Linear.Quaternion:$csinh from linear-1.19.1.3

Percentage Accurate: 99.9% → 99.9%

Time: 13.5s

Alternatives: 19

Speedup: 1.0×

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

Specification

Math

\[\begin{array}{l} \\ \cosh x \cdot \frac{\sin y}{y} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* (cosh x) (/ (sin y) y)))

C

double code(double x, double y) {
	return cosh(x) * (sin(y) / y);
}

Fortran

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

Java

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

Python

def code(x, y):
	return math.cosh(x) * (math.sin(y) / y)

Julia

function code(x, y)
	return Float64(cosh(x) * Float64(sin(y) / y))
end

MATLAB

function tmp = code(x, y)
	tmp = cosh(x) * (sin(y) / y);
end

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \cosh x \cdot \frac{\sin y}{y} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* (cosh x) (/ (sin y) y)))

C

double code(double x, double y) {
	return cosh(x) * (sin(y) / y);
}

Fortran

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

Java

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

Python

def code(x, y):
	return math.cosh(x) * (math.sin(y) / y)

Julia

function code(x, y)
	return Float64(cosh(x) * Float64(sin(y) / y))
end

MATLAB

function tmp = code(x, y)
	tmp = cosh(x) * (sin(y) / y);
end

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\cosh x \cdot \sin y}{y} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ (* (cosh x) (sin y)) y))

C

double code(double x, double y) {
	return (cosh(x) * sin(y)) / y;
}

Fortran

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

Java

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

Python

def code(x, y):
	return (math.cosh(x) * math.sin(y)) / y

Julia

function code(x, y)
	return Float64(Float64(cosh(x) * sin(y)) / y)
end

MATLAB

function tmp = code(x, y)
	tmp = (cosh(x) * sin(y)) / y;
end

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{\cosh x \cdot \sin y}{y}} \]

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

      \[\leadsto \color{blue}{\frac{\cosh x \cdot \sin y}{y}} \]

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

      \[\leadsto \frac{\color{blue}{\cosh x \cdot \sin y}}{y} \]

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

      \[\leadsto \frac{\color{blue}{\cosh x} \cdot \sin y}{y} \]

   5. sin-lowering-sin.f64 99.9

      \[\leadsto \frac{\cosh x \cdot \color{blue}{\sin y}}{y} \]

4. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{\frac{\cosh x \cdot \sin y}{y}} \]
5. Add Preprocessing

Alternative 2: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right)\\ t_1 := \frac{\sin y}{y}\\ t_2 := \cosh x \cdot t\_1\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{x \cdot x}{y}, t\_0, \frac{0.5}{y}\right), \frac{1}{y}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)\\ \mathbf{elif}\;t\_2 \leq 1.00001:\\ \;\;\;\;t\_1 \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot t\_0, 0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\cosh x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma (* x x) 0.001388888888888889 0.041666666666666664))
        (t_1 (/ (sin y) y))
        (t_2 (* (cosh x) t_1)))
   (if (<= t_2 (- INFINITY))
     (*
      (fma (* x x) (fma (/ (* x x) y) t_0 (/ 0.5 y)) (/ 1.0 y))
      (fma
       (fma
        (* y y)
        (fma (* y y) -0.0001984126984126984 0.008333333333333333)
        -0.16666666666666666)
       (* y (* y y))
       y))
     (if (<= t_2 1.00001)
       (* t_1 (fma (* x x) (fma x (* x t_0) 0.5) 1.0))
       (cosh x)))))

C

double code(double x, double y) {
	double t_0 = fma((x * x), 0.001388888888888889, 0.041666666666666664);
	double t_1 = sin(y) / y;
	double t_2 = cosh(x) * t_1;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = fma((x * x), fma(((x * x) / y), t_0, (0.5 / y)), (1.0 / y)) * fma(fma((y * y), fma((y * y), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666), (y * (y * y)), y);
	} else if (t_2 <= 1.00001) {
		tmp = t_1 * fma((x * x), fma(x, (x * t_0), 0.5), 1.0);
	} else {
		tmp = cosh(x);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(Float64(x * x), 0.001388888888888889, 0.041666666666666664)
	t_1 = Float64(sin(y) / y)
	t_2 = Float64(cosh(x) * t_1)
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(fma(Float64(x * x), fma(Float64(Float64(x * x) / y), t_0, Float64(0.5 / y)), Float64(1.0 / y)) * fma(fma(Float64(y * y), fma(Float64(y * y), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666), Float64(y * Float64(y * y)), y));
	elseif (t_2 <= 1.00001)
		tmp = Float64(t_1 * fma(Float64(x * x), fma(x, Float64(x * t_0), 0.5), 1.0));
	else
		tmp = cosh(x);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cosh[x], $MachinePrecision] * t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(N[(N[(x * x), $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] / y), $MachinePrecision] * t$95$0 + N[(0.5 / y), $MachinePrecision]), $MachinePrecision] + N[(1.0 / y), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1.00001], N[(t$95$1 * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * t$95$0), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[Cosh[x], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < -inf.0

   1. Initial program 100.0%

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\sin y}{y} \cdot \cosh x} \]

      2. div-inv N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

      7. div-inv N/A

         \[\leadsto \color{blue}{\frac{\cosh x}{y}} \cdot \sin y \]

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

         \[\leadsto \color{blue}{\frac{\cosh x}{y}} \cdot \sin y \]

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

         \[\leadsto \frac{\color{blue}{\cosh x}}{y} \cdot \sin y \]

      10. sin-lowering-sin.f64 100.0

          \[\leadsto \frac{\cosh x}{y} \cdot \color{blue}{\sin y} \]

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{\cosh x}{y} \cdot \sin y} \]

   5. Taylor expanded in x around 0

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

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

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

   7. Simplified 81.7%

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

   8. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. unpow2 N/A

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

      7. unpow3 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{x \cdot x}{y}, \mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right), \frac{\frac{1}{2}}{y}\right), \frac{1}{y}\right) \cdot \left(\left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) - \frac{1}{6}\right) \cdot \color{blue}{{y}^{3}} + y\right) \]

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

         \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{x \cdot x}{y}, \mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right), \frac{\frac{1}{2}}{y}\right), \frac{1}{y}\right) \cdot \color{blue}{\mathsf{fma}\left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) - \frac{1}{6}, {y}^{3}, y\right)} \]

   10. Simplified 100.0%

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

   if -inf.0 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < 1.0000100000000001

   1. Initial program 99.7%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

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

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

      3. unpow2 N/A

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

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

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

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

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

      10. *-commutative N/A

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

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

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

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

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

      15. unpow2 N/A

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

      16. *-lowering-*.f64 98.5

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

   5. Simplified 98.5%

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

   if 1.0000100000000001 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y))

   1. Initial program 100.0%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \cosh x \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 100.0%

      \[\leadsto \cosh x \cdot \color{blue}{1} \]
   6. Step-by-step derivation

      1. *-rgt-identity N/A

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

      2. cosh-lowering-cosh.f64 100.0

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

   7. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\cosh x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{x \cdot x}{y}, \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), \frac{0.5}{y}\right), \frac{1}{y}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)\\ \mathbf{elif}\;\cosh x \cdot \frac{\sin y}{y} \leq 1.00001:\\ \;\;\;\;\frac{\sin y}{y} \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\cosh x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ t_1 := \cosh x \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{x \cdot x}{y}, \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), \frac{0.5}{y}\right), \frac{1}{y}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)\\ \mathbf{elif}\;t\_1 \leq 1.00001:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\cosh x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)) (t_1 (* (cosh x) t_0)))
   (if (<= t_1 (- INFINITY))
     (*
      (fma
       (* x x)
       (fma
        (/ (* x x) y)
        (fma (* x x) 0.001388888888888889 0.041666666666666664)
        (/ 0.5 y))
       (/ 1.0 y))
      (fma
       (fma
        (* y y)
        (fma (* y y) -0.0001984126984126984 0.008333333333333333)
        -0.16666666666666666)
       (* y (* y y))
       y))
     (if (<= t_1 1.00001)
       (* t_0 (fma (* x x) (fma (* x x) 0.041666666666666664 0.5) 1.0))
       (cosh x)))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double t_1 = cosh(x) * t_0;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma((x * x), fma(((x * x) / y), fma((x * x), 0.001388888888888889, 0.041666666666666664), (0.5 / y)), (1.0 / y)) * fma(fma((y * y), fma((y * y), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666), (y * (y * y)), y);
	} else if (t_1 <= 1.00001) {
		tmp = t_0 * fma((x * x), fma((x * x), 0.041666666666666664, 0.5), 1.0);
	} else {
		tmp = cosh(x);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(y) / y)
	t_1 = Float64(cosh(x) * t_0)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(fma(Float64(x * x), fma(Float64(Float64(x * x) / y), fma(Float64(x * x), 0.001388888888888889, 0.041666666666666664), Float64(0.5 / y)), Float64(1.0 / y)) * fma(fma(Float64(y * y), fma(Float64(y * y), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666), Float64(y * Float64(y * y)), y));
	elseif (t_1 <= 1.00001)
		tmp = Float64(t_0 * fma(Float64(x * x), fma(Float64(x * x), 0.041666666666666664, 0.5), 1.0));
	else
		tmp = cosh(x);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cosh[x], $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(x * x), $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] / y), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision] + N[(0.5 / y), $MachinePrecision]), $MachinePrecision] + N[(1.0 / y), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1.00001], N[(t$95$0 * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[Cosh[x], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < -inf.0

   1. Initial program 100.0%

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\sin y}{y} \cdot \cosh x} \]

      2. div-inv N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

      7. div-inv N/A

         \[\leadsto \color{blue}{\frac{\cosh x}{y}} \cdot \sin y \]

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

         \[\leadsto \color{blue}{\frac{\cosh x}{y}} \cdot \sin y \]

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

         \[\leadsto \frac{\color{blue}{\cosh x}}{y} \cdot \sin y \]

      10. sin-lowering-sin.f64 100.0

          \[\leadsto \frac{\cosh x}{y} \cdot \color{blue}{\sin y} \]

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{\cosh x}{y} \cdot \sin y} \]

   5. Taylor expanded in x around 0

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

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

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

   7. Simplified 81.7%

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

   8. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. unpow2 N/A

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

      7. unpow3 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{x \cdot x}{y}, \mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right), \frac{\frac{1}{2}}{y}\right), \frac{1}{y}\right) \cdot \left(\left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) - \frac{1}{6}\right) \cdot \color{blue}{{y}^{3}} + y\right) \]

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

         \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{x \cdot x}{y}, \mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right), \frac{\frac{1}{2}}{y}\right), \frac{1}{y}\right) \cdot \color{blue}{\mathsf{fma}\left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) - \frac{1}{6}, {y}^{3}, y\right)} \]

   10. Simplified 100.0%

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

   if -inf.0 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < 1.0000100000000001

   1. Initial program 99.7%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

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

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

      3. unpow2 N/A

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

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

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 98.3

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

   5. Simplified 98.3%

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

   if 1.0000100000000001 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y))

   1. Initial program 100.0%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \cosh x \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 100.0%

      \[\leadsto \cosh x \cdot \color{blue}{1} \]
   6. Step-by-step derivation

      1. *-rgt-identity N/A

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

      2. cosh-lowering-cosh.f64 100.0

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

   7. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\cosh x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{x \cdot x}{y}, \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), \frac{0.5}{y}\right), \frac{1}{y}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)\\ \mathbf{elif}\;\cosh x \cdot \frac{\sin y}{y} \leq 1.00001:\\ \;\;\;\;\frac{\sin y}{y} \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\cosh x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cosh x \cdot \frac{\sin y}{y}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{x \cdot x}{y}, \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), \frac{0.5}{y}\right), \frac{1}{y}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)\\ \mathbf{elif}\;t\_0 \leq 0.9999835474605927:\\ \;\;\;\;\frac{\sin y \cdot \mathsf{fma}\left(0.5, x \cdot x, 1\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\cosh x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (cosh x) (/ (sin y) y))))
   (if (<= t_0 (- INFINITY))
     (*
      (fma
       (* x x)
       (fma
        (/ (* x x) y)
        (fma (* x x) 0.001388888888888889 0.041666666666666664)
        (/ 0.5 y))
       (/ 1.0 y))
      (fma
       (fma
        (* y y)
        (fma (* y y) -0.0001984126984126984 0.008333333333333333)
        -0.16666666666666666)
       (* y (* y y))
       y))
     (if (<= t_0 0.9999835474605927)
       (/ (* (sin y) (fma 0.5 (* x x) 1.0)) y)
       (cosh x)))))

C

double code(double x, double y) {
	double t_0 = cosh(x) * (sin(y) / y);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma((x * x), fma(((x * x) / y), fma((x * x), 0.001388888888888889, 0.041666666666666664), (0.5 / y)), (1.0 / y)) * fma(fma((y * y), fma((y * y), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666), (y * (y * y)), y);
	} else if (t_0 <= 0.9999835474605927) {
		tmp = (sin(y) * fma(0.5, (x * x), 1.0)) / y;
	} else {
		tmp = cosh(x);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cosh(x) * Float64(sin(y) / y))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(fma(Float64(x * x), fma(Float64(Float64(x * x) / y), fma(Float64(x * x), 0.001388888888888889, 0.041666666666666664), Float64(0.5 / y)), Float64(1.0 / y)) * fma(fma(Float64(y * y), fma(Float64(y * y), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666), Float64(y * Float64(y * y)), y));
	elseif (t_0 <= 0.9999835474605927)
		tmp = Float64(Float64(sin(y) * fma(0.5, Float64(x * x), 1.0)) / y);
	else
		tmp = cosh(x);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cosh[x], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(x * x), $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] / y), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision] + N[(0.5 / y), $MachinePrecision]), $MachinePrecision] + N[(1.0 / y), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.9999835474605927], N[(N[(N[Sin[y], $MachinePrecision] * N[(0.5 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], N[Cosh[x], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < -inf.0

   1. Initial program 100.0%

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\sin y}{y} \cdot \cosh x} \]

      2. div-inv N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

      7. div-inv N/A

         \[\leadsto \color{blue}{\frac{\cosh x}{y}} \cdot \sin y \]

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

         \[\leadsto \color{blue}{\frac{\cosh x}{y}} \cdot \sin y \]

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

         \[\leadsto \frac{\color{blue}{\cosh x}}{y} \cdot \sin y \]

      10. sin-lowering-sin.f64 100.0

          \[\leadsto \frac{\cosh x}{y} \cdot \color{blue}{\sin y} \]

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{\cosh x}{y} \cdot \sin y} \]

   5. Taylor expanded in x around 0

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

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

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

   7. Simplified 81.7%

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

   8. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. unpow2 N/A

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

      7. unpow3 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{x \cdot x}{y}, \mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right), \frac{\frac{1}{2}}{y}\right), \frac{1}{y}\right) \cdot \left(\left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) - \frac{1}{6}\right) \cdot \color{blue}{{y}^{3}} + y\right) \]

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

         \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{x \cdot x}{y}, \mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right), \frac{\frac{1}{2}}{y}\right), \frac{1}{y}\right) \cdot \color{blue}{\mathsf{fma}\left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) - \frac{1}{6}, {y}^{3}, y\right)} \]

   10. Simplified 100.0%

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

   if -inf.0 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < 0.9999835474605927

   1. Initial program 99.5%

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

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\cosh x \cdot \sin y}{y}} \]

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

         \[\leadsto \color{blue}{\frac{\cosh x \cdot \sin y}{y}} \]

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

         \[\leadsto \frac{\color{blue}{\cosh x \cdot \sin y}}{y} \]

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

         \[\leadsto \frac{\color{blue}{\cosh x} \cdot \sin y}{y} \]

      5. sin-lowering-sin.f64 99.6

         \[\leadsto \frac{\cosh x \cdot \color{blue}{\sin y}}{y} \]

   4. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\frac{\cosh x \cdot \sin y}{y}} \]

   5. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. +-commutative N/A

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

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

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

      5. +-commutative N/A

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

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

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

      7. unpow2 N/A

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

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

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

      9. sin-lowering-sin.f64 96.6

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

   7. Simplified 96.6%

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

   if 0.9999835474605927 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y))

   1. Initial program 100.0%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \cosh x \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 100.0%

      \[\leadsto \cosh x \cdot \color{blue}{1} \]
   6. Step-by-step derivation

      1. *-rgt-identity N/A

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

      2. cosh-lowering-cosh.f64 100.0

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

   7. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\cosh x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{x \cdot x}{y}, \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), \frac{0.5}{y}\right), \frac{1}{y}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)\\ \mathbf{elif}\;\cosh x \cdot \frac{\sin y}{y} \leq 0.9999835474605927:\\ \;\;\;\;\frac{\sin y \cdot \mathsf{fma}\left(0.5, x \cdot x, 1\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\cosh x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ t_1 := \cosh x \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{x \cdot x}{y}, \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), \frac{0.5}{y}\right), \frac{1}{y}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)\\ \mathbf{elif}\;t\_1 \leq 0.9999835474605927:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(0.5, x \cdot x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\cosh x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)) (t_1 (* (cosh x) t_0)))
   (if (<= t_1 (- INFINITY))
     (*
      (fma
       (* x x)
       (fma
        (/ (* x x) y)
        (fma (* x x) 0.001388888888888889 0.041666666666666664)
        (/ 0.5 y))
       (/ 1.0 y))
      (fma
       (fma
        (* y y)
        (fma (* y y) -0.0001984126984126984 0.008333333333333333)
        -0.16666666666666666)
       (* y (* y y))
       y))
     (if (<= t_1 0.9999835474605927) (* t_0 (fma 0.5 (* x x) 1.0)) (cosh x)))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double t_1 = cosh(x) * t_0;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma((x * x), fma(((x * x) / y), fma((x * x), 0.001388888888888889, 0.041666666666666664), (0.5 / y)), (1.0 / y)) * fma(fma((y * y), fma((y * y), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666), (y * (y * y)), y);
	} else if (t_1 <= 0.9999835474605927) {
		tmp = t_0 * fma(0.5, (x * x), 1.0);
	} else {
		tmp = cosh(x);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(y) / y)
	t_1 = Float64(cosh(x) * t_0)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(fma(Float64(x * x), fma(Float64(Float64(x * x) / y), fma(Float64(x * x), 0.001388888888888889, 0.041666666666666664), Float64(0.5 / y)), Float64(1.0 / y)) * fma(fma(Float64(y * y), fma(Float64(y * y), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666), Float64(y * Float64(y * y)), y));
	elseif (t_1 <= 0.9999835474605927)
		tmp = Float64(t_0 * fma(0.5, Float64(x * x), 1.0));
	else
		tmp = cosh(x);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cosh[x], $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(x * x), $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] / y), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision] + N[(0.5 / y), $MachinePrecision]), $MachinePrecision] + N[(1.0 / y), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.9999835474605927], N[(t$95$0 * N[(0.5 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[Cosh[x], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < -inf.0

   1. Initial program 100.0%

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\sin y}{y} \cdot \cosh x} \]

      2. div-inv N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

      7. div-inv N/A

         \[\leadsto \color{blue}{\frac{\cosh x}{y}} \cdot \sin y \]

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

         \[\leadsto \color{blue}{\frac{\cosh x}{y}} \cdot \sin y \]

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

         \[\leadsto \frac{\color{blue}{\cosh x}}{y} \cdot \sin y \]

      10. sin-lowering-sin.f64 100.0

          \[\leadsto \frac{\cosh x}{y} \cdot \color{blue}{\sin y} \]

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{\cosh x}{y} \cdot \sin y} \]

   5. Taylor expanded in x around 0

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

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

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

   7. Simplified 81.7%

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

   8. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. unpow2 N/A

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

      7. unpow3 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{x \cdot x}{y}, \mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right), \frac{\frac{1}{2}}{y}\right), \frac{1}{y}\right) \cdot \left(\left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) - \frac{1}{6}\right) \cdot \color{blue}{{y}^{3}} + y\right) \]

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

         \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{x \cdot x}{y}, \mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right), \frac{\frac{1}{2}}{y}\right), \frac{1}{y}\right) \cdot \color{blue}{\mathsf{fma}\left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) - \frac{1}{6}, {y}^{3}, y\right)} \]

   10. Simplified 100.0%

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

   if -inf.0 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < 0.9999835474605927

   1. Initial program 99.5%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. distribute-lft1-in N/A

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

      8. +-commutative N/A

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

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

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

      10. +-commutative N/A

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

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

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

      12. unpow2 N/A

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

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

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

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

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

      15. sin-lowering-sin.f64 96.6

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

   5. Simplified 96.6%

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

   if 0.9999835474605927 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y))

   1. Initial program 100.0%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \cosh x \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 100.0%

      \[\leadsto \cosh x \cdot \color{blue}{1} \]
   6. Step-by-step derivation

      1. *-rgt-identity N/A

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

      2. cosh-lowering-cosh.f64 100.0

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

   7. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\cosh x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{x \cdot x}{y}, \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), \frac{0.5}{y}\right), \frac{1}{y}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)\\ \mathbf{elif}\;\cosh x \cdot \frac{\sin y}{y} \leq 0.9999835474605927:\\ \;\;\;\;\frac{\sin y}{y} \cdot \mathsf{fma}\left(0.5, x \cdot x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\cosh x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ t_1 := \cosh x \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{x \cdot x}{y}, \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), \frac{0.5}{y}\right), \frac{1}{y}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)\\ \mathbf{elif}\;t\_1 \leq 0.9999835474605927:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\cosh x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)) (t_1 (* (cosh x) t_0)))
   (if (<= t_1 (- INFINITY))
     (*
      (fma
       (* x x)
       (fma
        (/ (* x x) y)
        (fma (* x x) 0.001388888888888889 0.041666666666666664)
        (/ 0.5 y))
       (/ 1.0 y))
      (fma
       (fma
        (* y y)
        (fma (* y y) -0.0001984126984126984 0.008333333333333333)
        -0.16666666666666666)
       (* y (* y y))
       y))
     (if (<= t_1 0.9999835474605927) t_0 (cosh x)))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double t_1 = cosh(x) * t_0;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma((x * x), fma(((x * x) / y), fma((x * x), 0.001388888888888889, 0.041666666666666664), (0.5 / y)), (1.0 / y)) * fma(fma((y * y), fma((y * y), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666), (y * (y * y)), y);
	} else if (t_1 <= 0.9999835474605927) {
		tmp = t_0;
	} else {
		tmp = cosh(x);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(y) / y)
	t_1 = Float64(cosh(x) * t_0)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(fma(Float64(x * x), fma(Float64(Float64(x * x) / y), fma(Float64(x * x), 0.001388888888888889, 0.041666666666666664), Float64(0.5 / y)), Float64(1.0 / y)) * fma(fma(Float64(y * y), fma(Float64(y * y), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666), Float64(y * Float64(y * y)), y));
	elseif (t_1 <= 0.9999835474605927)
		tmp = t_0;
	else
		tmp = cosh(x);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cosh[x], $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(x * x), $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] / y), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision] + N[(0.5 / y), $MachinePrecision]), $MachinePrecision] + N[(1.0 / y), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.9999835474605927], t$95$0, N[Cosh[x], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < -inf.0

   1. Initial program 100.0%

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\sin y}{y} \cdot \cosh x} \]

      2. div-inv N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

      7. div-inv N/A

         \[\leadsto \color{blue}{\frac{\cosh x}{y}} \cdot \sin y \]

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

         \[\leadsto \color{blue}{\frac{\cosh x}{y}} \cdot \sin y \]

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

         \[\leadsto \frac{\color{blue}{\cosh x}}{y} \cdot \sin y \]

      10. sin-lowering-sin.f64 100.0

          \[\leadsto \frac{\cosh x}{y} \cdot \color{blue}{\sin y} \]

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{\cosh x}{y} \cdot \sin y} \]

   5. Taylor expanded in x around 0

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

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

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

   7. Simplified 81.7%

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

   8. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. unpow2 N/A

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

      7. unpow3 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{x \cdot x}{y}, \mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right), \frac{\frac{1}{2}}{y}\right), \frac{1}{y}\right) \cdot \left(\left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) - \frac{1}{6}\right) \cdot \color{blue}{{y}^{3}} + y\right) \]

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

         \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{x \cdot x}{y}, \mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right), \frac{\frac{1}{2}}{y}\right), \frac{1}{y}\right) \cdot \color{blue}{\mathsf{fma}\left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) - \frac{1}{6}, {y}^{3}, y\right)} \]

   10. Simplified 100.0%

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

   if -inf.0 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < 0.9999835474605927

   1. Initial program 99.5%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

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

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

      2. sin-lowering-sin.f64 96.1

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

   5. Simplified 96.1%

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

   if 0.9999835474605927 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y))

   1. Initial program 100.0%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \cosh x \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 100.0%

      \[\leadsto \cosh x \cdot \color{blue}{1} \]
   6. Step-by-step derivation

      1. *-rgt-identity N/A

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

      2. cosh-lowering-cosh.f64 100.0

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

   7. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\cosh x} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 75.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -1 \cdot 10^{-146}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{x \cdot x}{y}, \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), \frac{0.5}{y}\right), \frac{1}{y}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)\\ \mathbf{else}:\\ \;\;\;\;\cosh x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cosh x) (/ (sin y) y)) -1e-146)
   (*
    (fma
     (* x x)
     (fma
      (/ (* x x) y)
      (fma (* x x) 0.001388888888888889 0.041666666666666664)
      (/ 0.5 y))
     (/ 1.0 y))
    (fma
     (fma
      (* y y)
      (fma (* y y) -0.0001984126984126984 0.008333333333333333)
      -0.16666666666666666)
     (* y (* y y))
     y))
   (cosh x)))

C

double code(double x, double y) {
	double tmp;
	if ((cosh(x) * (sin(y) / y)) <= -1e-146) {
		tmp = fma((x * x), fma(((x * x) / y), fma((x * x), 0.001388888888888889, 0.041666666666666664), (0.5 / y)), (1.0 / y)) * fma(fma((y * y), fma((y * y), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666), (y * (y * y)), y);
	} else {
		tmp = cosh(x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(cosh(x) * Float64(sin(y) / y)) <= -1e-146)
		tmp = Float64(fma(Float64(x * x), fma(Float64(Float64(x * x) / y), fma(Float64(x * x), 0.001388888888888889, 0.041666666666666664), Float64(0.5 / y)), Float64(1.0 / y)) * fma(fma(Float64(y * y), fma(Float64(y * y), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666), Float64(y * Float64(y * y)), y));
	else
		tmp = cosh(x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Cosh[x], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], -1e-146], N[(N[(N[(x * x), $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] / y), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision] + N[(0.5 / y), $MachinePrecision]), $MachinePrecision] + N[(1.0 / y), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision], N[Cosh[x], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < -1.00000000000000003e-146

   1. Initial program 99.8%

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\sin y}{y} \cdot \cosh x} \]

      2. div-inv N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

      7. div-inv N/A

         \[\leadsto \color{blue}{\frac{\cosh x}{y}} \cdot \sin y \]

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

         \[\leadsto \color{blue}{\frac{\cosh x}{y}} \cdot \sin y \]

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

         \[\leadsto \frac{\color{blue}{\cosh x}}{y} \cdot \sin y \]

      10. sin-lowering-sin.f64 99.7

          \[\leadsto \frac{\cosh x}{y} \cdot \color{blue}{\sin y} \]

   4. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\frac{\cosh x}{y} \cdot \sin y} \]

   5. Taylor expanded in x around 0

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

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

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

   7. Simplified 86.5%

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

   8. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. unpow2 N/A

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

      7. unpow3 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{x \cdot x}{y}, \mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right), \frac{\frac{1}{2}}{y}\right), \frac{1}{y}\right) \cdot \left(\left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) - \frac{1}{6}\right) \cdot \color{blue}{{y}^{3}} + y\right) \]

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

         \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{x \cdot x}{y}, \mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right), \frac{\frac{1}{2}}{y}\right), \frac{1}{y}\right) \cdot \color{blue}{\mathsf{fma}\left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) - \frac{1}{6}, {y}^{3}, y\right)} \]

   10. Simplified 73.4%

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

   if -1.00000000000000003e-146 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y))

   1. Initial program 99.9%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \cosh x \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 75.1%

      \[\leadsto \cosh x \cdot \color{blue}{1} \]
   6. Step-by-step derivation

      1. *-rgt-identity N/A

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

      2. cosh-lowering-cosh.f64 75.1

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

   7. Applied egg-rr 75.1%

      \[\leadsto \color{blue}{\cosh x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 75.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -1 \cdot 10^{-146}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right) \cdot \left(\left(y \cdot y\right) \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\cosh x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cosh x) (/ (sin y) y)) -1e-146)
   (*
    (fma
     (* x x)
     (fma x (* x (fma (* x x) 0.001388888888888889 0.041666666666666664)) 0.5)
     1.0)
    (* (* y y) -0.16666666666666666))
   (cosh x)))

C

double code(double x, double y) {
	double tmp;
	if ((cosh(x) * (sin(y) / y)) <= -1e-146) {
		tmp = fma((x * x), fma(x, (x * fma((x * x), 0.001388888888888889, 0.041666666666666664)), 0.5), 1.0) * ((y * y) * -0.16666666666666666);
	} else {
		tmp = cosh(x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(cosh(x) * Float64(sin(y) / y)) <= -1e-146)
		tmp = Float64(fma(Float64(x * x), fma(x, Float64(x * fma(Float64(x * x), 0.001388888888888889, 0.041666666666666664)), 0.5), 1.0) * Float64(Float64(y * y) * -0.16666666666666666));
	else
		tmp = cosh(x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Cosh[x], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], -1e-146], N[(N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision], N[Cosh[x], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < -1.00000000000000003e-146

   1. Initial program 99.8%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-rgt-identity N/A

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

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

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

      7. unpow2 N/A

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

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

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

      9. *-lowering-*.f64 73.7

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

   5. Simplified 73.7%

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

   6. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 73.7

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

   8. Simplified 73.7%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

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

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

       3. unpow2 N/A

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

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

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

       5. +-commutative N/A

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

       6. unpow2 N/A

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

       7. associate-*l* N/A

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

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

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

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

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

       10. +-commutative N/A

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

       11. *-commutative N/A

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

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

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

       13. unpow2 N/A

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

       14. *-lowering-*.f64 70.6

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

   11. Simplified 70.6%

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

   if -1.00000000000000003e-146 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y))

   1. Initial program 99.9%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \cosh x \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 75.1%

      \[\leadsto \cosh x \cdot \color{blue}{1} \]
   6. Step-by-step derivation

      1. *-rgt-identity N/A

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

      2. cosh-lowering-cosh.f64 75.1

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

   7. Applied egg-rr 75.1%

      \[\leadsto \color{blue}{\cosh x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 71.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right)\\ \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -1 \cdot 10^{-146}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, t\_0, 1\right) \cdot \left(\left(y \cdot y\right) \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\mathsf{fma}\left(x, x \cdot t\_0, 1\right)}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (fma
          x
          (* x (fma (* x x) 0.001388888888888889 0.041666666666666664))
          0.5)))
   (if (<= (* (cosh x) (/ (sin y) y)) -1e-146)
     (* (fma (* x x) t_0 1.0) (* (* y y) -0.16666666666666666))
     (* y (/ (fma x (* x t_0) 1.0) y)))))

C

double code(double x, double y) {
	double t_0 = fma(x, (x * fma((x * x), 0.001388888888888889, 0.041666666666666664)), 0.5);
	double tmp;
	if ((cosh(x) * (sin(y) / y)) <= -1e-146) {
		tmp = fma((x * x), t_0, 1.0) * ((y * y) * -0.16666666666666666);
	} else {
		tmp = y * (fma(x, (x * t_0), 1.0) / y);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(x, Float64(x * fma(Float64(x * x), 0.001388888888888889, 0.041666666666666664)), 0.5)
	tmp = 0.0
	if (Float64(cosh(x) * Float64(sin(y) / y)) <= -1e-146)
		tmp = Float64(fma(Float64(x * x), t_0, 1.0) * Float64(Float64(y * y) * -0.16666666666666666));
	else
		tmp = Float64(y * Float64(fma(x, Float64(x * t_0), 1.0) / y));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]}, If[LessEqual[N[(N[Cosh[x], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], -1e-146], N[(N[(N[(x * x), $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[(x * N[(x * t$95$0), $MachinePrecision] + 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < -1.00000000000000003e-146

   1. Initial program 99.8%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-rgt-identity N/A

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

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

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

      7. unpow2 N/A

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

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

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

      9. *-lowering-*.f64 73.7

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

   5. Simplified 73.7%

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

   6. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 73.7

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

   8. Simplified 73.7%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

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

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

       3. unpow2 N/A

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

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

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

       5. +-commutative N/A

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

       6. unpow2 N/A

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

       7. associate-*l* N/A

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

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

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

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

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

       10. +-commutative N/A

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

       11. *-commutative N/A

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

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

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

       13. unpow2 N/A

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

       14. *-lowering-*.f64 70.6

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

   11. Simplified 70.6%

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

   if -1.00000000000000003e-146 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y))

   1. Initial program 99.9%

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\sin y}{y} \cdot \cosh x} \]

      2. div-inv N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

      7. div-inv N/A

         \[\leadsto \color{blue}{\frac{\cosh x}{y}} \cdot \sin y \]

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

         \[\leadsto \color{blue}{\frac{\cosh x}{y}} \cdot \sin y \]

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

         \[\leadsto \frac{\color{blue}{\cosh x}}{y} \cdot \sin y \]

      10. sin-lowering-sin.f64 99.8

          \[\leadsto \frac{\cosh x}{y} \cdot \color{blue}{\sin y} \]

   4. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{\cosh x}{y} \cdot \sin y} \]

   5. Taylor expanded in x around 0

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

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

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

   7. Simplified 94.2%

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

   8. Taylor expanded in y around 0

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

   10. Simplified 71.6%

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

   11. Taylor expanded in y around 0

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

   13. Simplified 71.7%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)}{y}} \cdot y \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -1 \cdot 10^{-146}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right) \cdot \left(\left(y \cdot y\right) \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 69.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right)\\ \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -1 \cdot 10^{-146}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, t\_0, 1\right) \cdot \left(\left(y \cdot y\right) \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot t\_0, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (fma
          x
          (* x (fma (* x x) 0.001388888888888889 0.041666666666666664))
          0.5)))
   (if (<= (* (cosh x) (/ (sin y) y)) -1e-146)
     (* (fma (* x x) t_0 1.0) (* (* y y) -0.16666666666666666))
     (fma x (* x t_0) 1.0))))

C

double code(double x, double y) {
	double t_0 = fma(x, (x * fma((x * x), 0.001388888888888889, 0.041666666666666664)), 0.5);
	double tmp;
	if ((cosh(x) * (sin(y) / y)) <= -1e-146) {
		tmp = fma((x * x), t_0, 1.0) * ((y * y) * -0.16666666666666666);
	} else {
		tmp = fma(x, (x * t_0), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(x, Float64(x * fma(Float64(x * x), 0.001388888888888889, 0.041666666666666664)), 0.5)
	tmp = 0.0
	if (Float64(cosh(x) * Float64(sin(y) / y)) <= -1e-146)
		tmp = Float64(fma(Float64(x * x), t_0, 1.0) * Float64(Float64(y * y) * -0.16666666666666666));
	else
		tmp = fma(x, Float64(x * t_0), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]}, If[LessEqual[N[(N[Cosh[x], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], -1e-146], N[(N[(N[(x * x), $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision], N[(x * N[(x * t$95$0), $MachinePrecision] + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < -1.00000000000000003e-146

   1. Initial program 99.8%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-rgt-identity N/A

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

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

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

      7. unpow2 N/A

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

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

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

      9. *-lowering-*.f64 73.7

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

   5. Simplified 73.7%

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

   6. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 73.7

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

   8. Simplified 73.7%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

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

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

       3. unpow2 N/A

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

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

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

       5. +-commutative N/A

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

       6. unpow2 N/A

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

       7. associate-*l* N/A

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

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

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

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

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

       10. +-commutative N/A

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

       11. *-commutative N/A

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

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

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

       13. unpow2 N/A

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

       14. *-lowering-*.f64 70.6

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

   11. Simplified 70.6%

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

   if -1.00000000000000003e-146 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y))

   1. Initial program 99.9%

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\sin y}{y} \cdot \cosh x} \]

      2. div-inv N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

      7. div-inv N/A

         \[\leadsto \color{blue}{\frac{\cosh x}{y}} \cdot \sin y \]

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

         \[\leadsto \color{blue}{\frac{\cosh x}{y}} \cdot \sin y \]

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

         \[\leadsto \frac{\color{blue}{\cosh x}}{y} \cdot \sin y \]

      10. sin-lowering-sin.f64 99.8

          \[\leadsto \frac{\cosh x}{y} \cdot \color{blue}{\sin y} \]

   4. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{\cosh x}{y} \cdot \sin y} \]

   5. Taylor expanded in x around 0

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

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

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

   7. Simplified 94.2%

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

   8. Taylor expanded in y around 0

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

   10. Simplified 71.6%

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

   11. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

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

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

   13. Simplified 69.4%

       \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 69.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -1 \cdot 10^{-146}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right), 1\right) \cdot \left(\left(y \cdot y\right) \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cosh x) (/ (sin y) y)) -1e-146)
   (*
    (fma (* x x) (fma (* x x) 0.041666666666666664 0.5) 1.0)
    (* (* y y) -0.16666666666666666))
   (fma
    x
    (*
     x
     (fma x (* x (fma (* x x) 0.001388888888888889 0.041666666666666664)) 0.5))
    1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((cosh(x) * (sin(y) / y)) <= -1e-146) {
		tmp = fma((x * x), fma((x * x), 0.041666666666666664, 0.5), 1.0) * ((y * y) * -0.16666666666666666);
	} else {
		tmp = fma(x, (x * fma(x, (x * fma((x * x), 0.001388888888888889, 0.041666666666666664)), 0.5)), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(cosh(x) * Float64(sin(y) / y)) <= -1e-146)
		tmp = Float64(fma(Float64(x * x), fma(Float64(x * x), 0.041666666666666664, 0.5), 1.0) * Float64(Float64(y * y) * -0.16666666666666666));
	else
		tmp = fma(x, Float64(x * fma(x, Float64(x * fma(Float64(x * x), 0.001388888888888889, 0.041666666666666664)), 0.5)), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Cosh[x], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], -1e-146], N[(N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision], N[(x * N[(x * N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < -1.00000000000000003e-146

   1. Initial program 99.8%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-rgt-identity N/A

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

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

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

      7. unpow2 N/A

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

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

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

      9. *-lowering-*.f64 73.7

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

   5. Simplified 73.7%

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

   6. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 73.7

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

   8. Simplified 73.7%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

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

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

       3. unpow2 N/A

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

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

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

       5. +-commutative N/A

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

       6. *-commutative N/A

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

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

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

       8. unpow2 N/A

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

       9. *-lowering-*.f64 68.9

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

   11. Simplified 68.9%

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

   if -1.00000000000000003e-146 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y))

   1. Initial program 99.9%

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\sin y}{y} \cdot \cosh x} \]

      2. div-inv N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

      7. div-inv N/A

         \[\leadsto \color{blue}{\frac{\cosh x}{y}} \cdot \sin y \]

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

         \[\leadsto \color{blue}{\frac{\cosh x}{y}} \cdot \sin y \]

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

         \[\leadsto \frac{\color{blue}{\cosh x}}{y} \cdot \sin y \]

      10. sin-lowering-sin.f64 99.8

          \[\leadsto \frac{\cosh x}{y} \cdot \color{blue}{\sin y} \]

   4. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{\cosh x}{y} \cdot \sin y} \]

   5. Taylor expanded in x around 0

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

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

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

   7. Simplified 94.2%

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

   8. Taylor expanded in y around 0

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

   10. Simplified 71.6%

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

   11. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

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

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

   13. Simplified 69.4%

       \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 68.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -1 \cdot 10^{-146}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(0.5 \cdot \mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cosh x) (/ (sin y) y)) -1e-146)
   (* (* x x) (* 0.5 (fma (* y y) -0.16666666666666666 1.0)))
   (fma
    x
    (*
     x
     (fma x (* x (fma (* x x) 0.001388888888888889 0.041666666666666664)) 0.5))
    1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((cosh(x) * (sin(y) / y)) <= -1e-146) {
		tmp = (x * x) * (0.5 * fma((y * y), -0.16666666666666666, 1.0));
	} else {
		tmp = fma(x, (x * fma(x, (x * fma((x * x), 0.001388888888888889, 0.041666666666666664)), 0.5)), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(cosh(x) * Float64(sin(y) / y)) <= -1e-146)
		tmp = Float64(Float64(x * x) * Float64(0.5 * fma(Float64(y * y), -0.16666666666666666, 1.0)));
	else
		tmp = fma(x, Float64(x * fma(x, Float64(x * fma(Float64(x * x), 0.001388888888888889, 0.041666666666666664)), 0.5)), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Cosh[x], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], -1e-146], N[(N[(x * x), $MachinePrecision] * N[(0.5 * N[(N[(y * y), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(x * N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < -1.00000000000000003e-146

   1. Initial program 99.8%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-rgt-identity N/A

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

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

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

      7. unpow2 N/A

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

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

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

      9. *-lowering-*.f64 73.7

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

   5. Simplified 73.7%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 60.8

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

   8. Simplified 60.8%

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

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{{x}^{2} \cdot \left(y + \frac{-1}{6} \cdot {y}^{3}\right)}{y}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

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

       2. associate-/l* N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

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

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

       6. unpow2 N/A

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

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

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

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

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

       9. unpow3 N/A

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

       10. unpow2 N/A

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

       11. associate-*r* N/A

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

       12. *-lft-identity N/A

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

       13. distribute-rgt-in N/A

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

       14. *-rgt-identity N/A

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

       15. times-frac N/A

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

       16. *-inverses N/A

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

       17. /-rgt-identity N/A

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

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

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

       19. +-commutative N/A

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

       20. *-commutative N/A

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

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

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

   11. Simplified 61.0%

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

   if -1.00000000000000003e-146 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y))

   1. Initial program 99.9%

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\sin y}{y} \cdot \cosh x} \]

      2. div-inv N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

      7. div-inv N/A

         \[\leadsto \color{blue}{\frac{\cosh x}{y}} \cdot \sin y \]

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

         \[\leadsto \color{blue}{\frac{\cosh x}{y}} \cdot \sin y \]

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

         \[\leadsto \frac{\color{blue}{\cosh x}}{y} \cdot \sin y \]

      10. sin-lowering-sin.f64 99.8

          \[\leadsto \frac{\cosh x}{y} \cdot \color{blue}{\sin y} \]

   4. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{\cosh x}{y} \cdot \sin y} \]

   5. Taylor expanded in x around 0

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

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

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

   7. Simplified 94.2%

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

   8. Taylor expanded in y around 0

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

   10. Simplified 71.6%

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

   11. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

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

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

   13. Simplified 69.4%

       \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -1 \cdot 10^{-146}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(0.5 \cdot \mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 65.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -1 \cdot 10^{-146}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(0.5 \cdot \mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cosh x) (/ (sin y) y)) -1e-146)
   (* (* x x) (* 0.5 (fma (* y y) -0.16666666666666666 1.0)))
   (fma x (* x (fma x (* x 0.041666666666666664) 0.5)) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((cosh(x) * (sin(y) / y)) <= -1e-146) {
		tmp = (x * x) * (0.5 * fma((y * y), -0.16666666666666666, 1.0));
	} else {
		tmp = fma(x, (x * fma(x, (x * 0.041666666666666664), 0.5)), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(cosh(x) * Float64(sin(y) / y)) <= -1e-146)
		tmp = Float64(Float64(x * x) * Float64(0.5 * fma(Float64(y * y), -0.16666666666666666, 1.0)));
	else
		tmp = fma(x, Float64(x * fma(x, Float64(x * 0.041666666666666664), 0.5)), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Cosh[x], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], -1e-146], N[(N[(x * x), $MachinePrecision] * N[(0.5 * N[(N[(y * y), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(x * N[(x * N[(x * 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < -1.00000000000000003e-146

   1. Initial program 99.8%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-rgt-identity N/A

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

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

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

      7. unpow2 N/A

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

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

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

      9. *-lowering-*.f64 73.7

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

   5. Simplified 73.7%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 60.8

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

   8. Simplified 60.8%

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

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{{x}^{2} \cdot \left(y + \frac{-1}{6} \cdot {y}^{3}\right)}{y}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

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

       2. associate-/l* N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

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

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

       6. unpow2 N/A

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

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

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

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

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

       9. unpow3 N/A

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

       10. unpow2 N/A

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

       11. associate-*r* N/A

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

       12. *-lft-identity N/A

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

       13. distribute-rgt-in N/A

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

       14. *-rgt-identity N/A

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

       15. times-frac N/A

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

       16. *-inverses N/A

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

       17. /-rgt-identity N/A

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

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

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

       19. +-commutative N/A

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

       20. *-commutative N/A

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

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

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

   11. Simplified 61.0%

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

   if -1.00000000000000003e-146 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y))

   1. Initial program 99.9%

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\sin y}{y} \cdot \cosh x} \]

      2. div-inv N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

      7. div-inv N/A

         \[\leadsto \color{blue}{\frac{\cosh x}{y}} \cdot \sin y \]

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

         \[\leadsto \color{blue}{\frac{\cosh x}{y}} \cdot \sin y \]

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

         \[\leadsto \frac{\color{blue}{\cosh x}}{y} \cdot \sin y \]

      10. sin-lowering-sin.f64 99.8

          \[\leadsto \frac{\cosh x}{y} \cdot \color{blue}{\sin y} \]

   4. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{\cosh x}{y} \cdot \sin y} \]

   5. Taylor expanded in x around 0

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

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

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

   7. Simplified 94.2%

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

   8. Taylor expanded in y around 0

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

   10. Simplified 71.6%

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

   11. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

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

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

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

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

       6. +-commutative N/A

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

       7. *-commutative N/A

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

       8. unpow2 N/A

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

       9. associate-*l* N/A

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

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

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

       11. *-lowering-*.f64 66.5

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

   13. Simplified 66.5%

       \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right), 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -1 \cdot 10^{-146}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(0.5 \cdot \mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 65.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -1 \cdot 10^{-146}:\\ \;\;\;\;\left(y \cdot y\right) \cdot \mathsf{fma}\left(x, x \cdot -0.08333333333333333, -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cosh x) (/ (sin y) y)) -1e-146)
   (* (* y y) (fma x (* x -0.08333333333333333) -0.16666666666666666))
   (fma x (* x (fma x (* x 0.041666666666666664) 0.5)) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((cosh(x) * (sin(y) / y)) <= -1e-146) {
		tmp = (y * y) * fma(x, (x * -0.08333333333333333), -0.16666666666666666);
	} else {
		tmp = fma(x, (x * fma(x, (x * 0.041666666666666664), 0.5)), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(cosh(x) * Float64(sin(y) / y)) <= -1e-146)
		tmp = Float64(Float64(y * y) * fma(x, Float64(x * -0.08333333333333333), -0.16666666666666666));
	else
		tmp = fma(x, Float64(x * fma(x, Float64(x * 0.041666666666666664), 0.5)), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Cosh[x], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], -1e-146], N[(N[(y * y), $MachinePrecision] * N[(x * N[(x * -0.08333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision], N[(x * N[(x * N[(x * N[(x * 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < -1.00000000000000003e-146

   1. Initial program 99.8%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-rgt-identity N/A

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

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

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

      7. unpow2 N/A

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

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

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

      9. *-lowering-*.f64 73.7

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

   5. Simplified 73.7%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 60.8

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

   8. Simplified 60.8%

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

   9. Taylor expanded in y around inf

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

       1. associate-+r+ N/A

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

       2. distribute-lft-in N/A

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

       3. rgt-mult-inverse N/A

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

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

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

   11. Simplified 60.8%

       \[\leadsto \color{blue}{\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(x \cdot x, -0.08333333333333333 + \frac{0.5}{y \cdot y}, -0.16666666666666666\right), 1\right)} \]

   12. Taylor expanded in y around inf

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

       1. sub-neg N/A

          \[\leadsto {y}^{2} \cdot \color{blue}{\left(\frac{-1}{12} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)} \]

       2. metadata-eval N/A

          \[\leadsto {y}^{2} \cdot \left(\frac{-1}{12} \cdot {x}^{2} + \color{blue}{\frac{-1}{6}}\right) \]

       3. +-commutative N/A

          \[\leadsto {y}^{2} \cdot \color{blue}{\left(\frac{-1}{6} + \frac{-1}{12} \cdot {x}^{2}\right)} \]

       4. metadata-eval N/A

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

       5. associate-*r* N/A

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

       6. metadata-eval N/A

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

       7. distribute-lft-in N/A

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

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

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

       9. unpow2 N/A

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

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

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

       11. +-commutative N/A

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

       12. distribute-lft-in N/A

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

       13. associate-*r* N/A

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

       14. metadata-eval N/A

           \[\leadsto \left(y \cdot y\right) \cdot \left(\color{blue}{\frac{-1}{12}} \cdot {x}^{2} + \frac{-1}{6} \cdot 1\right) \]

       15. *-commutative N/A

           \[\leadsto \left(y \cdot y\right) \cdot \left(\color{blue}{{x}^{2} \cdot \frac{-1}{12}} + \frac{-1}{6} \cdot 1\right) \]

       16. unpow2 N/A

           \[\leadsto \left(y \cdot y\right) \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{-1}{12} + \frac{-1}{6} \cdot 1\right) \]

       17. associate-*l* N/A

           \[\leadsto \left(y \cdot y\right) \cdot \left(\color{blue}{x \cdot \left(x \cdot \frac{-1}{12}\right)} + \frac{-1}{6} \cdot 1\right) \]

       18. metadata-eval N/A

           \[\leadsto \left(y \cdot y\right) \cdot \left(x \cdot \left(x \cdot \frac{-1}{12}\right) + \color{blue}{\frac{-1}{6}}\right) \]

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

           \[\leadsto \left(y \cdot y\right) \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{-1}{12}, \frac{-1}{6}\right)} \]

       20. *-lowering-*.f64 60.8

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

   14. Simplified 60.8%

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

   if -1.00000000000000003e-146 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y))

   1. Initial program 99.9%

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\sin y}{y} \cdot \cosh x} \]

      2. div-inv N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

      7. div-inv N/A

         \[\leadsto \color{blue}{\frac{\cosh x}{y}} \cdot \sin y \]

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

         \[\leadsto \color{blue}{\frac{\cosh x}{y}} \cdot \sin y \]

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

         \[\leadsto \frac{\color{blue}{\cosh x}}{y} \cdot \sin y \]

      10. sin-lowering-sin.f64 99.8

          \[\leadsto \frac{\cosh x}{y} \cdot \color{blue}{\sin y} \]

   4. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{\cosh x}{y} \cdot \sin y} \]

   5. Taylor expanded in x around 0

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

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

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

   7. Simplified 94.2%

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

   8. Taylor expanded in y around 0

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

   10. Simplified 71.6%

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

   11. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

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

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

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

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

       6. +-commutative N/A

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

       7. *-commutative N/A

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

       8. unpow2 N/A

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

       9. associate-*l* N/A

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

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

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

       11. *-lowering-*.f64 66.5

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

   13. Simplified 66.5%

       \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right), 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 56.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -1 \cdot 10^{-146}:\\ \;\;\;\;\left(y \cdot y\right) \cdot \mathsf{fma}\left(x, x \cdot -0.08333333333333333, -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, x \cdot x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cosh x) (/ (sin y) y)) -1e-146)
   (* (* y y) (fma x (* x -0.08333333333333333) -0.16666666666666666))
   (fma 0.5 (* x x) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((cosh(x) * (sin(y) / y)) <= -1e-146) {
		tmp = (y * y) * fma(x, (x * -0.08333333333333333), -0.16666666666666666);
	} else {
		tmp = fma(0.5, (x * x), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(cosh(x) * Float64(sin(y) / y)) <= -1e-146)
		tmp = Float64(Float64(y * y) * fma(x, Float64(x * -0.08333333333333333), -0.16666666666666666));
	else
		tmp = fma(0.5, Float64(x * x), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Cosh[x], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], -1e-146], N[(N[(y * y), $MachinePrecision] * N[(x * N[(x * -0.08333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < -1.00000000000000003e-146

   1. Initial program 99.8%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-rgt-identity N/A

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

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

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

      7. unpow2 N/A

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

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

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

      9. *-lowering-*.f64 73.7

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

   5. Simplified 73.7%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 60.8

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

   8. Simplified 60.8%

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

   9. Taylor expanded in y around inf

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

       1. associate-+r+ N/A

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

       2. distribute-lft-in N/A

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

       3. rgt-mult-inverse N/A

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

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

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

   11. Simplified 60.8%

       \[\leadsto \color{blue}{\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(x \cdot x, -0.08333333333333333 + \frac{0.5}{y \cdot y}, -0.16666666666666666\right), 1\right)} \]

   12. Taylor expanded in y around inf

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

       1. sub-neg N/A

          \[\leadsto {y}^{2} \cdot \color{blue}{\left(\frac{-1}{12} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)} \]

       2. metadata-eval N/A

          \[\leadsto {y}^{2} \cdot \left(\frac{-1}{12} \cdot {x}^{2} + \color{blue}{\frac{-1}{6}}\right) \]

       3. +-commutative N/A

          \[\leadsto {y}^{2} \cdot \color{blue}{\left(\frac{-1}{6} + \frac{-1}{12} \cdot {x}^{2}\right)} \]

       4. metadata-eval N/A

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

       5. associate-*r* N/A

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

       6. metadata-eval N/A

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

       7. distribute-lft-in N/A

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

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

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

       9. unpow2 N/A

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

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

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

       11. +-commutative N/A

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

       12. distribute-lft-in N/A

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

       13. associate-*r* N/A

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

       14. metadata-eval N/A

           \[\leadsto \left(y \cdot y\right) \cdot \left(\color{blue}{\frac{-1}{12}} \cdot {x}^{2} + \frac{-1}{6} \cdot 1\right) \]

       15. *-commutative N/A

           \[\leadsto \left(y \cdot y\right) \cdot \left(\color{blue}{{x}^{2} \cdot \frac{-1}{12}} + \frac{-1}{6} \cdot 1\right) \]

       16. unpow2 N/A

           \[\leadsto \left(y \cdot y\right) \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{-1}{12} + \frac{-1}{6} \cdot 1\right) \]

       17. associate-*l* N/A

           \[\leadsto \left(y \cdot y\right) \cdot \left(\color{blue}{x \cdot \left(x \cdot \frac{-1}{12}\right)} + \frac{-1}{6} \cdot 1\right) \]

       18. metadata-eval N/A

           \[\leadsto \left(y \cdot y\right) \cdot \left(x \cdot \left(x \cdot \frac{-1}{12}\right) + \color{blue}{\frac{-1}{6}}\right) \]

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

           \[\leadsto \left(y \cdot y\right) \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{-1}{12}, \frac{-1}{6}\right)} \]

       20. *-lowering-*.f64 60.8

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

   14. Simplified 60.8%

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

   if -1.00000000000000003e-146 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y))

   1. Initial program 99.9%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \cosh x \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 75.1%

      \[\leadsto \cosh x \cdot \color{blue}{1} \]

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 54.2

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

   8. Simplified 54.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, x \cdot x, 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 52.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -1 \cdot 10^{-146}:\\ \;\;\;\;y \cdot \left(y \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, x \cdot x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cosh x) (/ (sin y) y)) -1e-146)
   (* y (* y -0.16666666666666666))
   (fma 0.5 (* x x) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((cosh(x) * (sin(y) / y)) <= -1e-146) {
		tmp = y * (y * -0.16666666666666666);
	} else {
		tmp = fma(0.5, (x * x), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(cosh(x) * Float64(sin(y) / y)) <= -1e-146)
		tmp = Float64(y * Float64(y * -0.16666666666666666));
	else
		tmp = fma(0.5, Float64(x * x), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Cosh[x], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], -1e-146], N[(y * N[(y * -0.16666666666666666), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < -1.00000000000000003e-146

   1. Initial program 99.8%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-rgt-identity N/A

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

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

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

      7. unpow2 N/A

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

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

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

      9. *-lowering-*.f64 73.7

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

   5. Simplified 73.7%

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

   6. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 73.7

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

   8. Simplified 73.7%

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

   9. Taylor expanded in x around 0

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

       1. *-commutative N/A

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

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

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

       3. unpow2 N/A

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

       4. *-lowering-*.f64 34.3

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

   11. Simplified 34.3%

       \[\leadsto \color{blue}{\left(y \cdot y\right) \cdot -0.16666666666666666} \]
   12. Step-by-step derivation

       1. associate-*l* N/A

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

       2. *-commutative N/A

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

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

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

       4. *-lowering-*.f64 34.3

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

   13. Applied egg-rr 34.3%

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

   if -1.00000000000000003e-146 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y))

   1. Initial program 99.9%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \cosh x \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 75.1%

      \[\leadsto \cosh x \cdot \color{blue}{1} \]

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 54.2

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

   8. Simplified 54.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, x \cdot x, 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -1 \cdot 10^{-146}:\\ \;\;\;\;y \cdot \left(y \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, x \cdot x, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 33.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -1 \cdot 10^{-146}:\\ \;\;\;\;y \cdot \left(y \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cosh x) (/ (sin y) y)) -1e-146)
   (* y (* y -0.16666666666666666))
   1.0))

C

double code(double x, double y) {
	double tmp;
	if ((cosh(x) * (sin(y) / y)) <= -1e-146) {
		tmp = y * (y * -0.16666666666666666);
	} 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 ((cosh(x) * (sin(y) / y)) <= (-1d-146)) then
        tmp = y * (y * (-0.16666666666666666d0))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((Math.cosh(x) * (Math.sin(y) / y)) <= -1e-146) {
		tmp = y * (y * -0.16666666666666666);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (math.cosh(x) * (math.sin(y) / y)) <= -1e-146:
		tmp = y * (y * -0.16666666666666666)
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(cosh(x) * Float64(sin(y) / y)) <= -1e-146)
		tmp = Float64(y * Float64(y * -0.16666666666666666));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((cosh(x) * (sin(y) / y)) <= -1e-146)
		tmp = y * (y * -0.16666666666666666);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Cosh[x], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], -1e-146], N[(y * N[(y * -0.16666666666666666), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < -1.00000000000000003e-146

   1. Initial program 99.8%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-rgt-identity N/A

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

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

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

      7. unpow2 N/A

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

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

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

      9. *-lowering-*.f64 73.7

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

   5. Simplified 73.7%

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

   6. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 73.7

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

   8. Simplified 73.7%

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

   9. Taylor expanded in x around 0

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

       1. *-commutative N/A

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

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

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

       3. unpow2 N/A

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

       4. *-lowering-*.f64 34.3

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

   11. Simplified 34.3%

       \[\leadsto \color{blue}{\left(y \cdot y\right) \cdot -0.16666666666666666} \]
   12. Step-by-step derivation

       1. associate-*l* N/A

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

       2. *-commutative N/A

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

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

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

       4. *-lowering-*.f64 34.3

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

   13. Applied egg-rr 34.3%

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

   if -1.00000000000000003e-146 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y))

   1. Initial program 99.9%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \cosh x \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 75.1%

      \[\leadsto \cosh x \cdot \color{blue}{1} \]

   6. Taylor expanded in x around 0

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

   8. Simplified 29.6%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 30.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -1 \cdot 10^{-146}:\\ \;\;\;\;y \cdot \left(y \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \cosh x \cdot \frac{\sin y}{y} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* (cosh x) (/ (sin y) y)))

C

double code(double x, double y) {
	return cosh(x) * (sin(y) / y);
}

Fortran

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

Java

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

Python

def code(x, y):
	return math.cosh(x) * (math.sin(y) / y)

Julia

function code(x, y)
	return Float64(cosh(x) * Float64(sin(y) / y))
end

MATLAB

function tmp = code(x, y)
	tmp = cosh(x) * (sin(y) / y);
end

Wolfram

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

Derivation

1. Initial program 99.9%

   \[\cosh x \cdot \frac{\sin y}{y} \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 19: 26.4% accurate, 217.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

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

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 99.9%

   \[\cosh x \cdot \frac{\sin y}{y} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \cosh x \cdot \color{blue}{1} \]
4. Step-by-step derivation

5. Simplified 58.2%

   \[\leadsto \cosh x \cdot \color{blue}{1} \]

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1} \]
7. Step-by-step derivation

8. Simplified 23.1%

   \[\leadsto \color{blue}{1} \]
9. Add Preprocessing

Developer Target 1: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\cosh x \cdot \sin y}{y} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ (* (cosh x) (sin y)) y))

C

double code(double x, double y) {
	return (cosh(x) * sin(y)) / y;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (cosh(x) * sin(y)) / y
end function

Java

public static double code(double x, double y) {
	return (Math.cosh(x) * Math.sin(y)) / y;
}

Python

def code(x, y):
	return (math.cosh(x) * math.sin(y)) / y

Julia

function code(x, y)
	return Float64(Float64(cosh(x) * sin(y)) / y)
end

MATLAB

function tmp = code(x, y)
	tmp = (cosh(x) * sin(y)) / y;
end

Wolfram

code[x_, y_] := N[(N[(N[Cosh[x], $MachinePrecision] * N[Sin[y], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]

Reproduce

herbie shell --seed 2024199 
(FPCore (x y)
  :name "Linear.Quaternion:$csinh from linear-1.19.1.3"
  :precision binary64

  :alt
  (! :herbie-platform default (/ (* (cosh x) (sin y)) y))

  (* (cosh x) (/ (sin y) y)))