], $MachinePrecision], -0.01], N[(x * N[(x * -0.5), $MachinePrecision]), $MachinePrecision], N[(1.0 * N[(0.16666666666666666 * N[(y * y), $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

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

      1. lower-cos.f64 42.5

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

   5. Applied rewrites 42.5%

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

   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 34.4%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 34.4%

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

   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

      \[\leadsto \cos x \cdot \color{blue}{\left(1 + \frac{1}{6} \cdot {y}^{2}\right)} \]
   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 73.6

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

   5. Applied rewrites 73.6%

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

   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 62.4%

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

Alternative 21: 29.7% accurate, 217.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

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

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in y around 0

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

   1. lower-cos.f64 48.2

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

5. Applied rewrites 48.2%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 28.8%

   \[\leadsto 1 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024223 
(FPCore (x y)
  :name "Linear.Quaternion:$csin from linear-1.19.1.3"
  :precision binary64
  (* (cos x) (/ (sinh y) y)))