Linear.Quaternion:$ccosh from linear-1.19.1.3

Percentage Accurate: 89.2% → 99.9%

Time: 14.2s

Alternatives: 17

Speedup: 1.0×

• 49.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 89.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.9%

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

   1. clear-num N/A

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

   2. associate-/r* N/A

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

   3. clear-num N/A

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

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

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

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

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

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

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

   7. sin-lowering-sin.f64 99.9

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

4. Applied egg-rr 99.9%

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

Alternative 2: 84.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y \cdot \sin x}{x}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(\left(y \cdot y\right) \cdot \mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right)\right) \cdot \left(y \cdot \left(\left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\\ \mathbf{elif}\;t\_0 \leq 10^{-40}:\\ \;\;\;\;y \cdot \left(\mathsf{fma}\left(y, y \cdot 0.16666666666666666, 1\right) \cdot \frac{\sin x}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sinh y) (sin x)) x)))
   (if (<= t_0 (- INFINITY))
     (*
      (* (* y y) (fma x (* x -0.16666666666666666) 1.0))
      (* y (* (* y y) 0.008333333333333333)))
     (if (<= t_0 1e-40)
       (* y (* (fma y (* y 0.16666666666666666) 1.0) (/ (sin x) x)))
       (sinh y)))))

C

double code(double x, double y) {
	double t_0 = (sinh(y) * sin(x)) / x;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = ((y * y) * fma(x, (x * -0.16666666666666666), 1.0)) * (y * ((y * y) * 0.008333333333333333));
	} else if (t_0 <= 1e-40) {
		tmp = y * (fma(y, (y * 0.16666666666666666), 1.0) * (sin(x) / x));
	} else {
		tmp = sinh(y);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(sinh(y) * sin(x)) / x)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(y * y) * fma(x, Float64(x * -0.16666666666666666), 1.0)) * Float64(y * Float64(Float64(y * y) * 0.008333333333333333)));
	elseif (t_0 <= 1e-40)
		tmp = Float64(y * Float64(fma(y, Float64(y * 0.16666666666666666), 1.0) * Float64(sin(x) / x)));
	else
		tmp = sinh(y);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sinh[y], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(y * y), $MachinePrecision] * N[(x * N[(x * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[(y * N[(N[(y * y), $MachinePrecision] * 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e-40], N[(y * N[(N[(y * N[(y * 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sinh[y], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

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

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

      2. +-commutative N/A

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

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

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

      4. unpow2 N/A

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

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

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 81.1

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

   5. Simplified 81.1%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. *-rgt-identity N/A

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

      7. distribute-lft-in N/A

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

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

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

   8. Simplified 64.6%

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

   9. Taylor expanded in y around inf

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

       1. metadata-eval N/A

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

       2. pow-sqr N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. unpow2 N/A

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

       6. associate-*l* N/A

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

       7. *-commutative N/A

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

       8. associate-*r* N/A

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

       9. unpow2 N/A

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

       10. unpow3 N/A

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

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

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

       12. unpow3 N/A

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

       13. unpow2 N/A

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

       14. associate-*r* N/A

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

       15. *-commutative N/A

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

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

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

       17. *-commutative N/A

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

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

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

       19. unpow2 N/A

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

       20. *-lowering-*.f64 64.6

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

   11. Simplified 64.6%

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

       1. *-commutative N/A

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

       2. associate-*l/ N/A

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

       3. clear-num N/A

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

       4. associate-/r* N/A

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

       5. *-inverses N/A

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

       6. flip3-+ N/A

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

       7. clear-num N/A

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

   13. Applied egg-rr 64.6%

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

   if -inf.0 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 9.9999999999999993e-41

   1. Initial program 83.6%

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

   3. Taylor expanded in y around 0

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

      1. distribute-lft-in N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. distribute-lft-in N/A

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

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

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. distribute-lft1-in N/A

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

      11. +-commutative N/A

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

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

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

   5. Simplified 99.0%

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

   if 9.9999999999999993e-41 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 77.6%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-inverses N/A

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

      4. *-rgt-identity N/A

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

      5. sinh-lowering-sinh.f64 77.6

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

   7. Applied egg-rr 77.6%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -\infty:\\ \;\;\;\;\left(\left(y \cdot y\right) \cdot \mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right)\right) \cdot \left(y \cdot \left(\left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\\ \mathbf{elif}\;\frac{\sinh y \cdot \sin x}{x} \leq 10^{-40}:\\ \;\;\;\;y \cdot \left(\mathsf{fma}\left(y, y \cdot 0.16666666666666666, 1\right) \cdot \frac{\sin x}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y \cdot \sin x}{x}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(\left(y \cdot y\right) \cdot \mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right)\right) \cdot \left(y \cdot \left(\left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\\ \mathbf{elif}\;t\_0 \leq 10^{-40}:\\ \;\;\;\;y \cdot \frac{\sin x}{x}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sinh y) (sin x)) x)))
   (if (<= t_0 (- INFINITY))
     (*
      (* (* y y) (fma x (* x -0.16666666666666666) 1.0))
      (* y (* (* y y) 0.008333333333333333)))
     (if (<= t_0 1e-40) (* y (/ (sin x) x)) (sinh y)))))

C

double code(double x, double y) {
	double t_0 = (sinh(y) * sin(x)) / x;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = ((y * y) * fma(x, (x * -0.16666666666666666), 1.0)) * (y * ((y * y) * 0.008333333333333333));
	} else if (t_0 <= 1e-40) {
		tmp = y * (sin(x) / x);
	} else {
		tmp = sinh(y);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(sinh(y) * sin(x)) / x)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(y * y) * fma(x, Float64(x * -0.16666666666666666), 1.0)) * Float64(y * Float64(Float64(y * y) * 0.008333333333333333)));
	elseif (t_0 <= 1e-40)
		tmp = Float64(y * Float64(sin(x) / x));
	else
		tmp = sinh(y);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sinh[y], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(y * y), $MachinePrecision] * N[(x * N[(x * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[(y * N[(N[(y * y), $MachinePrecision] * 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e-40], N[(y * N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[Sinh[y], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

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

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

      2. +-commutative N/A

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

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

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

      4. unpow2 N/A

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

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

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 81.1

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

   5. Simplified 81.1%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. *-rgt-identity N/A

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

      7. distribute-lft-in N/A

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

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

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

   8. Simplified 64.6%

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

   9. Taylor expanded in y around inf

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

       1. metadata-eval N/A

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

       2. pow-sqr N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. unpow2 N/A

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

       6. associate-*l* N/A

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

       7. *-commutative N/A

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

       8. associate-*r* N/A

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

       9. unpow2 N/A

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

       10. unpow3 N/A

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

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

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

       12. unpow3 N/A

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

       13. unpow2 N/A

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

       14. associate-*r* N/A

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

       15. *-commutative N/A

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

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

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

       17. *-commutative N/A

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

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

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

       19. unpow2 N/A

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

       20. *-lowering-*.f64 64.6

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

   11. Simplified 64.6%

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

       1. *-commutative N/A

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

       2. associate-*l/ N/A

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

       3. clear-num N/A

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

       4. associate-/r* N/A

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

       5. *-inverses N/A

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

       6. flip3-+ N/A

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

       7. clear-num N/A

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

   13. Applied egg-rr 64.6%

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

   if -inf.0 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 9.9999999999999993e-41

   1. Initial program 83.6%

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

   3. Taylor expanded in y around 0

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

      1. distribute-lft-in N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. distribute-lft-in N/A

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

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

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. distribute-lft1-in N/A

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

      11. +-commutative N/A

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

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

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

   5. Simplified 99.0%

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

   6. Taylor expanded in y around 0

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

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

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

      2. sin-lowering-sin.f64 98.6

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

   8. Simplified 98.6%

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

   if 9.9999999999999993e-41 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 77.6%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-inverses N/A

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

      4. *-rgt-identity N/A

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

      5. sinh-lowering-sinh.f64 77.6

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

   7. Applied egg-rr 77.6%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -\infty:\\ \;\;\;\;\left(\left(y \cdot y\right) \cdot \mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right)\right) \cdot \left(y \cdot \left(\left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\\ \mathbf{elif}\;\frac{\sinh y \cdot \sin x}{x} \leq 10^{-40}:\\ \;\;\;\;y \cdot \frac{\sin x}{x}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 65.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y \cdot \sin x}{x}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-271}:\\ \;\;\;\;\left(\left(y \cdot y\right) \cdot \mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right)\right) \cdot \left(y \cdot \left(\left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{-134}:\\ \;\;\;\;\frac{x}{\frac{x}{y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), 1\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sinh y) (sin x)) x)))
   (if (<= t_0 -2e-271)
     (*
      (* (* y y) (fma x (* x -0.16666666666666666) 1.0))
      (* y (* (* y y) 0.008333333333333333)))
     (if (<= t_0 4e-134)
       (/
        x
        (/
         x
         (*
          y
          (fma
           y
           (* y (fma (* y y) 0.008333333333333333 0.16666666666666666))
           1.0))))
       (sinh y)))))

C

double code(double x, double y) {
	double t_0 = (sinh(y) * sin(x)) / x;
	double tmp;
	if (t_0 <= -2e-271) {
		tmp = ((y * y) * fma(x, (x * -0.16666666666666666), 1.0)) * (y * ((y * y) * 0.008333333333333333));
	} else if (t_0 <= 4e-134) {
		tmp = x / (x / (y * fma(y, (y * fma((y * y), 0.008333333333333333, 0.16666666666666666)), 1.0)));
	} else {
		tmp = sinh(y);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(sinh(y) * sin(x)) / x)
	tmp = 0.0
	if (t_0 <= -2e-271)
		tmp = Float64(Float64(Float64(y * y) * fma(x, Float64(x * -0.16666666666666666), 1.0)) * Float64(y * Float64(Float64(y * y) * 0.008333333333333333)));
	elseif (t_0 <= 4e-134)
		tmp = Float64(x / Float64(x / Float64(y * fma(y, Float64(y * fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666)), 1.0))));
	else
		tmp = sinh(y);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sinh[y], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-271], N[(N[(N[(y * y), $MachinePrecision] * N[(x * N[(x * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[(y * N[(N[(y * y), $MachinePrecision] * 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 4e-134], N[(x / N[(x / N[(y * N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sinh[y], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -1.99999999999999993e-271

   1. Initial program 98.6%

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

   3. Taylor expanded in y around 0

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

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

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

      2. +-commutative N/A

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

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

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

      4. unpow2 N/A

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

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

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 87.3

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

   5. Simplified 87.3%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. *-rgt-identity N/A

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

      7. distribute-lft-in N/A

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

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

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

   8. Simplified 59.7%

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

   9. Taylor expanded in y around inf

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

       1. metadata-eval N/A

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

       2. pow-sqr N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. unpow2 N/A

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

       6. associate-*l* N/A

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

       7. *-commutative N/A

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

       8. associate-*r* N/A

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

       9. unpow2 N/A

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

       10. unpow3 N/A

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

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

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

       12. unpow3 N/A

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

       13. unpow2 N/A

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

       14. associate-*r* N/A

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

       15. *-commutative N/A

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

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

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

       17. *-commutative N/A

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

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

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

       19. unpow2 N/A

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

       20. *-lowering-*.f64 40.6

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

   11. Simplified 40.6%

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

       1. *-commutative N/A

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

       2. associate-*l/ N/A

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

       3. clear-num N/A

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

       4. associate-/r* N/A

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

       5. *-inverses N/A

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

       6. flip3-+ N/A

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

       7. clear-num N/A

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

   13. Applied egg-rr 40.7%

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

   if -1.99999999999999993e-271 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 4.00000000000000016e-134

   1. Initial program 74.4%

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

   3. Taylor expanded in y around 0

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

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

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

      2. +-commutative N/A

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

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

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

      4. unpow2 N/A

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

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

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 73.3

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

   5. Simplified 73.3%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

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

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

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

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

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

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

      8. associate-*l* N/A

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

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

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

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

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

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

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

      12. *-lowering-*.f64 96.4

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

   7. Applied egg-rr 96.4%

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

   8. Taylor expanded in x around 0

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

   10. Simplified 75.8%

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

   if 4.00000000000000016e-134 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 99.2%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 78.2%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-inverses N/A

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

      4. *-rgt-identity N/A

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

      5. sinh-lowering-sinh.f64 79.0

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

   7. Applied egg-rr 79.0%

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

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -2 \cdot 10^{-271}:\\ \;\;\;\;\left(\left(y \cdot y\right) \cdot \mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right)\right) \cdot \left(y \cdot \left(\left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\\ \mathbf{elif}\;\frac{\sinh y \cdot \sin x}{x} \leq 4 \cdot 10^{-134}:\\ \;\;\;\;\frac{x}{\frac{x}{y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), 1\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 62.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -2 \cdot 10^{-271}:\\ \;\;\;\;\left(\left(y \cdot y\right) \cdot \mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right)\right) \cdot \left(y \cdot \left(\left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{x}{y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), 1\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (sinh y) (sin x)) x) -2e-271)
   (*
    (* (* y y) (fma x (* x -0.16666666666666666) 1.0))
    (* y (* (* y y) 0.008333333333333333)))
   (/
    x
    (/
     x
     (*
      y
      (fma
       y
       (* y (fma (* y y) 0.008333333333333333 0.16666666666666666))
       1.0))))))

C

double code(double x, double y) {
	double tmp;
	if (((sinh(y) * sin(x)) / x) <= -2e-271) {
		tmp = ((y * y) * fma(x, (x * -0.16666666666666666), 1.0)) * (y * ((y * y) * 0.008333333333333333));
	} else {
		tmp = x / (x / (y * fma(y, (y * fma((y * y), 0.008333333333333333, 0.16666666666666666)), 1.0)));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(sinh(y) * sin(x)) / x) <= -2e-271)
		tmp = Float64(Float64(Float64(y * y) * fma(x, Float64(x * -0.16666666666666666), 1.0)) * Float64(y * Float64(Float64(y * y) * 0.008333333333333333)));
	else
		tmp = Float64(x / Float64(x / Float64(y * fma(y, Float64(y * fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666)), 1.0))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[Sinh[y], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], -2e-271], N[(N[(N[(y * y), $MachinePrecision] * N[(x * N[(x * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[(y * N[(N[(y * y), $MachinePrecision] * 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(x / N[(y * N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -1.99999999999999993e-271

   1. Initial program 98.6%

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

   3. Taylor expanded in y around 0

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

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

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

      2. +-commutative N/A

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

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

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

      4. unpow2 N/A

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

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

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 87.3

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

   5. Simplified 87.3%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. *-rgt-identity N/A

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

      7. distribute-lft-in N/A

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

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

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

   8. Simplified 59.7%

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

   9. Taylor expanded in y around inf

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

       1. metadata-eval N/A

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

       2. pow-sqr N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. unpow2 N/A

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

       6. associate-*l* N/A

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

       7. *-commutative N/A

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

       8. associate-*r* N/A

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

       9. unpow2 N/A

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

       10. unpow3 N/A

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

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

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

       12. unpow3 N/A

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

       13. unpow2 N/A

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

       14. associate-*r* N/A

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

       15. *-commutative N/A

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

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

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

       17. *-commutative N/A

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

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

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

       19. unpow2 N/A

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

       20. *-lowering-*.f64 40.6

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

   11. Simplified 40.6%

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

       1. *-commutative N/A

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

       2. associate-*l/ N/A

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

       3. clear-num N/A

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

       4. associate-/r* N/A

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

       5. *-inverses N/A

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

       6. flip3-+ N/A

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

       7. clear-num N/A

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

   13. Applied egg-rr 40.7%

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

   if -1.99999999999999993e-271 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 86.5%

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

   3. Taylor expanded in y around 0

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

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

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

      2. +-commutative N/A

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

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

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

      4. unpow2 N/A

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

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

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 81.3

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

   5. Simplified 81.3%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

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

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

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

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

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

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

      8. associate-*l* N/A

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

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

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

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

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

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

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

      12. *-lowering-*.f64 94.1

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

   7. Applied egg-rr 94.1%

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

   8. Taylor expanded in x around 0

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

   10. Simplified 74.5%

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

4. Final simplification 62.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -2 \cdot 10^{-271}:\\ \;\;\;\;\left(\left(y \cdot y\right) \cdot \mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right)\right) \cdot \left(y \cdot \left(\left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{x}{y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), 1\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 51.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -2 \cdot 10^{-271}:\\ \;\;\;\;\left(\left(y \cdot y\right) \cdot \mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right)\right) \cdot \left(y \cdot \left(\left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (sinh y) (sin x)) x) -2e-271)
   (*
    (* (* y y) (fma x (* x -0.16666666666666666) 1.0))
    (* y (* (* y y) 0.008333333333333333)))
   (*
    y
    (fma
     (* y y)
     (fma
      y
      (* y (fma (* y y) 0.0001984126984126984 0.008333333333333333))
      0.16666666666666666)
     1.0))))

C

double code(double x, double y) {
	double tmp;
	if (((sinh(y) * sin(x)) / x) <= -2e-271) {
		tmp = ((y * y) * fma(x, (x * -0.16666666666666666), 1.0)) * (y * ((y * y) * 0.008333333333333333));
	} else {
		tmp = y * fma((y * y), fma(y, (y * fma((y * y), 0.0001984126984126984, 0.008333333333333333)), 0.16666666666666666), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(sinh(y) * sin(x)) / x) <= -2e-271)
		tmp = Float64(Float64(Float64(y * y) * fma(x, Float64(x * -0.16666666666666666), 1.0)) * Float64(y * Float64(Float64(y * y) * 0.008333333333333333)));
	else
		tmp = Float64(y * fma(Float64(y * y), fma(y, Float64(y * fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333)), 0.16666666666666666), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[Sinh[y], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], -2e-271], N[(N[(N[(y * y), $MachinePrecision] * N[(x * N[(x * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[(y * N[(N[(y * y), $MachinePrecision] * 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -1.99999999999999993e-271

   1. Initial program 98.6%

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

   3. Taylor expanded in y around 0

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

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

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

      2. +-commutative N/A

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

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

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

      4. unpow2 N/A

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

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

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 87.3

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

   5. Simplified 87.3%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. *-rgt-identity N/A

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

      7. distribute-lft-in N/A

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

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

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

   8. Simplified 59.7%

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

   9. Taylor expanded in y around inf

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

       1. metadata-eval N/A

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

       2. pow-sqr N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. unpow2 N/A

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

       6. associate-*l* N/A

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

       7. *-commutative N/A

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

       8. associate-*r* N/A

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

       9. unpow2 N/A

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

       10. unpow3 N/A

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

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

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

       12. unpow3 N/A

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

       13. unpow2 N/A

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

       14. associate-*r* N/A

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

       15. *-commutative N/A

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

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

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

       17. *-commutative N/A

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

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

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

       19. unpow2 N/A

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

       20. *-lowering-*.f64 40.6

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

   11. Simplified 40.6%

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

       1. *-commutative N/A

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

       2. associate-*l/ N/A

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

       3. clear-num N/A

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

       4. associate-/r* N/A

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

       5. *-inverses N/A

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

       6. flip3-+ N/A

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

       7. clear-num N/A

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

   13. Applied egg-rr 40.7%

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

   if -1.99999999999999993e-271 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 86.5%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 46.9%

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

   6. Taylor expanded in y around 0

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

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

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

      2. +-commutative N/A

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

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

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

      4. unpow2 N/A

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

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

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

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

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

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

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

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

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 57.6

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

   8. Simplified 57.6%

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

4. Final simplification 51.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -2 \cdot 10^{-271}:\\ \;\;\;\;\left(\left(y \cdot y\right) \cdot \mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right)\right) \cdot \left(y \cdot \left(\left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 55.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -5 \cdot 10^{-214}:\\ \;\;\;\;y \cdot \left(\mathsf{fma}\left(y, y \cdot 0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (sinh y) (sin x)) x) -5e-214)
   (*
    y
    (*
     (fma y (* y 0.16666666666666666) 1.0)
     (fma (* x x) -0.16666666666666666 1.0)))
   (*
    y
    (fma
     (* y y)
     (fma
      y
      (* y (fma (* y y) 0.0001984126984126984 0.008333333333333333))
      0.16666666666666666)
     1.0))))

C

double code(double x, double y) {
	double tmp;
	if (((sinh(y) * sin(x)) / x) <= -5e-214) {
		tmp = y * (fma(y, (y * 0.16666666666666666), 1.0) * fma((x * x), -0.16666666666666666, 1.0));
	} else {
		tmp = y * fma((y * y), fma(y, (y * fma((y * y), 0.0001984126984126984, 0.008333333333333333)), 0.16666666666666666), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(sinh(y) * sin(x)) / x) <= -5e-214)
		tmp = Float64(y * Float64(fma(y, Float64(y * 0.16666666666666666), 1.0) * fma(Float64(x * x), -0.16666666666666666, 1.0)));
	else
		tmp = Float64(y * fma(Float64(y * y), fma(y, Float64(y * fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333)), 0.16666666666666666), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[Sinh[y], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], -5e-214], N[(y * N[(N[(y * N[(y * 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -4.9999999999999998e-214

   1. Initial program 98.6%

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

   3. Taylor expanded in y around 0

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

      1. distribute-lft-in N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. distribute-lft-in N/A

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

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

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. distribute-lft1-in N/A

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

      11. +-commutative N/A

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

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

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

   5. Simplified 78.0%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt1-in N/A

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

      5. +-commutative N/A

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

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

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

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

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

      10. unpow2 N/A

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

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

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. associate-*l* N/A

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

      16. *-commutative N/A

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

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

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

      18. *-commutative N/A

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

      19. *-lowering-*.f64 58.2

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

   8. Simplified 58.2%

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

   if -4.9999999999999998e-214 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 87.1%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 46.1%

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

   6. Taylor expanded in y around 0

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

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

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

      2. +-commutative N/A

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

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

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

      4. unpow2 N/A

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

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

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

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

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

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

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

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

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 56.2

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

   8. Simplified 56.2%

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

4. Final simplification 56.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -5 \cdot 10^{-214}:\\ \;\;\;\;y \cdot \left(\mathsf{fma}\left(y, y \cdot 0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 55.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -5 \cdot 10^{-214}:\\ \;\;\;\;y \cdot \left(\mathsf{fma}\left(y, y \cdot 0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \left(\left(y \cdot y\right) \cdot 0.0001984126984126984\right), 0.16666666666666666\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (sinh y) (sin x)) x) -5e-214)
   (*
    y
    (*
     (fma y (* y 0.16666666666666666) 1.0)
     (fma (* x x) -0.16666666666666666 1.0)))
   (*
    y
    (fma
     (* y y)
     (fma y (* y (* (* y y) 0.0001984126984126984)) 0.16666666666666666)
     1.0))))

C

double code(double x, double y) {
	double tmp;
	if (((sinh(y) * sin(x)) / x) <= -5e-214) {
		tmp = y * (fma(y, (y * 0.16666666666666666), 1.0) * fma((x * x), -0.16666666666666666, 1.0));
	} else {
		tmp = y * fma((y * y), fma(y, (y * ((y * y) * 0.0001984126984126984)), 0.16666666666666666), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(sinh(y) * sin(x)) / x) <= -5e-214)
		tmp = Float64(y * Float64(fma(y, Float64(y * 0.16666666666666666), 1.0) * fma(Float64(x * x), -0.16666666666666666, 1.0)));
	else
		tmp = Float64(y * fma(Float64(y * y), fma(y, Float64(y * Float64(Float64(y * y) * 0.0001984126984126984)), 0.16666666666666666), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[Sinh[y], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], -5e-214], N[(y * N[(N[(y * N[(y * 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -4.9999999999999998e-214

   1. Initial program 98.6%

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

   3. Taylor expanded in y around 0

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

      1. distribute-lft-in N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. distribute-lft-in N/A

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

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

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. distribute-lft1-in N/A

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

      11. +-commutative N/A

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

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

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

   5. Simplified 78.0%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt1-in N/A

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

      5. +-commutative N/A

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

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

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

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

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

      10. unpow2 N/A

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

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

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. associate-*l* N/A

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

      16. *-commutative N/A

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

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

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

      18. *-commutative N/A

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

      19. *-lowering-*.f64 58.2

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

   8. Simplified 58.2%

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

   if -4.9999999999999998e-214 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 87.1%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 46.1%

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

   6. Taylor expanded in y around 0

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

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

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

      2. +-commutative N/A

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

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

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

      4. unpow2 N/A

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

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

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

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

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

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

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

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

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 56.2

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

   8. Simplified 56.2%

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

   9. Taylor expanded in y around inf

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

       1. *-commutative N/A

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

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

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

       3. unpow2 N/A

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

       4. *-lowering-*.f64 56.2

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

   11. Simplified 56.2%

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

4. Final simplification 56.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -5 \cdot 10^{-214}:\\ \;\;\;\;y \cdot \left(\mathsf{fma}\left(y, y \cdot 0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \left(\left(y \cdot y\right) \cdot 0.0001984126984126984\right), 0.16666666666666666\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 55.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -5 \cdot 10^{-214}:\\ \;\;\;\;y \cdot \left(\mathsf{fma}\left(y, y \cdot 0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(y \cdot y, y \cdot \left(0.0001984126984126984 \cdot \left(y \cdot \left(y \cdot y\right)\right)\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (sinh y) (sin x)) x) -5e-214)
   (*
    y
    (*
     (fma y (* y 0.16666666666666666) 1.0)
     (fma (* x x) -0.16666666666666666 1.0)))
   (* y (fma (* y y) (* y (* 0.0001984126984126984 (* y (* y y)))) 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (((sinh(y) * sin(x)) / x) <= -5e-214) {
		tmp = y * (fma(y, (y * 0.16666666666666666), 1.0) * fma((x * x), -0.16666666666666666, 1.0));
	} else {
		tmp = y * fma((y * y), (y * (0.0001984126984126984 * (y * (y * y)))), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(sinh(y) * sin(x)) / x) <= -5e-214)
		tmp = Float64(y * Float64(fma(y, Float64(y * 0.16666666666666666), 1.0) * fma(Float64(x * x), -0.16666666666666666, 1.0)));
	else
		tmp = Float64(y * fma(Float64(y * y), Float64(y * Float64(0.0001984126984126984 * Float64(y * Float64(y * y)))), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[Sinh[y], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], -5e-214], N[(y * N[(N[(y * N[(y * 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[(y * y), $MachinePrecision] * N[(y * N[(0.0001984126984126984 * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -4.9999999999999998e-214

   1. Initial program 98.6%

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

   3. Taylor expanded in y around 0

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

      1. distribute-lft-in N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. distribute-lft-in N/A

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

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

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. distribute-lft1-in N/A

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

      11. +-commutative N/A

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

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

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

   5. Simplified 78.0%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt1-in N/A

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

      5. +-commutative N/A

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

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

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

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

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

      10. unpow2 N/A

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

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

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. associate-*l* N/A

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

      16. *-commutative N/A

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

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

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

      18. *-commutative N/A

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

      19. *-lowering-*.f64 58.2

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

   8. Simplified 58.2%

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

   if -4.9999999999999998e-214 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 87.1%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 46.1%

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

   6. Taylor expanded in y around 0

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

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

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

      2. +-commutative N/A

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

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

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

      4. unpow2 N/A

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

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

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

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

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

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

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

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

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 56.2

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

   8. Simplified 56.2%

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

   9. Taylor expanded in y around inf

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

       1. metadata-eval N/A

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

       2. pow-sqr N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. unpow2 N/A

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

       6. associate-*l* N/A

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

       7. *-commutative N/A

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

       8. associate-*r* N/A

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

       9. unpow2 N/A

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

       10. unpow3 N/A

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

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

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

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

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

       13. cube-mult N/A

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

       14. unpow2 N/A

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

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

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

       16. unpow2 N/A

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

       17. *-lowering-*.f64 56.2

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

   11. Simplified 56.2%

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

4. Final simplification 56.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -5 \cdot 10^{-214}:\\ \;\;\;\;y \cdot \left(\mathsf{fma}\left(y, y \cdot 0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(y \cdot y, y \cdot \left(0.0001984126984126984 \cdot \left(y \cdot \left(y \cdot y\right)\right)\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 54.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -5 \cdot 10^{-214}:\\ \;\;\;\;y \cdot \left(\mathsf{fma}\left(y, y \cdot 0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (sinh y) (sin x)) x) -5e-214)
   (*
    y
    (*
     (fma y (* y 0.16666666666666666) 1.0)
     (fma (* x x) -0.16666666666666666 1.0)))
   (*
    y
    (fma y (* y (fma y (* y 0.008333333333333333) 0.16666666666666666)) 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (((sinh(y) * sin(x)) / x) <= -5e-214) {
		tmp = y * (fma(y, (y * 0.16666666666666666), 1.0) * fma((x * x), -0.16666666666666666, 1.0));
	} else {
		tmp = y * fma(y, (y * fma(y, (y * 0.008333333333333333), 0.16666666666666666)), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(sinh(y) * sin(x)) / x) <= -5e-214)
		tmp = Float64(y * Float64(fma(y, Float64(y * 0.16666666666666666), 1.0) * fma(Float64(x * x), -0.16666666666666666, 1.0)));
	else
		tmp = Float64(y * fma(y, Float64(y * fma(y, Float64(y * 0.008333333333333333), 0.16666666666666666)), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[Sinh[y], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], -5e-214], N[(y * N[(N[(y * N[(y * 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(y * N[(y * N[(y * N[(y * 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -4.9999999999999998e-214

   1. Initial program 98.6%

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

   3. Taylor expanded in y around 0

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

      1. distribute-lft-in N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. distribute-lft-in N/A

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

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

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. distribute-lft1-in N/A

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

      11. +-commutative N/A

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

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

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

   5. Simplified 78.0%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt1-in N/A

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

      5. +-commutative N/A

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

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

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

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

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

      10. unpow2 N/A

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

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

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. associate-*l* N/A

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

      16. *-commutative N/A

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

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

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

      18. *-commutative N/A

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

      19. *-lowering-*.f64 58.2

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

   8. Simplified 58.2%

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

   if -4.9999999999999998e-214 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 87.1%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 46.1%

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

   6. Taylor expanded in y around 0

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

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

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto y \cdot \color{blue}{\mathsf{fma}\left(y, y \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right), 1\right)} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto y \cdot \mathsf{fma}\left(y, \color{blue}{y \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)}, 1\right) \]

      7. +-commutative N/A

         \[\leadsto y \cdot \mathsf{fma}\left(y, y \cdot \color{blue}{\left(\frac{1}{120} \cdot {y}^{2} + \frac{1}{6}\right)}, 1\right) \]

      8. *-commutative N/A

         \[\leadsto y \cdot \mathsf{fma}\left(y, y \cdot \left(\color{blue}{{y}^{2} \cdot \frac{1}{120}} + \frac{1}{6}\right), 1\right) \]

      9. unpow2 N/A

         \[\leadsto y \cdot \mathsf{fma}\left(y, y \cdot \left(\color{blue}{\left(y \cdot y\right)} \cdot \frac{1}{120} + \frac{1}{6}\right), 1\right) \]

      10. associate-*l* N/A

          \[\leadsto y \cdot \mathsf{fma}\left(y, y \cdot \left(\color{blue}{y \cdot \left(y \cdot \frac{1}{120}\right)} + \frac{1}{6}\right), 1\right) \]

      11. *-commutative N/A

          \[\leadsto y \cdot \mathsf{fma}\left(y, y \cdot \left(y \cdot \color{blue}{\left(\frac{1}{120} \cdot y\right)} + \frac{1}{6}\right), 1\right) \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto y \cdot \mathsf{fma}\left(y, y \cdot \color{blue}{\mathsf{fma}\left(y, \frac{1}{120} \cdot y, \frac{1}{6}\right)}, 1\right) \]

      13. *-commutative N/A

          \[\leadsto y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, \color{blue}{y \cdot \frac{1}{120}}, \frac{1}{6}\right), 1\right) \]

      14. *-lowering-*.f64 55.6

          \[\leadsto y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, \color{blue}{y \cdot 0.008333333333333333}, 0.16666666666666666\right), 1\right) \]

   8. Simplified 55.6%

      \[\leadsto \color{blue}{y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -5 \cdot 10^{-214}:\\ \;\;\;\;y \cdot \left(\mathsf{fma}\left(y, y \cdot 0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 46.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -5 \cdot 10^{-214}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(-0.16666666666666666, x \cdot x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (sinh y) (sin x)) x) -5e-214)
   (* y (fma -0.16666666666666666 (* x x) 1.0))
   (*
    y
    (fma y (* y (fma y (* y 0.008333333333333333) 0.16666666666666666)) 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (((sinh(y) * sin(x)) / x) <= -5e-214) {
		tmp = y * fma(-0.16666666666666666, (x * x), 1.0);
	} else {
		tmp = y * fma(y, (y * fma(y, (y * 0.008333333333333333), 0.16666666666666666)), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(sinh(y) * sin(x)) / x) <= -5e-214)
		tmp = Float64(y * fma(-0.16666666666666666, Float64(x * x), 1.0));
	else
		tmp = Float64(y * fma(y, Float64(y * fma(y, Float64(y * 0.008333333333333333), 0.16666666666666666)), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[Sinh[y], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], -5e-214], N[(y * N[(-0.16666666666666666 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(y * N[(y * N[(y * N[(y * N[(y * 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -4.9999999999999998e-214

   1. Initial program 98.6%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\sin x \cdot \color{blue}{\left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)}}{x} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \frac{\sin x \cdot \color{blue}{\left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)}}{x} \]

      2. +-commutative N/A

         \[\leadsto \frac{\sin x \cdot \left(y \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) + 1\right)}\right)}{x} \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\sin x \cdot \left(y \cdot \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{1}{6} + \frac{1}{120} \cdot {y}^{2}, 1\right)}\right)}{x} \]

      4. unpow2 N/A

         \[\leadsto \frac{\sin x \cdot \left(y \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{6} + \frac{1}{120} \cdot {y}^{2}, 1\right)\right)}{x} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{\sin x \cdot \left(y \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{6} + \frac{1}{120} \cdot {y}^{2}, 1\right)\right)}{x} \]

      6. +-commutative N/A

         \[\leadsto \frac{\sin x \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{\frac{1}{120} \cdot {y}^{2} + \frac{1}{6}}, 1\right)\right)}{x} \]

      7. *-commutative N/A

         \[\leadsto \frac{\sin x \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{{y}^{2} \cdot \frac{1}{120}} + \frac{1}{6}, 1\right)\right)}{x} \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\sin x \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{1}{120}, \frac{1}{6}\right)}, 1\right)\right)}{x} \]

      9. unpow2 N/A

         \[\leadsto \frac{\sin x \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{120}, \frac{1}{6}\right), 1\right)\right)}{x} \]

      10. *-lowering-*.f64 86.2

          \[\leadsto \frac{\sin x \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, 0.008333333333333333, 0.16666666666666666\right), 1\right)\right)}{x} \]

   5. Simplified 86.2%

      \[\leadsto \frac{\sin x \cdot \color{blue}{\left(y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), 1\right)\right)}}{x} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{x \cdot \left(\frac{-1}{6} \cdot \left({x}^{2} \cdot \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right) + y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)}}{x} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) + \frac{-1}{6} \cdot \left({x}^{2} \cdot \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right)\right)}}{x} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\color{blue}{x \cdot \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) + x \cdot \left(\frac{-1}{6} \cdot \left({x}^{2} \cdot \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right)\right)}}{x} \]

      3. associate-*r* N/A

         \[\leadsto \frac{x \cdot \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) + x \cdot \color{blue}{\left(\left(\frac{-1}{6} \cdot {x}^{2}\right) \cdot \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right)}}{x} \]

      4. associate-*r* N/A

         \[\leadsto \frac{x \cdot \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) + \color{blue}{\left(x \cdot \left(\frac{-1}{6} \cdot {x}^{2}\right)\right) \cdot \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)}}{x} \]

      5. distribute-rgt-out N/A

         \[\leadsto \frac{\color{blue}{\left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) \cdot \left(x + x \cdot \left(\frac{-1}{6} \cdot {x}^{2}\right)\right)}}{x} \]

      6. *-rgt-identity N/A

         \[\leadsto \frac{\left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) \cdot \left(\color{blue}{x \cdot 1} + x \cdot \left(\frac{-1}{6} \cdot {x}^{2}\right)\right)}{x} \]

      7. distribute-lft-in N/A

         \[\leadsto \frac{\left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) \cdot \color{blue}{\left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)}}{x} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)}}{x} \]

   8. Simplified 62.9%

      \[\leadsto \frac{\color{blue}{\left(y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)\right) \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right)\right)}}{x} \]

   9. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{y \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)} \]

       2. +-commutative N/A

          \[\leadsto y \cdot \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2} + 1\right)} \]

       3. accelerator-lowering-fma.f64 N/A

          \[\leadsto y \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{6}, {x}^{2}, 1\right)} \]

       4. unpow2 N/A

          \[\leadsto y \cdot \mathsf{fma}\left(\frac{-1}{6}, \color{blue}{x \cdot x}, 1\right) \]

       5. *-lowering-*.f64 36.5

          \[\leadsto y \cdot \mathsf{fma}\left(-0.16666666666666666, \color{blue}{x \cdot x}, 1\right) \]

   11. Simplified 36.5%

       \[\leadsto \color{blue}{y \cdot \mathsf{fma}\left(-0.16666666666666666, x \cdot x, 1\right)} \]

   if -4.9999999999999998e-214 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 87.1%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{x} \cdot \sinh y}{x} \]
   4. Step-by-step derivation

   5. Simplified 46.1%

      \[\leadsto \frac{\color{blue}{x} \cdot \sinh y}{x} \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto y \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) + 1\right)} \]

      3. unpow2 N/A

         \[\leadsto y \cdot \left(\color{blue}{\left(y \cdot y\right)} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) + 1\right) \]

      4. associate-*l* N/A

         \[\leadsto y \cdot \left(\color{blue}{y \cdot \left(y \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} + 1\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto y \cdot \color{blue}{\mathsf{fma}\left(y, y \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right), 1\right)} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto y \cdot \mathsf{fma}\left(y, \color{blue}{y \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)}, 1\right) \]

      7. +-commutative N/A

         \[\leadsto y \cdot \mathsf{fma}\left(y, y \cdot \color{blue}{\left(\frac{1}{120} \cdot {y}^{2} + \frac{1}{6}\right)}, 1\right) \]

      8. *-commutative N/A

         \[\leadsto y \cdot \mathsf{fma}\left(y, y \cdot \left(\color{blue}{{y}^{2} \cdot \frac{1}{120}} + \frac{1}{6}\right), 1\right) \]

      9. unpow2 N/A

         \[\leadsto y \cdot \mathsf{fma}\left(y, y \cdot \left(\color{blue}{\left(y \cdot y\right)} \cdot \frac{1}{120} + \frac{1}{6}\right), 1\right) \]

      10. associate-*l* N/A

          \[\leadsto y \cdot \mathsf{fma}\left(y, y \cdot \left(\color{blue}{y \cdot \left(y \cdot \frac{1}{120}\right)} + \frac{1}{6}\right), 1\right) \]

      11. *-commutative N/A

          \[\leadsto y \cdot \mathsf{fma}\left(y, y \cdot \left(y \cdot \color{blue}{\left(\frac{1}{120} \cdot y\right)} + \frac{1}{6}\right), 1\right) \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto y \cdot \mathsf{fma}\left(y, y \cdot \color{blue}{\mathsf{fma}\left(y, \frac{1}{120} \cdot y, \frac{1}{6}\right)}, 1\right) \]

      13. *-commutative N/A

          \[\leadsto y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, \color{blue}{y \cdot \frac{1}{120}}, \frac{1}{6}\right), 1\right) \]

      14. *-lowering-*.f64 55.6

          \[\leadsto y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, \color{blue}{y \cdot 0.008333333333333333}, 0.16666666666666666\right), 1\right) \]

   8. Simplified 55.6%

      \[\leadsto \color{blue}{y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -5 \cdot 10^{-214}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(-0.16666666666666666, x \cdot x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 44.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -5 \cdot 10^{-214}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(-0.16666666666666666, x \cdot x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(y, y \cdot 0.16666666666666666, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (sinh y) (sin x)) x) -5e-214)
   (* y (fma -0.16666666666666666 (* x x) 1.0))
   (* y (fma y (* y 0.16666666666666666) 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (((sinh(y) * sin(x)) / x) <= -5e-214) {
		tmp = y * fma(-0.16666666666666666, (x * x), 1.0);
	} else {
		tmp = y * fma(y, (y * 0.16666666666666666), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(sinh(y) * sin(x)) / x) <= -5e-214)
		tmp = Float64(y * fma(-0.16666666666666666, Float64(x * x), 1.0));
	else
		tmp = Float64(y * fma(y, Float64(y * 0.16666666666666666), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[Sinh[y], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], -5e-214], N[(y * N[(-0.16666666666666666 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(y * N[(y * N[(y * 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -4.9999999999999998e-214

   1. Initial program 98.6%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\sin x \cdot \color{blue}{\left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)}}{x} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \frac{\sin x \cdot \color{blue}{\left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)}}{x} \]

      2. +-commutative N/A

         \[\leadsto \frac{\sin x \cdot \left(y \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) + 1\right)}\right)}{x} \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\sin x \cdot \left(y \cdot \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{1}{6} + \frac{1}{120} \cdot {y}^{2}, 1\right)}\right)}{x} \]

      4. unpow2 N/A

         \[\leadsto \frac{\sin x \cdot \left(y \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{6} + \frac{1}{120} \cdot {y}^{2}, 1\right)\right)}{x} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{\sin x \cdot \left(y \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{6} + \frac{1}{120} \cdot {y}^{2}, 1\right)\right)}{x} \]

      6. +-commutative N/A

         \[\leadsto \frac{\sin x \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{\frac{1}{120} \cdot {y}^{2} + \frac{1}{6}}, 1\right)\right)}{x} \]

      7. *-commutative N/A

         \[\leadsto \frac{\sin x \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{{y}^{2} \cdot \frac{1}{120}} + \frac{1}{6}, 1\right)\right)}{x} \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\sin x \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{1}{120}, \frac{1}{6}\right)}, 1\right)\right)}{x} \]

      9. unpow2 N/A

         \[\leadsto \frac{\sin x \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{120}, \frac{1}{6}\right), 1\right)\right)}{x} \]

      10. *-lowering-*.f64 86.2

          \[\leadsto \frac{\sin x \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, 0.008333333333333333, 0.16666666666666666\right), 1\right)\right)}{x} \]

   5. Simplified 86.2%

      \[\leadsto \frac{\sin x \cdot \color{blue}{\left(y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), 1\right)\right)}}{x} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{x \cdot \left(\frac{-1}{6} \cdot \left({x}^{2} \cdot \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right) + y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)}}{x} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) + \frac{-1}{6} \cdot \left({x}^{2} \cdot \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right)\right)}}{x} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\color{blue}{x \cdot \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) + x \cdot \left(\frac{-1}{6} \cdot \left({x}^{2} \cdot \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right)\right)}}{x} \]

      3. associate-*r* N/A

         \[\leadsto \frac{x \cdot \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) + x \cdot \color{blue}{\left(\left(\frac{-1}{6} \cdot {x}^{2}\right) \cdot \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right)}}{x} \]

      4. associate-*r* N/A

         \[\leadsto \frac{x \cdot \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) + \color{blue}{\left(x \cdot \left(\frac{-1}{6} \cdot {x}^{2}\right)\right) \cdot \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)}}{x} \]

      5. distribute-rgt-out N/A

         \[\leadsto \frac{\color{blue}{\left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) \cdot \left(x + x \cdot \left(\frac{-1}{6} \cdot {x}^{2}\right)\right)}}{x} \]

      6. *-rgt-identity N/A

         \[\leadsto \frac{\left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) \cdot \left(\color{blue}{x \cdot 1} + x \cdot \left(\frac{-1}{6} \cdot {x}^{2}\right)\right)}{x} \]

      7. distribute-lft-in N/A

         \[\leadsto \frac{\left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) \cdot \color{blue}{\left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)}}{x} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)}}{x} \]

   8. Simplified 62.9%

      \[\leadsto \frac{\color{blue}{\left(y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)\right) \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right)\right)}}{x} \]

   9. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{y \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)} \]

       2. +-commutative N/A

          \[\leadsto y \cdot \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2} + 1\right)} \]

       3. accelerator-lowering-fma.f64 N/A

          \[\leadsto y \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{6}, {x}^{2}, 1\right)} \]

       4. unpow2 N/A

          \[\leadsto y \cdot \mathsf{fma}\left(\frac{-1}{6}, \color{blue}{x \cdot x}, 1\right) \]

       5. *-lowering-*.f64 36.5

          \[\leadsto y \cdot \mathsf{fma}\left(-0.16666666666666666, \color{blue}{x \cdot x}, 1\right) \]

   11. Simplified 36.5%

       \[\leadsto \color{blue}{y \cdot \mathsf{fma}\left(-0.16666666666666666, x \cdot x, 1\right)} \]

   if -4.9999999999999998e-214 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 87.1%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{x} \cdot \sinh y}{x} \]
   4. Step-by-step derivation

   5. Simplified 46.1%

      \[\leadsto \frac{\color{blue}{x} \cdot \sinh y}{x} \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{y \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)} \]

      2. +-commutative N/A

         \[\leadsto y \cdot \color{blue}{\left(\frac{1}{6} \cdot {y}^{2} + 1\right)} \]

      3. *-commutative N/A

         \[\leadsto y \cdot \left(\color{blue}{{y}^{2} \cdot \frac{1}{6}} + 1\right) \]

      4. unpow2 N/A

         \[\leadsto y \cdot \left(\color{blue}{\left(y \cdot y\right)} \cdot \frac{1}{6} + 1\right) \]

      5. associate-*l* N/A

         \[\leadsto y \cdot \left(\color{blue}{y \cdot \left(y \cdot \frac{1}{6}\right)} + 1\right) \]

      6. *-commutative N/A

         \[\leadsto y \cdot \left(y \cdot \color{blue}{\left(\frac{1}{6} \cdot y\right)} + 1\right) \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto y \cdot \color{blue}{\mathsf{fma}\left(y, \frac{1}{6} \cdot y, 1\right)} \]

      8. *-commutative N/A

         \[\leadsto y \cdot \mathsf{fma}\left(y, \color{blue}{y \cdot \frac{1}{6}}, 1\right) \]

      9. *-lowering-*.f64 51.4

         \[\leadsto y \cdot \mathsf{fma}\left(y, \color{blue}{y \cdot 0.16666666666666666}, 1\right) \]

   8. Simplified 51.4%

      \[\leadsto \color{blue}{y \cdot \mathsf{fma}\left(y, y \cdot 0.16666666666666666, 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 46.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -5 \cdot 10^{-214}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(-0.16666666666666666, x \cdot x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(y, y \cdot 0.16666666666666666, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 26.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -5 \cdot 10^{-214}:\\ \;\;\;\;y \cdot \left(x \cdot \left(x \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (sinh y) (sin x)) x) -5e-214)
   (* y (* x (* x -0.16666666666666666)))
   y))

C

double code(double x, double y) {
	double tmp;
	if (((sinh(y) * sin(x)) / x) <= -5e-214) {
		tmp = y * (x * (x * -0.16666666666666666));
	} else {
		tmp = y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (((sinh(y) * sin(x)) / x) <= (-5d-214)) then
        tmp = y * (x * (x * (-0.16666666666666666d0)))
    else
        tmp = y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (((Math.sinh(y) * Math.sin(x)) / x) <= -5e-214) {
		tmp = y * (x * (x * -0.16666666666666666));
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((math.sinh(y) * math.sin(x)) / x) <= -5e-214:
		tmp = y * (x * (x * -0.16666666666666666))
	else:
		tmp = y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(sinh(y) * sin(x)) / x) <= -5e-214)
		tmp = Float64(y * Float64(x * Float64(x * -0.16666666666666666)));
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((sinh(y) * sin(x)) / x) <= -5e-214)
		tmp = y * (x * (x * -0.16666666666666666));
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[Sinh[y], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], -5e-214], N[(y * N[(x * N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], y]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -4.9999999999999998e-214

   1. Initial program 98.6%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\sin x \cdot \color{blue}{\left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)}}{x} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \frac{\sin x \cdot \color{blue}{\left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)}}{x} \]

      2. +-commutative N/A

         \[\leadsto \frac{\sin x \cdot \left(y \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) + 1\right)}\right)}{x} \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\sin x \cdot \left(y \cdot \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{1}{6} + \frac{1}{120} \cdot {y}^{2}, 1\right)}\right)}{x} \]

      4. unpow2 N/A

         \[\leadsto \frac{\sin x \cdot \left(y \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{6} + \frac{1}{120} \cdot {y}^{2}, 1\right)\right)}{x} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{\sin x \cdot \left(y \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{6} + \frac{1}{120} \cdot {y}^{2}, 1\right)\right)}{x} \]

      6. +-commutative N/A

         \[\leadsto \frac{\sin x \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{\frac{1}{120} \cdot {y}^{2} + \frac{1}{6}}, 1\right)\right)}{x} \]

      7. *-commutative N/A

         \[\leadsto \frac{\sin x \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{{y}^{2} \cdot \frac{1}{120}} + \frac{1}{6}, 1\right)\right)}{x} \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\sin x \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{1}{120}, \frac{1}{6}\right)}, 1\right)\right)}{x} \]

      9. unpow2 N/A

         \[\leadsto \frac{\sin x \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{120}, \frac{1}{6}\right), 1\right)\right)}{x} \]

      10. *-lowering-*.f64 86.2

          \[\leadsto \frac{\sin x \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, 0.008333333333333333, 0.16666666666666666\right), 1\right)\right)}{x} \]

   5. Simplified 86.2%

      \[\leadsto \frac{\sin x \cdot \color{blue}{\left(y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), 1\right)\right)}}{x} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{x \cdot \left(\frac{-1}{6} \cdot \left({x}^{2} \cdot \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right) + y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)}}{x} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) + \frac{-1}{6} \cdot \left({x}^{2} \cdot \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right)\right)}}{x} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\color{blue}{x \cdot \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) + x \cdot \left(\frac{-1}{6} \cdot \left({x}^{2} \cdot \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right)\right)}}{x} \]

      3. associate-*r* N/A

         \[\leadsto \frac{x \cdot \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) + x \cdot \color{blue}{\left(\left(\frac{-1}{6} \cdot {x}^{2}\right) \cdot \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right)}}{x} \]

      4. associate-*r* N/A

         \[\leadsto \frac{x \cdot \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) + \color{blue}{\left(x \cdot \left(\frac{-1}{6} \cdot {x}^{2}\right)\right) \cdot \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)}}{x} \]

      5. distribute-rgt-out N/A

         \[\leadsto \frac{\color{blue}{\left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) \cdot \left(x + x \cdot \left(\frac{-1}{6} \cdot {x}^{2}\right)\right)}}{x} \]

      6. *-rgt-identity N/A

         \[\leadsto \frac{\left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) \cdot \left(\color{blue}{x \cdot 1} + x \cdot \left(\frac{-1}{6} \cdot {x}^{2}\right)\right)}{x} \]

      7. distribute-lft-in N/A

         \[\leadsto \frac{\left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) \cdot \color{blue}{\left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)}}{x} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)}}{x} \]

   8. Simplified 62.9%

      \[\leadsto \frac{\color{blue}{\left(y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)\right) \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right)\right)}}{x} \]

   9. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{y \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)} \]

       2. +-commutative N/A

          \[\leadsto y \cdot \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2} + 1\right)} \]

       3. accelerator-lowering-fma.f64 N/A

          \[\leadsto y \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{6}, {x}^{2}, 1\right)} \]

       4. unpow2 N/A

          \[\leadsto y \cdot \mathsf{fma}\left(\frac{-1}{6}, \color{blue}{x \cdot x}, 1\right) \]

       5. *-lowering-*.f64 36.5

          \[\leadsto y \cdot \mathsf{fma}\left(-0.16666666666666666, \color{blue}{x \cdot x}, 1\right) \]

   11. Simplified 36.5%

       \[\leadsto \color{blue}{y \cdot \mathsf{fma}\left(-0.16666666666666666, x \cdot x, 1\right)} \]

   12. Taylor expanded in x around inf

       \[\leadsto y \cdot \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2}\right)} \]
   13. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto y \cdot \color{blue}{\left({x}^{2} \cdot \frac{-1}{6}\right)} \]

       2. unpow2 N/A

          \[\leadsto y \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{-1}{6}\right) \]

       3. associate-*l* N/A

          \[\leadsto y \cdot \color{blue}{\left(x \cdot \left(x \cdot \frac{-1}{6}\right)\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto y \cdot \color{blue}{\left(x \cdot \left(x \cdot \frac{-1}{6}\right)\right)} \]

       5. *-lowering-*.f64 15.1

          \[\leadsto y \cdot \left(x \cdot \color{blue}{\left(x \cdot -0.16666666666666666\right)}\right) \]

   14. Simplified 15.1%

       \[\leadsto y \cdot \color{blue}{\left(x \cdot \left(x \cdot -0.16666666666666666\right)\right)} \]

   if -4.9999999999999998e-214 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 87.1%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{x} \cdot \sinh y}{x} \]
   4. Step-by-step derivation

   5. Simplified 46.1%

      \[\leadsto \frac{\color{blue}{x} \cdot \sinh y}{x} \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y} \]
   7. Step-by-step derivation

   8. Simplified 36.5%

      \[\leadsto \color{blue}{y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 29.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -5 \cdot 10^{-214}:\\ \;\;\;\;y \cdot \left(x \cdot \left(x \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sin x \cdot \frac{\sinh y}{x} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* (sin x) (/ (sinh y) x)))

C

double code(double x, double y) {
	return sin(x) * (sinh(y) / x);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = sin(x) * (sinh(y) / x)
end function

Java

public static double code(double x, double y) {
	return Math.sin(x) * (Math.sinh(y) / x);
}

Python

def code(x, y):
	return math.sin(x) * (math.sinh(y) / x)

Julia

function code(x, y)
	return Float64(sin(x) * Float64(sinh(y) / x))
end

MATLAB

function tmp = code(x, y)
	tmp = sin(x) * (sinh(y) / x);
end

Wolfram

code[x_, y_] := N[(N[Sin[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 90.9%

   \[\frac{\sin x \cdot \sinh y}{x} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. associate-/l* N/A

      \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\frac{\sinh y}{x} \cdot \sin x} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{\frac{\sinh y}{x} \cdot \sin x} \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\sinh y}{x}} \cdot \sin x \]

   5. sinh-lowering-sinh.f64 N/A

      \[\leadsto \frac{\color{blue}{\sinh y}}{x} \cdot \sin x \]

   6. sin-lowering-sin.f64 99.8

      \[\leadsto \frac{\sinh y}{x} \cdot \color{blue}{\sin x} \]

4. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{\frac{\sinh y}{x} \cdot \sin x} \]

5. Final simplification 99.8%

   \[\leadsto \sin x \cdot \frac{\sinh y}{x} \]
6. Add Preprocessing

Alternative 15: 79.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.6 \cdot 10^{-21}:\\ \;\;\;\;\sinh y\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin x \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 3.6e-21)
   (sinh y)
   (/
    (*
     (sin x)
     (*
      y
      (fma
       (* y y)
       (fma
        (* y y)
        (fma (* y y) 0.0001984126984126984 0.008333333333333333)
        0.16666666666666666)
       1.0)))
    x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 3.6e-21) {
		tmp = sinh(y);
	} else {
		tmp = (sin(x) * (y * fma((y * y), fma((y * y), fma((y * y), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666), 1.0))) / x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 3.6e-21)
		tmp = sinh(y);
	else
		tmp = Float64(Float64(sin(x) * Float64(y * fma(Float64(y * y), fma(Float64(y * y), fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666), 1.0))) / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 3.6e-21], N[Sinh[y], $MachinePrecision], N[(N[(N[Sin[x], $MachinePrecision] * N[(y * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.59999999999999989e-21

   1. Initial program 87.6%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{x} \cdot \sinh y}{x} \]
   4. Step-by-step derivation

   5. Simplified 58.9%

      \[\leadsto \frac{\color{blue}{x} \cdot \sinh y}{x} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sinh y \cdot x}}{x} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\sinh y \cdot \frac{x}{x}} \]

      3. *-inverses N/A

         \[\leadsto \sinh y \cdot \color{blue}{1} \]

      4. *-rgt-identity N/A

         \[\leadsto \color{blue}{\sinh y} \]

      5. sinh-lowering-sinh.f64 71.2

         \[\leadsto \color{blue}{\sinh y} \]

   7. Applied egg-rr 71.2%

      \[\leadsto \color{blue}{\sinh y} \]

   if 3.59999999999999989e-21 < x

   1. Initial program 99.8%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\sin x \cdot \color{blue}{\left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)}}{x} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \frac{\sin x \cdot \color{blue}{\left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)}}{x} \]

      2. +-commutative N/A

         \[\leadsto \frac{\sin x \cdot \left(y \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) + 1\right)}\right)}{x} \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\sin x \cdot \left(y \cdot \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right), 1\right)}\right)}{x} \]

      4. unpow2 N/A

         \[\leadsto \frac{\sin x \cdot \left(y \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right), 1\right)\right)}{x} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{\sin x \cdot \left(y \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right), 1\right)\right)}{x} \]

      6. +-commutative N/A

         \[\leadsto \frac{\sin x \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{{y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right) + \frac{1}{6}}, 1\right)\right)}{x} \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\sin x \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}, \frac{1}{6}\right)}, 1\right)\right)}{x} \]

      8. unpow2 N/A

         \[\leadsto \frac{\sin x \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}, \frac{1}{6}\right), 1\right)\right)}{x} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \frac{\sin x \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}, \frac{1}{6}\right), 1\right)\right)}{x} \]

      10. +-commutative N/A

          \[\leadsto \frac{\sin x \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \color{blue}{\frac{1}{5040} \cdot {y}^{2} + \frac{1}{120}}, \frac{1}{6}\right), 1\right)\right)}{x} \]

      11. *-commutative N/A

          \[\leadsto \frac{\sin x \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \color{blue}{{y}^{2} \cdot \frac{1}{5040}} + \frac{1}{120}, \frac{1}{6}\right), 1\right)\right)}{x} \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\sin x \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{1}{5040}, \frac{1}{120}\right)}, \frac{1}{6}\right), 1\right)\right)}{x} \]

      13. unpow2 N/A

          \[\leadsto \frac{\sin x \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right)\right)}{x} \]

      14. *-lowering-*.f64 98.5

          \[\leadsto \frac{\sin x \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)\right)}{x} \]

   5. Simplified 98.5%

      \[\leadsto \frac{\sin x \cdot \color{blue}{\left(y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)\right)}}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 36.6% accurate, 12.8× speedup

Math

\[\begin{array}{l} \\ y \cdot \mathsf{fma}\left(-0.16666666666666666, x \cdot x, 1\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* y (fma -0.16666666666666666 (* x x) 1.0)))

C

double code(double x, double y) {
	return y * fma(-0.16666666666666666, (x * x), 1.0);
}

Julia

function code(x, y)
	return Float64(y * fma(-0.16666666666666666, Float64(x * x), 1.0))
end

Wolfram

code[x_, y_] := N[(y * N[(-0.16666666666666666 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 90.9%

   \[\frac{\sin x \cdot \sinh y}{x} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \frac{\sin x \cdot \color{blue}{\left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)}}{x} \]
4. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \frac{\sin x \cdot \color{blue}{\left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)}}{x} \]

   2. +-commutative N/A

      \[\leadsto \frac{\sin x \cdot \left(y \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) + 1\right)}\right)}{x} \]

   3. accelerator-lowering-fma.f64 N/A

      \[\leadsto \frac{\sin x \cdot \left(y \cdot \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{1}{6} + \frac{1}{120} \cdot {y}^{2}, 1\right)}\right)}{x} \]

   4. unpow2 N/A

      \[\leadsto \frac{\sin x \cdot \left(y \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{6} + \frac{1}{120} \cdot {y}^{2}, 1\right)\right)}{x} \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \frac{\sin x \cdot \left(y \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{6} + \frac{1}{120} \cdot {y}^{2}, 1\right)\right)}{x} \]

   6. +-commutative N/A

      \[\leadsto \frac{\sin x \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{\frac{1}{120} \cdot {y}^{2} + \frac{1}{6}}, 1\right)\right)}{x} \]

   7. *-commutative N/A

      \[\leadsto \frac{\sin x \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{{y}^{2} \cdot \frac{1}{120}} + \frac{1}{6}, 1\right)\right)}{x} \]

   8. accelerator-lowering-fma.f64 N/A

      \[\leadsto \frac{\sin x \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{1}{120}, \frac{1}{6}\right)}, 1\right)\right)}{x} \]

   9. unpow2 N/A

      \[\leadsto \frac{\sin x \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{120}, \frac{1}{6}\right), 1\right)\right)}{x} \]

   10. *-lowering-*.f64 83.5

       \[\leadsto \frac{\sin x \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, 0.008333333333333333, 0.16666666666666666\right), 1\right)\right)}{x} \]

5. Simplified 83.5%

   \[\leadsto \frac{\sin x \cdot \color{blue}{\left(y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), 1\right)\right)}}{x} \]

6. Taylor expanded in x around 0

   \[\leadsto \frac{\color{blue}{x \cdot \left(\frac{-1}{6} \cdot \left({x}^{2} \cdot \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right) + y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)}}{x} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{x \cdot \color{blue}{\left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) + \frac{-1}{6} \cdot \left({x}^{2} \cdot \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right)\right)}}{x} \]

   2. distribute-lft-in N/A

      \[\leadsto \frac{\color{blue}{x \cdot \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) + x \cdot \left(\frac{-1}{6} \cdot \left({x}^{2} \cdot \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right)\right)}}{x} \]

   3. associate-*r* N/A

      \[\leadsto \frac{x \cdot \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) + x \cdot \color{blue}{\left(\left(\frac{-1}{6} \cdot {x}^{2}\right) \cdot \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right)}}{x} \]

   4. associate-*r* N/A

      \[\leadsto \frac{x \cdot \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) + \color{blue}{\left(x \cdot \left(\frac{-1}{6} \cdot {x}^{2}\right)\right) \cdot \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)}}{x} \]

   5. distribute-rgt-out N/A

      \[\leadsto \frac{\color{blue}{\left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) \cdot \left(x + x \cdot \left(\frac{-1}{6} \cdot {x}^{2}\right)\right)}}{x} \]

   6. *-rgt-identity N/A

      \[\leadsto \frac{\left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) \cdot \left(\color{blue}{x \cdot 1} + x \cdot \left(\frac{-1}{6} \cdot {x}^{2}\right)\right)}{x} \]

   7. distribute-lft-in N/A

      \[\leadsto \frac{\left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) \cdot \color{blue}{\left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)}}{x} \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)}}{x} \]

8. Simplified 49.1%

   \[\leadsto \frac{\color{blue}{\left(y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)\right) \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right)\right)}}{x} \]

9. Taylor expanded in y around 0

   \[\leadsto \color{blue}{y \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)} \]
10. Step-by-step derivation

    1. *-lowering-*.f64 N/A

       \[\leadsto \color{blue}{y \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)} \]

    2. +-commutative N/A

       \[\leadsto y \cdot \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2} + 1\right)} \]

    3. accelerator-lowering-fma.f64 N/A

       \[\leadsto y \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{6}, {x}^{2}, 1\right)} \]

    4. unpow2 N/A

       \[\leadsto y \cdot \mathsf{fma}\left(\frac{-1}{6}, \color{blue}{x \cdot x}, 1\right) \]

    5. *-lowering-*.f64 40.3

       \[\leadsto y \cdot \mathsf{fma}\left(-0.16666666666666666, \color{blue}{x \cdot x}, 1\right) \]

11. Simplified 40.3%

    \[\leadsto \color{blue}{y \cdot \mathsf{fma}\left(-0.16666666666666666, x \cdot x, 1\right)} \]
12. Add Preprocessing

Alternative 17: 28.1% accurate, 217.0× speedup

Math

\[\begin{array}{l} \\ y \end{array} \]

FPCore

(FPCore (x y) :precision binary64 y)

C

double code(double x, double y) {
	return y;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = y
end function

Java

public static double code(double x, double y) {
	return y;
}

Python

def code(x, y):
	return y

Julia

function code(x, y)
	return y
end

MATLAB

function tmp = code(x, y)
	tmp = y;
end

Wolfram

code[x_, y_] := y

Derivation

1. Initial program 90.9%

   \[\frac{\sin x \cdot \sinh y}{x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \frac{\color{blue}{x} \cdot \sinh y}{x} \]
4. Step-by-step derivation

5. Simplified 53.0%

   \[\leadsto \frac{\color{blue}{x} \cdot \sinh y}{x} \]

6. Taylor expanded in y around 0

   \[\leadsto \color{blue}{y} \]
7. Step-by-step derivation

8. Simplified 32.4%

   \[\leadsto \color{blue}{y} \]
9. Add Preprocessing

Developer Target 1: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sin x \cdot \frac{\sinh y}{x} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* (sin x) (/ (sinh y) x)))

C

double code(double x, double y) {
	return sin(x) * (sinh(y) / x);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = sin(x) * (sinh(y) / x)
end function

Java

public static double code(double x, double y) {
	return Math.sin(x) * (Math.sinh(y) / x);
}

Python

def code(x, y):
	return math.sin(x) * (math.sinh(y) / x)

Julia

function code(x, y)
	return Float64(sin(x) * Float64(sinh(y) / x))
end

MATLAB

function tmp = code(x, y)
	tmp = sin(x) * (sinh(y) / x);
end

Wolfram

code[x_, y_] := N[(N[Sin[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024196 
(FPCore (x y)
  :name "Linear.Quaternion:$ccosh from linear-1.19.1.3"
  :precision binary64

  :alt
  (! :herbie-platform default (* (sin x) (/ (sinh y) x)))

  (/ (* (sin x) (sinh y)) x))