Linear.Quaternion:$ccosh from linear-1.19.1.3

Percentage Accurate: 88.9% → 99.8%

Time: 13.3s

Alternatives: 17

Speedup: 1.4×

• 49.8% 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: 88.9% 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: 99.8% accurate, 1.0× speedup

Math

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

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(Float64(sinh(y) / x) * sin(x))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.6%

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

   1. lift-sin.f64 N/A

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

   2. lift-sinh.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-*l/ N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 99.9

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

4. Applied egg-rr 99.9%

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

Alternative 2: 84.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y \cdot \sin x}{x}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(y \cdot \left(0.008333333333333333 \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right) \cdot \mathsf{fma}\left(y, \left(x \cdot x\right) \cdot -0.16666666666666666, y\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sin x}{x} \cdot \mathsf{fma}\left(y \cdot y, y \cdot 0.16666666666666666, y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sinh y \cdot \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.008333333333333333, -0.16666666666666666\right), x \cdot \left(x \cdot x\right), x\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sinh y) (sin x)) x)))
   (if (<= t_0 (- INFINITY))
     (*
      (* y (* 0.008333333333333333 (* y (* y y))))
      (fma y (* (* x x) -0.16666666666666666) y))
     (if (<= t_0 5e-5)
       (* (/ (sin x) x) (fma (* y y) (* y 0.16666666666666666) y))
       (/
        (*
         (sinh y)
         (fma
          (fma (* x x) 0.008333333333333333 -0.16666666666666666)
          (* x (* x x))
          x))
        x)))))

C

double code(double x, double y) {
	double t_0 = (sinh(y) * sin(x)) / x;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (y * (0.008333333333333333 * (y * (y * y)))) * fma(y, ((x * x) * -0.16666666666666666), y);
	} else if (t_0 <= 5e-5) {
		tmp = (sin(x) / x) * fma((y * y), (y * 0.16666666666666666), y);
	} else {
		tmp = (sinh(y) * fma(fma((x * x), 0.008333333333333333, -0.16666666666666666), (x * (x * x)), x)) / x;
	}
	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(y * Float64(0.008333333333333333 * Float64(y * Float64(y * y)))) * fma(y, Float64(Float64(x * x) * -0.16666666666666666), y));
	elseif (t_0 <= 5e-5)
		tmp = Float64(Float64(sin(x) / x) * fma(Float64(y * y), Float64(y * 0.16666666666666666), y));
	else
		tmp = Float64(Float64(sinh(y) * fma(fma(Float64(x * x), 0.008333333333333333, -0.16666666666666666), Float64(x * Float64(x * x)), x)) / x);
	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[(y * N[(0.008333333333333333 * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y * N[(N[(x * x), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e-5], N[(N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(y * 0.16666666666666666), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sinh[y], $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision] * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision] / x), $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

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

      1. lower-*.f64 N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. distribute-lft-in N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. distribute-lft-in N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

   5. Simplified 90.8%

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

   6. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

   8. Simplified 79.1%

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

   9. Taylor expanded in y around inf

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

       1. metadata-eval N/A

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

       2. pow-sqr N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. unpow2 N/A

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

       6. associate-*l* N/A

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

       7. *-commutative N/A

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

       8. associate-*r* N/A

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

       9. unpow2 N/A

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

       10. unpow3 N/A

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

       11. lower-*.f64 N/A

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

       12. lower-*.f64 N/A

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

       13. cube-mult N/A

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

       14. unpow2 N/A

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

       15. lower-*.f64 N/A

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

       16. unpow2 N/A

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

       17. lower-*.f64 79.1

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

   11. Simplified 79.1%

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

   if -inf.0 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 5.00000000000000024e-5

   1. Initial program 77.7%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*r/ N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. associate-*r* N/A

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

      11. associate-/l* N/A

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

      12. distribute-rgt-out N/A

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

      13. *-rgt-identity N/A

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

      14. distribute-lft-in N/A

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

   5. Simplified 99.6%

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

   if 5.00000000000000024e-5 < (/.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

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. sub-neg N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 77.4

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

   5. Simplified 77.4%

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

4. Final simplification 90.1%

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

Alternative 3: 81.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y \cdot \sin x}{x}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(y \cdot \left(0.008333333333333333 \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right) \cdot \mathsf{fma}\left(y, \left(x \cdot x\right) \cdot -0.16666666666666666, y\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sin x}{x} \cdot \mathsf{fma}\left(y \cdot y, y \cdot 0.16666666666666666, y\right)\\ \mathbf{else}:\\ \;\;\;\;0.0001984126984126984 \cdot {y}^{7}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sinh y) (sin x)) x)))
   (if (<= t_0 (- INFINITY))
     (*
      (* y (* 0.008333333333333333 (* y (* y y))))
      (fma y (* (* x x) -0.16666666666666666) y))
     (if (<= t_0 5e-5)
       (* (/ (sin x) x) (fma (* y y) (* y 0.16666666666666666) y))
       (* 0.0001984126984126984 (pow y 7.0))))))

C

double code(double x, double y) {
	double t_0 = (sinh(y) * sin(x)) / x;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (y * (0.008333333333333333 * (y * (y * y)))) * fma(y, ((x * x) * -0.16666666666666666), y);
	} else if (t_0 <= 5e-5) {
		tmp = (sin(x) / x) * fma((y * y), (y * 0.16666666666666666), y);
	} else {
		tmp = 0.0001984126984126984 * pow(y, 7.0);
	}
	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(y * Float64(0.008333333333333333 * Float64(y * Float64(y * y)))) * fma(y, Float64(Float64(x * x) * -0.16666666666666666), y));
	elseif (t_0 <= 5e-5)
		tmp = Float64(Float64(sin(x) / x) * fma(Float64(y * y), Float64(y * 0.16666666666666666), y));
	else
		tmp = Float64(0.0001984126984126984 * (y ^ 7.0));
	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[(y * N[(0.008333333333333333 * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y * N[(N[(x * x), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e-5], N[(N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(y * 0.16666666666666666), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision], N[(0.0001984126984126984 * N[Power[y, 7.0], $MachinePrecision]), $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

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

      1. lower-*.f64 N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. distribute-lft-in N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. distribute-lft-in N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

   5. Simplified 90.8%

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

   6. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

   8. Simplified 79.1%

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

   9. Taylor expanded in y around inf

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

       1. metadata-eval N/A

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

       2. pow-sqr N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. unpow2 N/A

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

       6. associate-*l* N/A

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

       7. *-commutative N/A

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

       8. associate-*r* N/A

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

       9. unpow2 N/A

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

       10. unpow3 N/A

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

       11. lower-*.f64 N/A

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

       12. lower-*.f64 N/A

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

       13. cube-mult N/A

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

       14. unpow2 N/A

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

       15. lower-*.f64 N/A

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

       16. unpow2 N/A

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

       17. lower-*.f64 79.1

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

   11. Simplified 79.1%

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

   if -inf.0 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 5.00000000000000024e-5

   1. Initial program 77.7%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*r/ N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. associate-*r* N/A

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

      11. associate-/l* N/A

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

      12. distribute-rgt-out N/A

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

      13. *-rgt-identity N/A

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

      14. distribute-lft-in N/A

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

   5. Simplified 99.6%

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

   if 5.00000000000000024e-5 < (/.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. Step-by-step derivation

      1. lift-sin.f64 N/A

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

      2. lift-sinh.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l/ N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. unpow2 N/A

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

      6. unpow3 N/A

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

      7. *-lft-identity N/A

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

      8. lower-fma.f64 N/A

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

   7. Simplified 92.2%

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

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{y + {y}^{3} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)} \]
   9. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow3 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

   10. Simplified 68.5%

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

   11. Taylor expanded in y around inf

       \[\leadsto \color{blue}{\frac{1}{5040} \cdot {y}^{7}} \]
   12. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{5040} \cdot {y}^{7}} \]

       2. lower-pow.f64 69.9

          \[\leadsto 0.0001984126984126984 \cdot \color{blue}{{y}^{7}} \]

   13. Simplified 69.9%

       \[\leadsto \color{blue}{0.0001984126984126984 \cdot {y}^{7}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -\infty:\\ \;\;\;\;\left(y \cdot \left(0.008333333333333333 \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right) \cdot \mathsf{fma}\left(y, \left(x \cdot x\right) \cdot -0.16666666666666666, y\right)\\ \mathbf{elif}\;\frac{\sinh y \cdot \sin x}{x} \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sin x}{x} \cdot \mathsf{fma}\left(y \cdot y, y \cdot 0.16666666666666666, y\right)\\ \mathbf{else}:\\ \;\;\;\;0.0001984126984126984 \cdot {y}^{7}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y \cdot \sin x}{x}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(y \cdot \left(0.008333333333333333 \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right) \cdot \mathsf{fma}\left(y, \left(x \cdot x\right) \cdot -0.16666666666666666, y\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-5}:\\ \;\;\;\;y \cdot \frac{\sin x}{x}\\ \mathbf{else}:\\ \;\;\;\;0.0001984126984126984 \cdot {y}^{7}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sinh y) (sin x)) x)))
   (if (<= t_0 (- INFINITY))
     (*
      (* y (* 0.008333333333333333 (* y (* y y))))
      (fma y (* (* x x) -0.16666666666666666) y))
     (if (<= t_0 5e-5)
       (* y (/ (sin x) x))
       (* 0.0001984126984126984 (pow y 7.0))))))

C

double code(double x, double y) {
	double t_0 = (sinh(y) * sin(x)) / x;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (y * (0.008333333333333333 * (y * (y * y)))) * fma(y, ((x * x) * -0.16666666666666666), y);
	} else if (t_0 <= 5e-5) {
		tmp = y * (sin(x) / x);
	} else {
		tmp = 0.0001984126984126984 * pow(y, 7.0);
	}
	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(y * Float64(0.008333333333333333 * Float64(y * Float64(y * y)))) * fma(y, Float64(Float64(x * x) * -0.16666666666666666), y));
	elseif (t_0 <= 5e-5)
		tmp = Float64(y * Float64(sin(x) / x));
	else
		tmp = Float64(0.0001984126984126984 * (y ^ 7.0));
	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[(y * N[(0.008333333333333333 * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y * N[(N[(x * x), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e-5], N[(y * N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(0.0001984126984126984 * N[Power[y, 7.0], $MachinePrecision]), $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

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

      1. lower-*.f64 N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. distribute-lft-in N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. distribute-lft-in N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

   5. Simplified 90.8%

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

   6. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

   8. Simplified 79.1%

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

   9. Taylor expanded in y around inf

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

       1. metadata-eval N/A

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

       2. pow-sqr N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. unpow2 N/A

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

       6. associate-*l* N/A

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

       7. *-commutative N/A

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

       8. associate-*r* N/A

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

       9. unpow2 N/A

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

       10. unpow3 N/A

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

       11. lower-*.f64 N/A

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

       12. lower-*.f64 N/A

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

       13. cube-mult N/A

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

       14. unpow2 N/A

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

       15. lower-*.f64 N/A

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

       16. unpow2 N/A

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

       17. lower-*.f64 79.1

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

   11. Simplified 79.1%

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

   if -inf.0 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 5.00000000000000024e-5

   1. Initial program 77.7%

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

   3. Taylor expanded in y around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sin.f64 99.2

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

   5. Simplified 99.2%

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

   if 5.00000000000000024e-5 < (/.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. Step-by-step derivation

      1. lift-sin.f64 N/A

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

      2. lift-sinh.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l/ N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. unpow2 N/A

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

      6. unpow3 N/A

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

      7. *-lft-identity N/A

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

      8. lower-fma.f64 N/A

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

   7. Simplified 92.2%

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

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{y + {y}^{3} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)} \]
   9. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow3 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

   10. Simplified 68.5%

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

   11. Taylor expanded in y around inf

       \[\leadsto \color{blue}{\frac{1}{5040} \cdot {y}^{7}} \]
   12. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{5040} \cdot {y}^{7}} \]

       2. lower-pow.f64 69.9

          \[\leadsto 0.0001984126984126984 \cdot \color{blue}{{y}^{7}} \]

   13. Simplified 69.9%

       \[\leadsto \color{blue}{0.0001984126984126984 \cdot {y}^{7}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -\infty:\\ \;\;\;\;\left(y \cdot \left(0.008333333333333333 \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right) \cdot \mathsf{fma}\left(y, \left(x \cdot x\right) \cdot -0.16666666666666666, y\right)\\ \mathbf{elif}\;\frac{\sinh y \cdot \sin x}{x} \leq 5 \cdot 10^{-5}:\\ \;\;\;\;y \cdot \frac{\sin x}{x}\\ \mathbf{else}:\\ \;\;\;\;0.0001984126984126984 \cdot {y}^{7}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 90.9% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 5.00000000000000024e-5

   1. Initial program 83.7%

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

      1. lift-sin.f64 N/A

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

      2. lift-sinh.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l/ N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. unpow2 N/A

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

      6. unpow3 N/A

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

      7. *-lft-identity N/A

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

      8. lower-fma.f64 N/A

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

   7. Simplified 98.4%

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

   if 5.00000000000000024e-5 < (/.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

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. sub-neg N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 77.4

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

   5. Simplified 77.4%

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

4. Final simplification 93.3%

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

Alternative 6: 52.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -1 \cdot 10^{-322}:\\ \;\;\;\;\left(y \cdot \left(0.008333333333333333 \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right) \cdot \mathsf{fma}\left(y, \left(x \cdot x\right) \cdot -0.16666666666666666, y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (sinh y) (sin x)) x) -1e-322)
   (*
    (* y (* 0.008333333333333333 (* y (* y y))))
    (fma y (* (* x x) -0.16666666666666666) y))
   (fma
    (* y y)
    (*
     y
     (fma
      (* y y)
      (fma (* y y) 0.0001984126984126984 0.008333333333333333)
      0.16666666666666666))
    y)))

C

double code(double x, double y) {
	double tmp;
	if (((sinh(y) * sin(x)) / x) <= -1e-322) {
		tmp = (y * (0.008333333333333333 * (y * (y * y)))) * fma(y, ((x * x) * -0.16666666666666666), y);
	} else {
		tmp = fma((y * y), (y * fma((y * y), fma((y * y), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666)), y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(sinh(y) * sin(x)) / x) <= -1e-322)
		tmp = Float64(Float64(y * Float64(0.008333333333333333 * Float64(y * Float64(y * y)))) * fma(y, Float64(Float64(x * x) * -0.16666666666666666), y));
	else
		tmp = fma(Float64(y * y), Float64(y * fma(Float64(y * y), fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666)), y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[Sinh[y], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], -1e-322], N[(N[(y * N[(0.008333333333333333 * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y * N[(N[(x * x), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision], N[(N[(y * y), $MachinePrecision] * N[(y * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -9.88131e-323

   1. Initial program 98.3%

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

   3. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. distribute-lft-in N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. distribute-lft-in N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

   5. Simplified 93.1%

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

   6. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

   8. Simplified 67.1%

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

   9. Taylor expanded in y around inf

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

       1. metadata-eval N/A

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

       2. pow-sqr N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. unpow2 N/A

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

       6. associate-*l* N/A

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

       7. *-commutative N/A

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

       8. associate-*r* N/A

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

       9. unpow2 N/A

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

       10. unpow3 N/A

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

       11. lower-*.f64 N/A

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

       12. lower-*.f64 N/A

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

       13. cube-mult N/A

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

       14. unpow2 N/A

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

       15. lower-*.f64 N/A

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

       16. unpow2 N/A

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

       17. lower-*.f64 46.8

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

   11. Simplified 46.8%

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

   if -9.88131e-323 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 81.6%

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

      1. lift-sin.f64 N/A

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

      2. lift-sinh.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l/ N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. unpow2 N/A

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

      6. unpow3 N/A

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

      7. *-lft-identity N/A

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

      8. lower-fma.f64 N/A

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

   7. Simplified 97.0%

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

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{y + {y}^{3} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)} \]
   9. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow3 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

   10. Simplified 60.2%

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

4. Final simplification 55.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -1 \cdot 10^{-322}:\\ \;\;\;\;\left(y \cdot \left(0.008333333333333333 \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right) \cdot \mathsf{fma}\left(y, \left(x \cdot x\right) \cdot -0.16666666666666666, y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 56.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -2 \cdot 10^{-237}:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(y, \left(y \cdot y\right) \cdot 0.16666666666666666, y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (sinh y) (sin x)) x) -2e-237)
   (*
    (fma x (* x -0.16666666666666666) 1.0)
    (fma y (* (* y y) 0.16666666666666666) y))
   (fma
    (* y y)
    (*
     y
     (fma
      (* y y)
      (fma (* y y) 0.0001984126984126984 0.008333333333333333)
      0.16666666666666666))
    y)))

C

double code(double x, double y) {
	double tmp;
	if (((sinh(y) * sin(x)) / x) <= -2e-237) {
		tmp = fma(x, (x * -0.16666666666666666), 1.0) * fma(y, ((y * y) * 0.16666666666666666), y);
	} else {
		tmp = fma((y * y), (y * fma((y * y), fma((y * y), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666)), y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(sinh(y) * sin(x)) / x) <= -2e-237)
		tmp = Float64(fma(x, Float64(x * -0.16666666666666666), 1.0) * fma(y, Float64(Float64(y * y) * 0.16666666666666666), y));
	else
		tmp = fma(Float64(y * y), Float64(y * fma(Float64(y * y), fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666)), y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[Sinh[y], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], -2e-237], N[(N[(x * N[(x * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * N[(y * N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision], N[(N[(y * y), $MachinePrecision] * N[(y * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -2e-237

   1. Initial program 99.1%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-lft-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 78.0

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

   5. Simplified 78.0%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. pow-plus N/A

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

      7. metadata-eval N/A

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

      8. cube-unmult N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. sub-neg N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 60.6

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

   8. Simplified 60.6%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. associate-+r+ N/A

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

       3. associate-*r* N/A

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

       4. distribute-rgt1-in N/A

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

       5. +-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. +-commutative N/A

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

       8. *-commutative N/A

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

       9. unpow2 N/A

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

       10. associate-*l* N/A

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

       11. *-commutative N/A

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

       12. lower-fma.f64 N/A

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

       13. *-commutative N/A

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

       14. lower-*.f64 N/A

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

       15. +-commutative N/A

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

       16. unpow3 N/A

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

       17. unpow2 N/A

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

       18. associate-*r* N/A

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

       19. *-commutative N/A

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

       20. lower-fma.f64 N/A

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

   11. Simplified 65.4%

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

   if -2e-237 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 82.3%

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

      1. lift-sin.f64 N/A

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

      2. lift-sinh.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l/ N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. unpow2 N/A

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

      6. unpow3 N/A

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

      7. *-lft-identity N/A

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

      8. lower-fma.f64 N/A

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

   7. Simplified 97.1%

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

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{y + {y}^{3} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)} \]
   9. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow3 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

   10. Simplified 57.8%

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

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -2 \cdot 10^{-237}:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(y, \left(y \cdot y\right) \cdot 0.16666666666666666, y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 56.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -2 \cdot 10^{-237}:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(y, \left(y \cdot y\right) \cdot 0.16666666666666666, y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, y \cdot \mathsf{fma}\left(y \cdot y, \left(y \cdot y\right) \cdot 0.0001984126984126984, 0.16666666666666666\right), y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (sinh y) (sin x)) x) -2e-237)
   (*
    (fma x (* x -0.16666666666666666) 1.0)
    (fma y (* (* y y) 0.16666666666666666) y))
   (fma
    (* y y)
    (* y (fma (* y y) (* (* y y) 0.0001984126984126984) 0.16666666666666666))
    y)))

C

double code(double x, double y) {
	double tmp;
	if (((sinh(y) * sin(x)) / x) <= -2e-237) {
		tmp = fma(x, (x * -0.16666666666666666), 1.0) * fma(y, ((y * y) * 0.16666666666666666), y);
	} else {
		tmp = fma((y * y), (y * fma((y * y), ((y * y) * 0.0001984126984126984), 0.16666666666666666)), y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(sinh(y) * sin(x)) / x) <= -2e-237)
		tmp = Float64(fma(x, Float64(x * -0.16666666666666666), 1.0) * fma(y, Float64(Float64(y * y) * 0.16666666666666666), y));
	else
		tmp = fma(Float64(y * y), Float64(y * fma(Float64(y * y), Float64(Float64(y * y) * 0.0001984126984126984), 0.16666666666666666)), y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[Sinh[y], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], -2e-237], N[(N[(x * N[(x * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * N[(y * N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision], N[(N[(y * y), $MachinePrecision] * N[(y * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -2e-237

   1. Initial program 99.1%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-lft-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 78.0

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

   5. Simplified 78.0%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. pow-plus N/A

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

      7. metadata-eval N/A

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

      8. cube-unmult N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. sub-neg N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 60.6

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

   8. Simplified 60.6%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. associate-+r+ N/A

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

       3. associate-*r* N/A

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

       4. distribute-rgt1-in N/A

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

       5. +-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. +-commutative N/A

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

       8. *-commutative N/A

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

       9. unpow2 N/A

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

       10. associate-*l* N/A

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

       11. *-commutative N/A

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

       12. lower-fma.f64 N/A

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

       13. *-commutative N/A

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

       14. lower-*.f64 N/A

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

       15. +-commutative N/A

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

       16. unpow3 N/A

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

       17. unpow2 N/A

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

       18. associate-*r* N/A

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

       19. *-commutative N/A

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

       20. lower-fma.f64 N/A

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

   11. Simplified 65.4%

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

   if -2e-237 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 82.3%

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

      1. lift-sin.f64 N/A

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

      2. lift-sinh.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l/ N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. unpow2 N/A

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

      6. unpow3 N/A

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

      7. *-lft-identity N/A

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

      8. lower-fma.f64 N/A

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

   7. Simplified 97.1%

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

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{y + {y}^{3} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)} \]
   9. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow3 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

   10. Simplified 57.8%

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

   11. Taylor expanded in y around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 57.8

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

   13. Simplified 57.8%

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

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -2 \cdot 10^{-237}:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(y, \left(y \cdot y\right) \cdot 0.16666666666666666, y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, y \cdot \mathsf{fma}\left(y \cdot y, \left(y \cdot y\right) \cdot 0.0001984126984126984, 0.16666666666666666\right), y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 56.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -2 \cdot 10^{-237}:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(y, \left(y \cdot y\right) \cdot 0.16666666666666666, y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, y \cdot \left(y \cdot \left(y \cdot \left(\left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right)\right), y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (sinh y) (sin x)) x) -2e-237)
   (*
    (fma x (* x -0.16666666666666666) 1.0)
    (fma y (* (* y y) 0.16666666666666666) y))
   (fma (* y y) (* y (* y (* y (* (* y y) 0.0001984126984126984)))) y)))

C

double code(double x, double y) {
	double tmp;
	if (((sinh(y) * sin(x)) / x) <= -2e-237) {
		tmp = fma(x, (x * -0.16666666666666666), 1.0) * fma(y, ((y * y) * 0.16666666666666666), y);
	} else {
		tmp = fma((y * y), (y * (y * (y * ((y * y) * 0.0001984126984126984)))), y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(sinh(y) * sin(x)) / x) <= -2e-237)
		tmp = Float64(fma(x, Float64(x * -0.16666666666666666), 1.0) * fma(y, Float64(Float64(y * y) * 0.16666666666666666), y));
	else
		tmp = fma(Float64(y * y), Float64(y * Float64(y * Float64(y * Float64(Float64(y * y) * 0.0001984126984126984)))), y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[Sinh[y], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], -2e-237], N[(N[(x * N[(x * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * N[(y * N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision], N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -2e-237

   1. Initial program 99.1%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-lft-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 78.0

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

   5. Simplified 78.0%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. pow-plus N/A

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

      7. metadata-eval N/A

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

      8. cube-unmult N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. sub-neg N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 60.6

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

   8. Simplified 60.6%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. associate-+r+ N/A

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

       3. associate-*r* N/A

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

       4. distribute-rgt1-in N/A

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

       5. +-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. +-commutative N/A

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

       8. *-commutative N/A

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

       9. unpow2 N/A

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

       10. associate-*l* N/A

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

       11. *-commutative N/A

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

       12. lower-fma.f64 N/A

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

       13. *-commutative N/A

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

       14. lower-*.f64 N/A

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

       15. +-commutative N/A

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

       16. unpow3 N/A

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

       17. unpow2 N/A

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

       18. associate-*r* N/A

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

       19. *-commutative N/A

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

       20. lower-fma.f64 N/A

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

   11. Simplified 65.4%

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

   if -2e-237 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 82.3%

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

      1. lift-sin.f64 N/A

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

      2. lift-sinh.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l/ N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. unpow2 N/A

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

      6. unpow3 N/A

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

      7. *-lft-identity N/A

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

      8. lower-fma.f64 N/A

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

   7. Simplified 97.1%

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

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{y + {y}^{3} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)} \]
   9. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow3 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

   10. Simplified 57.8%

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

   11. Taylor expanded in y around inf

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

       1. metadata-eval N/A

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

       2. pow-sqr N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. unpow2 N/A

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

       6. associate-*l* N/A

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

       7. lower-*.f64 N/A

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

       8. lower-*.f64 N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 N/A

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

       11. unpow2 N/A

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

       12. lower-*.f64 57.6

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

   13. Simplified 57.6%

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

4. Final simplification 60.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -2 \cdot 10^{-237}:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(y, \left(y \cdot y\right) \cdot 0.16666666666666666, y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, y \cdot \left(y \cdot \left(y \cdot \left(\left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right)\right), y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 55.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -2 \cdot 10^{-237}:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(y, \left(y \cdot y\right) \cdot 0.16666666666666666, y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, y \cdot \mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (sinh y) (sin x)) x) -2e-237)
   (*
    (fma x (* x -0.16666666666666666) 1.0)
    (fma y (* (* y y) 0.16666666666666666) y))
   (fma
    (* y y)
    (* y (fma (* y y) 0.008333333333333333 0.16666666666666666))
    y)))

C

double code(double x, double y) {
	double tmp;
	if (((sinh(y) * sin(x)) / x) <= -2e-237) {
		tmp = fma(x, (x * -0.16666666666666666), 1.0) * fma(y, ((y * y) * 0.16666666666666666), y);
	} else {
		tmp = fma((y * y), (y * fma((y * y), 0.008333333333333333, 0.16666666666666666)), y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(sinh(y) * sin(x)) / x) <= -2e-237)
		tmp = Float64(fma(x, Float64(x * -0.16666666666666666), 1.0) * fma(y, Float64(Float64(y * y) * 0.16666666666666666), y));
	else
		tmp = fma(Float64(y * y), Float64(y * fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666)), y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[Sinh[y], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], -2e-237], N[(N[(x * N[(x * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * N[(y * N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision], N[(N[(y * y), $MachinePrecision] * N[(y * N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -2e-237

   1. Initial program 99.1%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-lft-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 78.0

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

   5. Simplified 78.0%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. pow-plus N/A

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

      7. metadata-eval N/A

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

      8. cube-unmult N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. sub-neg N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 60.6

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

   8. Simplified 60.6%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. associate-+r+ N/A

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

       3. associate-*r* N/A

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

       4. distribute-rgt1-in N/A

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

       5. +-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. +-commutative N/A

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

       8. *-commutative N/A

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

       9. unpow2 N/A

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

       10. associate-*l* N/A

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

       11. *-commutative N/A

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

       12. lower-fma.f64 N/A

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

       13. *-commutative N/A

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

       14. lower-*.f64 N/A

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

       15. +-commutative N/A

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

       16. unpow3 N/A

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

       17. unpow2 N/A

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

       18. associate-*r* N/A

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

       19. *-commutative N/A

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

       20. lower-fma.f64 N/A

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

   11. Simplified 65.4%

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

   if -2e-237 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 82.3%

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

      1. lift-sin.f64 N/A

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

      2. lift-sinh.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l/ N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. unpow2 N/A

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

      6. unpow3 N/A

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

      7. *-lft-identity N/A

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

      8. lower-fma.f64 N/A

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

   7. Simplified 97.1%

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

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{y + {y}^{3} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)} \]
   9. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow3 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

   10. Simplified 57.8%

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

   11. Taylor expanded in y around 0

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

   13. Simplified 55.0%

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

4. Final simplification 58.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -2 \cdot 10^{-237}:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(y, \left(y \cdot y\right) \cdot 0.16666666666666666, y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, y \cdot \mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 52.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq 10^{-117}:\\ \;\;\;\;\frac{y}{x} \cdot \mathsf{fma}\left(-0.16666666666666666, x \cdot \left(x \cdot x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, y \cdot \mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (sinh y) (sin x)) x) 1e-117)
   (* (/ y x) (fma -0.16666666666666666 (* x (* x x)) x))
   (fma
    (* y y)
    (* y (fma (* y y) 0.008333333333333333 0.16666666666666666))
    y)))

C

double code(double x, double y) {
	double tmp;
	if (((sinh(y) * sin(x)) / x) <= 1e-117) {
		tmp = (y / x) * fma(-0.16666666666666666, (x * (x * x)), x);
	} else {
		tmp = fma((y * y), (y * fma((y * y), 0.008333333333333333, 0.16666666666666666)), y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(sinh(y) * sin(x)) / x) <= 1e-117)
		tmp = Float64(Float64(y / x) * fma(-0.16666666666666666, Float64(x * Float64(x * x)), x));
	else
		tmp = fma(Float64(y * y), Float64(y * fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666)), y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[Sinh[y], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], 1e-117], N[(N[(y / x), $MachinePrecision] * N[(-0.16666666666666666 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], N[(N[(y * y), $MachinePrecision] * N[(y * N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 1.00000000000000003e-117

   1. Initial program 82.1%

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

      1. lift-sin.f64 N/A

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

      2. lift-sinh.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l/ N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in y around 0

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

      1. lower-/.f64 79.4

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

   7. Simplified 79.4%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-*l* N/A

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

      4. pow-plus N/A

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

      5. metadata-eval N/A

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

      6. cube-unmult N/A

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

      7. unpow2 N/A

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

      8. *-lft-identity N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 48.8

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

   10. Simplified 48.8%

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

   if 1.00000000000000003e-117 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 99.8%

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

      1. lift-sin.f64 N/A

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

      2. lift-sinh.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l/ N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. unpow2 N/A

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

      6. unpow3 N/A

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

      7. *-lft-identity N/A

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

      8. lower-fma.f64 N/A

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

   7. Simplified 93.9%

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

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{y + {y}^{3} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)} \]
   9. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow3 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

   10. Simplified 73.0%

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

   11. Taylor expanded in y around 0

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

   13. Simplified 67.0%

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

4. Final simplification 54.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq 10^{-117}:\\ \;\;\;\;\frac{y}{x} \cdot \mathsf{fma}\left(-0.16666666666666666, x \cdot \left(x \cdot x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, y \cdot \mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 46.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -2 \cdot 10^{-237}:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot \left(y \cdot -0.16666666666666666\right), y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, y \cdot \mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (sinh y) (sin x)) x) -2e-237)
   (fma x (* x (* y -0.16666666666666666)) y)
   (fma
    (* y y)
    (* y (fma (* y y) 0.008333333333333333 0.16666666666666666))
    y)))

C

double code(double x, double y) {
	double tmp;
	if (((sinh(y) * sin(x)) / x) <= -2e-237) {
		tmp = fma(x, (x * (y * -0.16666666666666666)), y);
	} else {
		tmp = fma((y * y), (y * fma((y * y), 0.008333333333333333, 0.16666666666666666)), y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(sinh(y) * sin(x)) / x) <= -2e-237)
		tmp = fma(x, Float64(x * Float64(y * -0.16666666666666666)), y);
	else
		tmp = fma(Float64(y * y), Float64(y * fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666)), y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[Sinh[y], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], -2e-237], N[(x * N[(x * N[(y * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision], N[(N[(y * y), $MachinePrecision] * N[(y * N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -2e-237

   1. Initial program 99.1%

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

   3. Taylor expanded in y around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sin.f64 38.8

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

   5. Simplified 38.8%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-out N/A

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

      8. +-commutative N/A

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

      9. metadata-eval N/A

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

      10. sub-neg N/A

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

      11. lower-*.f64 N/A

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

      12. sub-neg N/A

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

      13. metadata-eval N/A

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

      14. lower-fma.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 32.4

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

   8. Simplified 32.4%

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

   9. Taylor expanded in x around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 33.9

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

   11. Simplified 33.9%

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

   if -2e-237 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 82.3%

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

      1. lift-sin.f64 N/A

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

      2. lift-sinh.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l/ N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. unpow2 N/A

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

      6. unpow3 N/A

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

      7. *-lft-identity N/A

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

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right), {y}^{3}, y\right)}}{x} \cdot \sin x \]

   7. Simplified 97.1%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)}}{x} \cdot \sin x \]

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{y + {y}^{3} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{y}^{3} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) + y} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot {y}^{3}} + y \]

      3. unpow3 N/A

         \[\leadsto \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot \color{blue}{\left(\left(y \cdot y\right) \cdot y\right)} + y \]

      4. unpow2 N/A

         \[\leadsto \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot \left(\color{blue}{{y}^{2}} \cdot y\right) + y \]

      5. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot {y}^{2}\right) \cdot y} + y \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)} \cdot y + y \]

      7. associate-*l* N/A

         \[\leadsto \color{blue}{{y}^{2} \cdot \left(\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot y\right)} + y \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({y}^{2}, \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot y, y\right)} \]

   10. Simplified 57.8%

       \[\leadsto \color{blue}{\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right) \cdot y, y\right)} \]

   11. Taylor expanded in y around 0

       \[\leadsto \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \color{blue}{\frac{1}{120}}, \frac{1}{6}\right) \cdot y, y\right) \]
   12. Step-by-step derivation

   13. Simplified 55.0%

       \[\leadsto \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \color{blue}{0.008333333333333333}, 0.16666666666666666\right) \cdot y, y\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 48.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -2 \cdot 10^{-237}:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot \left(y \cdot -0.16666666666666666\right), y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, y \cdot \mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 44.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -2 \cdot 10^{-237}:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot \left(y \cdot -0.16666666666666666\right), y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \left(y \cdot y\right) \cdot 0.16666666666666666, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (sinh y) (sin x)) x) -2e-237)
   (fma x (* x (* y -0.16666666666666666)) y)
   (fma y (* (* y y) 0.16666666666666666) y)))

C

double code(double x, double y) {
	double tmp;
	if (((sinh(y) * sin(x)) / x) <= -2e-237) {
		tmp = fma(x, (x * (y * -0.16666666666666666)), y);
	} else {
		tmp = fma(y, ((y * y) * 0.16666666666666666), y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(sinh(y) * sin(x)) / x) <= -2e-237)
		tmp = fma(x, Float64(x * Float64(y * -0.16666666666666666)), y);
	else
		tmp = fma(y, Float64(Float64(y * y) * 0.16666666666666666), y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[Sinh[y], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], -2e-237], N[(x * N[(x * N[(y * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision], N[(y * N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -2e-237

   1. Initial program 99.1%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y \cdot \sin x}{x}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{y \cdot \frac{\sin x}{x}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{y \cdot \frac{\sin x}{x}} \]

      3. lower-/.f64 N/A

         \[\leadsto y \cdot \color{blue}{\frac{\sin x}{x}} \]

      4. lower-sin.f64 38.8

         \[\leadsto y \cdot \frac{\color{blue}{\sin x}}{x} \]

   5. Simplified 38.8%

      \[\leadsto \color{blue}{y \cdot \frac{\sin x}{x}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{y + {x}^{2} \cdot \left(\frac{-1}{6} \cdot y + \frac{1}{120} \cdot \left({x}^{2} \cdot y\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{-1}{6} \cdot y + \frac{1}{120} \cdot \left({x}^{2} \cdot y\right)\right) + y} \]

      2. unpow2 N/A

         \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{-1}{6} \cdot y + \frac{1}{120} \cdot \left({x}^{2} \cdot y\right)\right) + y \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(\frac{-1}{6} \cdot y + \frac{1}{120} \cdot \left({x}^{2} \cdot y\right)\right)\right)} + y \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot \left(\frac{-1}{6} \cdot y + \frac{1}{120} \cdot \left({x}^{2} \cdot y\right)\right), y\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{-1}{6} \cdot y + \frac{1}{120} \cdot \left({x}^{2} \cdot y\right)\right)}, y\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(x, x \cdot \left(\frac{-1}{6} \cdot y + \color{blue}{\left(\frac{1}{120} \cdot {x}^{2}\right) \cdot y}\right), y\right) \]

      7. distribute-rgt-out N/A

         \[\leadsto \mathsf{fma}\left(x, x \cdot \color{blue}{\left(y \cdot \left(\frac{-1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right)}, y\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, x \cdot \left(y \cdot \color{blue}{\left(\frac{1}{120} \cdot {x}^{2} + \frac{-1}{6}\right)}\right), y\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(x, x \cdot \left(y \cdot \left(\frac{1}{120} \cdot {x}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}\right)\right), y\right) \]

      10. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(x, x \cdot \left(y \cdot \color{blue}{\left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)}\right), y\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x, x \cdot \color{blue}{\left(y \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right)}, y\right) \]

      12. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(x, x \cdot \left(y \cdot \color{blue}{\left(\frac{1}{120} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)}\right), y\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(x, x \cdot \left(y \cdot \left(\frac{1}{120} \cdot {x}^{2} + \color{blue}{\frac{-1}{6}}\right)\right), y\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(x, x \cdot \left(y \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{120}, {x}^{2}, \frac{-1}{6}\right)}\right), y\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(x, x \cdot \left(y \cdot \mathsf{fma}\left(\frac{1}{120}, \color{blue}{x \cdot x}, \frac{-1}{6}\right)\right), y\right) \]

      16. lower-*.f64 32.4

          \[\leadsto \mathsf{fma}\left(x, x \cdot \left(y \cdot \mathsf{fma}\left(0.008333333333333333, \color{blue}{x \cdot x}, -0.16666666666666666\right)\right), y\right) \]

   8. Simplified 32.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot \left(y \cdot \mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right)\right), y\right)} \]

   9. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(x, x \cdot \color{blue}{\left(\frac{-1}{6} \cdot y\right)}, y\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x, x \cdot \color{blue}{\left(y \cdot \frac{-1}{6}\right)}, y\right) \]

       2. lower-*.f64 33.9

          \[\leadsto \mathsf{fma}\left(x, x \cdot \color{blue}{\left(y \cdot -0.16666666666666666\right)}, y\right) \]

   11. Simplified 33.9%

       \[\leadsto \mathsf{fma}\left(x, x \cdot \color{blue}{\left(y \cdot -0.16666666666666666\right)}, y\right) \]

   if -2e-237 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 82.3%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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. +-commutative N/A

         \[\leadsto y \cdot \color{blue}{\left(\frac{\sin x}{x} + \frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x}\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{y \cdot \frac{\sin x}{x} + y \cdot \left(\frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x}\right)} \]

      3. associate-*r/ N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + y \cdot \color{blue}{\frac{\frac{1}{6} \cdot \left({y}^{2} \cdot \sin x\right)}{x}} \]

      4. *-commutative N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + y \cdot \frac{\frac{1}{6} \cdot \color{blue}{\left(\sin x \cdot {y}^{2}\right)}}{x} \]

      5. associate-*r* N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + y \cdot \frac{\color{blue}{\left(\frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}}}{x} \]

      6. associate-*r/ N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + \color{blue}{\frac{y \cdot \left(\left(\frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}\right)}{x}} \]

      7. associate-*r* N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + \frac{y \cdot \color{blue}{\left(\frac{1}{6} \cdot \left(\sin x \cdot {y}^{2}\right)\right)}}{x} \]

      8. *-commutative N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + \frac{y \cdot \left(\frac{1}{6} \cdot \color{blue}{\left({y}^{2} \cdot \sin x\right)}\right)}{x} \]

      9. associate-*r* N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + \frac{y \cdot \color{blue}{\left(\left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \sin x\right)}}{x} \]

      10. associate-*r* N/A

          \[\leadsto y \cdot \frac{\sin x}{x} + \frac{\color{blue}{\left(y \cdot \left(\frac{1}{6} \cdot {y}^{2}\right)\right) \cdot \sin x}}{x} \]

      11. associate-/l* N/A

          \[\leadsto y \cdot \frac{\sin x}{x} + \color{blue}{\left(y \cdot \left(\frac{1}{6} \cdot {y}^{2}\right)\right) \cdot \frac{\sin x}{x}} \]

      12. distribute-rgt-out N/A

          \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot \left(y + y \cdot \left(\frac{1}{6} \cdot {y}^{2}\right)\right)} \]

      13. *-rgt-identity N/A

          \[\leadsto \frac{\sin x}{x} \cdot \left(\color{blue}{y \cdot 1} + y \cdot \left(\frac{1}{6} \cdot {y}^{2}\right)\right) \]

      14. distribute-lft-in N/A

          \[\leadsto \frac{\sin x}{x} \cdot \color{blue}{\left(y \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)} \]

   5. Simplified 86.5%

      \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot \mathsf{fma}\left(y \cdot y, y \cdot 0.16666666666666666, y\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{y + \frac{1}{6} \cdot {y}^{3}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{6} \cdot {y}^{3} + y} \]

      2. cube-mult N/A

         \[\leadsto \frac{1}{6} \cdot \color{blue}{\left(y \cdot \left(y \cdot y\right)\right)} + y \]

      3. unpow2 N/A

         \[\leadsto \frac{1}{6} \cdot \left(y \cdot \color{blue}{{y}^{2}}\right) + y \]

      4. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{6} \cdot y\right) \cdot {y}^{2}} + y \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left(y \cdot \frac{1}{6}\right)} \cdot {y}^{2} + y \]

      6. associate-*r* N/A

         \[\leadsto \color{blue}{y \cdot \left(\frac{1}{6} \cdot {y}^{2}\right)} + y \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{1}{6} \cdot {y}^{2}, y\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{1}{6} \cdot {y}^{2}}, y\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{1}{6} \cdot \color{blue}{\left(y \cdot y\right)}, y\right) \]

      10. lower-*.f64 50.7

          \[\leadsto \mathsf{fma}\left(y, 0.16666666666666666 \cdot \color{blue}{\left(y \cdot y\right)}, y\right) \]

   8. Simplified 50.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, 0.16666666666666666 \cdot \left(y \cdot y\right), y\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 45.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -2 \cdot 10^{-237}:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot \left(y \cdot -0.16666666666666666\right), y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \left(y \cdot y\right) \cdot 0.16666666666666666, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 89.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 35000:\\ \;\;\;\;\frac{\sin x}{x} \cdot \mathsf{fma}\left(y \cdot y, y \cdot 0.16666666666666666, y\right)\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{+62}:\\ \;\;\;\;\frac{\sinh y \cdot \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot -0.16666666666666666, x\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(\sin x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 35000.0)
   (* (/ (sin x) x) (fma (* y y) (* y 0.16666666666666666) y))
   (if (<= y 1.2e+62)
     (/ (* (sinh y) (fma x (* (* x x) -0.16666666666666666) x)) x)
     (/
      (*
       y
       (*
        (sin x)
        (fma
         (* y y)
         (fma y (* y 0.008333333333333333) 0.16666666666666666)
         1.0)))
      x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 35000.0) {
		tmp = (sin(x) / x) * fma((y * y), (y * 0.16666666666666666), y);
	} else if (y <= 1.2e+62) {
		tmp = (sinh(y) * fma(x, ((x * x) * -0.16666666666666666), x)) / x;
	} else {
		tmp = (y * (sin(x) * fma((y * y), fma(y, (y * 0.008333333333333333), 0.16666666666666666), 1.0))) / x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 35000.0)
		tmp = Float64(Float64(sin(x) / x) * fma(Float64(y * y), Float64(y * 0.16666666666666666), y));
	elseif (y <= 1.2e+62)
		tmp = Float64(Float64(sinh(y) * fma(x, Float64(Float64(x * x) * -0.16666666666666666), x)) / x);
	else
		tmp = Float64(Float64(y * Float64(sin(x) * fma(Float64(y * y), fma(y, Float64(y * 0.008333333333333333), 0.16666666666666666), 1.0))) / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 35000.0], N[(N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(y * 0.16666666666666666), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.2e+62], N[(N[(N[Sinh[y], $MachinePrecision] * N[(x * N[(N[(x * x), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(y * N[(N[Sin[x], $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 35000

   1. Initial program 83.8%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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. +-commutative N/A

         \[\leadsto y \cdot \color{blue}{\left(\frac{\sin x}{x} + \frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x}\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{y \cdot \frac{\sin x}{x} + y \cdot \left(\frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x}\right)} \]

      3. associate-*r/ N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + y \cdot \color{blue}{\frac{\frac{1}{6} \cdot \left({y}^{2} \cdot \sin x\right)}{x}} \]

      4. *-commutative N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + y \cdot \frac{\frac{1}{6} \cdot \color{blue}{\left(\sin x \cdot {y}^{2}\right)}}{x} \]

      5. associate-*r* N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + y \cdot \frac{\color{blue}{\left(\frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}}}{x} \]

      6. associate-*r/ N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + \color{blue}{\frac{y \cdot \left(\left(\frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}\right)}{x}} \]

      7. associate-*r* N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + \frac{y \cdot \color{blue}{\left(\frac{1}{6} \cdot \left(\sin x \cdot {y}^{2}\right)\right)}}{x} \]

      8. *-commutative N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + \frac{y \cdot \left(\frac{1}{6} \cdot \color{blue}{\left({y}^{2} \cdot \sin x\right)}\right)}{x} \]

      9. associate-*r* N/A

         \[\leadsto y \cdot \frac{\sin x}{x} + \frac{y \cdot \color{blue}{\left(\left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \sin x\right)}}{x} \]

      10. associate-*r* N/A

          \[\leadsto y \cdot \frac{\sin x}{x} + \frac{\color{blue}{\left(y \cdot \left(\frac{1}{6} \cdot {y}^{2}\right)\right) \cdot \sin x}}{x} \]

      11. associate-/l* N/A

          \[\leadsto y \cdot \frac{\sin x}{x} + \color{blue}{\left(y \cdot \left(\frac{1}{6} \cdot {y}^{2}\right)\right) \cdot \frac{\sin x}{x}} \]

      12. distribute-rgt-out N/A

          \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot \left(y + y \cdot \left(\frac{1}{6} \cdot {y}^{2}\right)\right)} \]

      13. *-rgt-identity N/A

          \[\leadsto \frac{\sin x}{x} \cdot \left(\color{blue}{y \cdot 1} + y \cdot \left(\frac{1}{6} \cdot {y}^{2}\right)\right) \]

      14. distribute-lft-in N/A

          \[\leadsto \frac{\sin x}{x} \cdot \color{blue}{\left(y \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)} \]

   5. Simplified 91.5%

      \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot \mathsf{fma}\left(y \cdot y, y \cdot 0.16666666666666666, y\right)} \]

   if 35000 < y < 1.2e62

   1. Initial program 100.0%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)} \cdot \sinh y}{x} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\left(x \cdot \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2} + 1\right)}\right) \cdot \sinh y}{x} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot \left(\frac{-1}{6} \cdot {x}^{2}\right) + x \cdot 1\right)} \cdot \sinh y}{x} \]

      3. *-rgt-identity N/A

         \[\leadsto \frac{\left(x \cdot \left(\frac{-1}{6} \cdot {x}^{2}\right) + \color{blue}{x}\right) \cdot \sinh y}{x} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, \frac{-1}{6} \cdot {x}^{2}, x\right)} \cdot \sinh y}{x} \]

      5. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, \color{blue}{{x}^{2} \cdot \frac{-1}{6}}, x\right) \cdot \sinh y}{x} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, \color{blue}{{x}^{2} \cdot \frac{-1}{6}}, x\right) \cdot \sinh y}{x} \]

      7. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, \color{blue}{\left(x \cdot x\right)} \cdot \frac{-1}{6}, x\right) \cdot \sinh y}{x} \]

      8. lower-*.f64 66.7

         \[\leadsto \frac{\mathsf{fma}\left(x, \color{blue}{\left(x \cdot x\right)} \cdot -0.16666666666666666, x\right) \cdot \sinh y}{x} \]

   5. Simplified 66.7%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot -0.16666666666666666, x\right)} \cdot \sinh y}{x} \]

   if 1.2e62 < y

   1. Initial program 100.0%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{y \cdot \left(\sin x + {y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{6} \cdot \sin x\right)\right)}}{x} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(\sin x + {y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{6} \cdot \sin x\right)\right)}}{x} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{y \cdot \left(\sin x + \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right)\right) + {y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x\right)\right)}\right)}{x} \]

      3. *-rgt-identity N/A

         \[\leadsto \frac{y \cdot \left(\color{blue}{\sin x \cdot 1} + \left({y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right)\right) + {y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x\right)\right)\right)}{x} \]

      4. distribute-lft-in N/A

         \[\leadsto \frac{y \cdot \left(\sin x \cdot 1 + \color{blue}{{y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{6} \cdot \sin x\right)}\right)}{x} \]

      5. +-commutative N/A

         \[\leadsto \frac{y \cdot \left(\sin x \cdot 1 + {y}^{2} \cdot \color{blue}{\left(\frac{1}{6} \cdot \sin x + \frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right)\right)}\right)}{x} \]

      6. *-commutative N/A

         \[\leadsto \frac{y \cdot \left(\sin x \cdot 1 + {y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x + \color{blue}{\left({y}^{2} \cdot \sin x\right) \cdot \frac{1}{120}}\right)\right)}{x} \]

      7. associate-*r* N/A

         \[\leadsto \frac{y \cdot \left(\sin x \cdot 1 + {y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x + \color{blue}{{y}^{2} \cdot \left(\sin x \cdot \frac{1}{120}\right)}\right)\right)}{x} \]

      8. *-commutative N/A

         \[\leadsto \frac{y \cdot \left(\sin x \cdot 1 + {y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x + {y}^{2} \cdot \color{blue}{\left(\frac{1}{120} \cdot \sin x\right)}\right)\right)}{x} \]

      9. distribute-lft-in N/A

         \[\leadsto \frac{y \cdot \left(\sin x \cdot 1 + \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x\right) + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{120} \cdot \sin x\right)\right)\right)}\right)}{x} \]

      10. associate-*r* N/A

          \[\leadsto \frac{y \cdot \left(\sin x \cdot 1 + \left(\color{blue}{\left({y}^{2} \cdot \frac{1}{6}\right) \cdot \sin x} + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{120} \cdot \sin x\right)\right)\right)\right)}{x} \]

      11. *-commutative N/A

          \[\leadsto \frac{y \cdot \left(\sin x \cdot 1 + \left(\color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right)} \cdot \sin x + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{120} \cdot \sin x\right)\right)\right)\right)}{x} \]

   5. Simplified 100.0%

      \[\leadsto \frac{\color{blue}{y \cdot \left(\sin x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)\right)}}{x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 35000:\\ \;\;\;\;\frac{\sin x}{x} \cdot \mathsf{fma}\left(y \cdot y, y \cdot 0.16666666666666666, y\right)\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{+62}:\\ \;\;\;\;\frac{\sinh y \cdot \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot -0.16666666666666666, x\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(\sin x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 64.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.6 \cdot 10^{-153}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, y \cdot \left(y \cdot \left(y \cdot \left(\left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right)\right), y\right)\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{+52}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right) \cdot \mathsf{fma}\left(y, -0.16666666666666666, \frac{y}{x \cdot x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.0001984126984126984 \cdot {y}^{7}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1.6e-153)
   (fma (* y y) (* y (* y (* y (* (* y y) 0.0001984126984126984)))) y)
   (if (<= x 3.9e+52)
     (*
      (* x x)
      (*
       (fma (* y y) (fma y (* y 0.008333333333333333) 0.16666666666666666) 1.0)
       (fma y -0.16666666666666666 (/ y (* x x)))))
     (* 0.0001984126984126984 (pow y 7.0)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.6e-153) {
		tmp = fma((y * y), (y * (y * (y * ((y * y) * 0.0001984126984126984)))), y);
	} else if (x <= 3.9e+52) {
		tmp = (x * x) * (fma((y * y), fma(y, (y * 0.008333333333333333), 0.16666666666666666), 1.0) * fma(y, -0.16666666666666666, (y / (x * x))));
	} else {
		tmp = 0.0001984126984126984 * pow(y, 7.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.6e-153)
		tmp = fma(Float64(y * y), Float64(y * Float64(y * Float64(y * Float64(Float64(y * y) * 0.0001984126984126984)))), y);
	elseif (x <= 3.9e+52)
		tmp = Float64(Float64(x * x) * Float64(fma(Float64(y * y), fma(y, Float64(y * 0.008333333333333333), 0.16666666666666666), 1.0) * fma(y, -0.16666666666666666, Float64(y / Float64(x * x)))));
	else
		tmp = Float64(0.0001984126984126984 * (y ^ 7.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.6e-153], N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision], If[LessEqual[x, 3.9e+52], N[(N[(x * x), $MachinePrecision] * N[(N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * N[(y * -0.16666666666666666 + N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.0001984126984126984 * N[Power[y, 7.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.6e-153

   1. Initial program 83.2%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \frac{\color{blue}{\sin x} \cdot \sinh y}{x} \]

      2. lift-sinh.f64 N/A

         \[\leadsto \frac{\sin x \cdot \color{blue}{\sinh y}}{x} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sinh y \cdot \sin x}}{x} \]

      4. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sinh y}{x} \cdot \sin x} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sinh y}{x} \cdot \sin x} \]

      6. lower-/.f64 99.9

         \[\leadsto \color{blue}{\frac{\sinh y}{x}} \cdot \sin x \]

   4. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\frac{\sinh y}{x} \cdot \sin x} \]

   5. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)}}{x} \cdot \sin x \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{y \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) + 1\right)}}{x} \cdot \sin x \]

      2. distribute-rgt-in N/A

         \[\leadsto \frac{\color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right) \cdot y + 1 \cdot y}}{x} \cdot \sin x \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot {y}^{2}\right)} \cdot y + 1 \cdot y}{x} \cdot \sin x \]

      4. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot \left({y}^{2} \cdot y\right)} + 1 \cdot y}{x} \cdot \sin x \]

      5. unpow2 N/A

         \[\leadsto \frac{\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot \left(\color{blue}{\left(y \cdot y\right)} \cdot y\right) + 1 \cdot y}{x} \cdot \sin x \]

      6. unpow3 N/A

         \[\leadsto \frac{\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot \color{blue}{{y}^{3}} + 1 \cdot y}{x} \cdot \sin x \]

      7. *-lft-identity N/A

         \[\leadsto \frac{\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot {y}^{3} + \color{blue}{y}}{x} \cdot \sin x \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right), {y}^{3}, y\right)}}{x} \cdot \sin x \]

   7. Simplified 98.8%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)}}{x} \cdot \sin x \]

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{y + {y}^{3} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{y}^{3} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) + y} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot {y}^{3}} + y \]

      3. unpow3 N/A

         \[\leadsto \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot \color{blue}{\left(\left(y \cdot y\right) \cdot y\right)} + y \]

      4. unpow2 N/A

         \[\leadsto \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot \left(\color{blue}{{y}^{2}} \cdot y\right) + y \]

      5. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot {y}^{2}\right) \cdot y} + y \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)} \cdot y + y \]

      7. associate-*l* N/A

         \[\leadsto \color{blue}{{y}^{2} \cdot \left(\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot y\right)} + y \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({y}^{2}, \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot y, y\right)} \]

   10. Simplified 68.0%

       \[\leadsto \color{blue}{\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right) \cdot y, y\right)} \]

   11. Taylor expanded in y around inf

       \[\leadsto \mathsf{fma}\left(y \cdot y, \color{blue}{\left(\frac{1}{5040} \cdot {y}^{4}\right)} \cdot y, y\right) \]
   12. Step-by-step derivation

       1. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(y \cdot y, \left(\frac{1}{5040} \cdot {y}^{\color{blue}{\left(2 \cdot 2\right)}}\right) \cdot y, y\right) \]

       2. pow-sqr N/A

          \[\leadsto \mathsf{fma}\left(y \cdot y, \left(\frac{1}{5040} \cdot \color{blue}{\left({y}^{2} \cdot {y}^{2}\right)}\right) \cdot y, y\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(y \cdot y, \color{blue}{\left(\left(\frac{1}{5040} \cdot {y}^{2}\right) \cdot {y}^{2}\right)} \cdot y, y\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(y \cdot y, \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{5040} \cdot {y}^{2}\right)\right)} \cdot y, y\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(y \cdot y, \left(\color{blue}{\left(y \cdot y\right)} \cdot \left(\frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot y, y\right) \]

       6. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(y \cdot y, \color{blue}{\left(y \cdot \left(y \cdot \left(\frac{1}{5040} \cdot {y}^{2}\right)\right)\right)} \cdot y, y\right) \]

       7. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(y \cdot y, \color{blue}{\left(y \cdot \left(y \cdot \left(\frac{1}{5040} \cdot {y}^{2}\right)\right)\right)} \cdot y, y\right) \]

       8. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(y \cdot y, \left(y \cdot \color{blue}{\left(y \cdot \left(\frac{1}{5040} \cdot {y}^{2}\right)\right)}\right) \cdot y, y\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(y \cdot y, \left(y \cdot \left(y \cdot \color{blue}{\left({y}^{2} \cdot \frac{1}{5040}\right)}\right)\right) \cdot y, y\right) \]

       10. lower-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(y \cdot y, \left(y \cdot \left(y \cdot \color{blue}{\left({y}^{2} \cdot \frac{1}{5040}\right)}\right)\right) \cdot y, y\right) \]

       11. unpow2 N/A

           \[\leadsto \mathsf{fma}\left(y \cdot y, \left(y \cdot \left(y \cdot \left(\color{blue}{\left(y \cdot y\right)} \cdot \frac{1}{5040}\right)\right)\right) \cdot y, y\right) \]

       12. lower-*.f64 68.0

           \[\leadsto \mathsf{fma}\left(y \cdot y, \left(y \cdot \left(y \cdot \left(\color{blue}{\left(y \cdot y\right)} \cdot 0.0001984126984126984\right)\right)\right) \cdot y, y\right) \]

   13. Simplified 68.0%

       \[\leadsto \mathsf{fma}\left(y \cdot y, \color{blue}{\left(y \cdot \left(y \cdot \left(\left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right)\right)} \cdot y, y\right) \]

   if 1.6e-153 < x < 3.9e52

   1. Initial program 92.1%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{y \cdot \left(\sin x + {y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{6} \cdot \sin x\right)\right)}}{x} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(\sin x + {y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{6} \cdot \sin x\right)\right)}}{x} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{y \cdot \left(\sin x + \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right)\right) + {y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x\right)\right)}\right)}{x} \]

      3. *-rgt-identity N/A

         \[\leadsto \frac{y \cdot \left(\color{blue}{\sin x \cdot 1} + \left({y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right)\right) + {y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x\right)\right)\right)}{x} \]

      4. distribute-lft-in N/A

         \[\leadsto \frac{y \cdot \left(\sin x \cdot 1 + \color{blue}{{y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{6} \cdot \sin x\right)}\right)}{x} \]

      5. +-commutative N/A

         \[\leadsto \frac{y \cdot \left(\sin x \cdot 1 + {y}^{2} \cdot \color{blue}{\left(\frac{1}{6} \cdot \sin x + \frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right)\right)}\right)}{x} \]

      6. *-commutative N/A

         \[\leadsto \frac{y \cdot \left(\sin x \cdot 1 + {y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x + \color{blue}{\left({y}^{2} \cdot \sin x\right) \cdot \frac{1}{120}}\right)\right)}{x} \]

      7. associate-*r* N/A

         \[\leadsto \frac{y \cdot \left(\sin x \cdot 1 + {y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x + \color{blue}{{y}^{2} \cdot \left(\sin x \cdot \frac{1}{120}\right)}\right)\right)}{x} \]

      8. *-commutative N/A

         \[\leadsto \frac{y \cdot \left(\sin x \cdot 1 + {y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x + {y}^{2} \cdot \color{blue}{\left(\frac{1}{120} \cdot \sin x\right)}\right)\right)}{x} \]

      9. distribute-lft-in N/A

         \[\leadsto \frac{y \cdot \left(\sin x \cdot 1 + \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x\right) + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{120} \cdot \sin x\right)\right)\right)}\right)}{x} \]

      10. associate-*r* N/A

          \[\leadsto \frac{y \cdot \left(\sin x \cdot 1 + \left(\color{blue}{\left({y}^{2} \cdot \frac{1}{6}\right) \cdot \sin x} + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{120} \cdot \sin x\right)\right)\right)\right)}{x} \]

      11. *-commutative N/A

          \[\leadsto \frac{y \cdot \left(\sin x \cdot 1 + \left(\color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right)} \cdot \sin x + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{120} \cdot \sin x\right)\right)\right)\right)}{x} \]

   5. Simplified 89.8%

      \[\leadsto \frac{\color{blue}{y \cdot \left(\sin x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)\right)}}{x} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-1}{6} \cdot \left({x}^{2} \cdot \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right) + y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{-1}{6} \cdot \color{blue}{\left(\left({x}^{2} \cdot y\right) \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)} + y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{6} \cdot \left({x}^{2} \cdot y\right)\right) \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} + y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) \]

      3. distribute-rgt-out N/A

         \[\leadsto \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) \cdot \left(\frac{-1}{6} \cdot \left({x}^{2} \cdot y\right) + y\right)} \]

      4. +-commutative N/A

         \[\leadsto \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) \cdot \color{blue}{\left(y + \frac{-1}{6} \cdot \left({x}^{2} \cdot y\right)\right)} \]

      5. *-commutative N/A

         \[\leadsto \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) \cdot \left(y + \frac{-1}{6} \cdot \color{blue}{\left(y \cdot {x}^{2}\right)}\right) \]

      6. associate-*r* N/A

         \[\leadsto \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) \cdot \left(y + \color{blue}{\left(\frac{-1}{6} \cdot y\right) \cdot {x}^{2}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) \cdot \left(y + \left(\frac{-1}{6} \cdot y\right) \cdot {x}^{2}\right)} \]

   8. Simplified 83.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), 1\right) \cdot \mathsf{fma}\left(y, \left(x \cdot x\right) \cdot -0.16666666666666666, y\right)} \]

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{-1}{6} \cdot \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) + \frac{y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)}{{x}^{2}}\right)} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{-1}{6} \cdot \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) + \frac{y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)}{{x}^{2}}\right)} \]

       2. unpow2 N/A

          \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{-1}{6} \cdot \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) + \frac{y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)}{{x}^{2}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{-1}{6} \cdot \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) + \frac{y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)}{{x}^{2}}\right) \]

       4. associate-*r* N/A

          \[\leadsto \left(x \cdot x\right) \cdot \left(\color{blue}{\left(\frac{-1}{6} \cdot y\right) \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} + \frac{y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)}{{x}^{2}}\right) \]

       5. *-commutative N/A

          \[\leadsto \left(x \cdot x\right) \cdot \left(\color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) \cdot \left(\frac{-1}{6} \cdot y\right)} + \frac{y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)}{{x}^{2}}\right) \]

       6. *-commutative N/A

          \[\leadsto \left(x \cdot x\right) \cdot \left(\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) \cdot \left(\frac{-1}{6} \cdot y\right) + \frac{\color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) \cdot y}}{{x}^{2}}\right) \]

       7. associate-/l* N/A

          \[\leadsto \left(x \cdot x\right) \cdot \left(\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) \cdot \left(\frac{-1}{6} \cdot y\right) + \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) \cdot \frac{y}{{x}^{2}}}\right) \]

       8. distribute-lft-out N/A

          \[\leadsto \left(x \cdot x\right) \cdot \color{blue}{\left(\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) \cdot \left(\frac{-1}{6} \cdot y + \frac{y}{{x}^{2}}\right)\right)} \]

       9. lower-*.f64 N/A

          \[\leadsto \left(x \cdot x\right) \cdot \color{blue}{\left(\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) \cdot \left(\frac{-1}{6} \cdot y + \frac{y}{{x}^{2}}\right)\right)} \]

   11. Simplified 83.4%

       \[\leadsto \color{blue}{\left(x \cdot x\right) \cdot \left(\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right) \cdot \mathsf{fma}\left(y, -0.16666666666666666, \frac{y}{x \cdot x}\right)\right)} \]

   if 3.9e52 < x

   1. Initial program 99.9%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \frac{\color{blue}{\sin x} \cdot \sinh y}{x} \]

      2. lift-sinh.f64 N/A

         \[\leadsto \frac{\sin x \cdot \color{blue}{\sinh y}}{x} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sinh y \cdot \sin x}}{x} \]

      4. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sinh y}{x} \cdot \sin x} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sinh y}{x} \cdot \sin x} \]

      6. lower-/.f64 99.8

         \[\leadsto \color{blue}{\frac{\sinh y}{x}} \cdot \sin x \]

   4. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{\sinh y}{x} \cdot \sin x} \]

   5. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)}}{x} \cdot \sin x \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{y \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) + 1\right)}}{x} \cdot \sin x \]

      2. distribute-rgt-in N/A

         \[\leadsto \frac{\color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right) \cdot y + 1 \cdot y}}{x} \cdot \sin x \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot {y}^{2}\right)} \cdot y + 1 \cdot y}{x} \cdot \sin x \]

      4. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot \left({y}^{2} \cdot y\right)} + 1 \cdot y}{x} \cdot \sin x \]

      5. unpow2 N/A

         \[\leadsto \frac{\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot \left(\color{blue}{\left(y \cdot y\right)} \cdot y\right) + 1 \cdot y}{x} \cdot \sin x \]

      6. unpow3 N/A

         \[\leadsto \frac{\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot \color{blue}{{y}^{3}} + 1 \cdot y}{x} \cdot \sin x \]

      7. *-lft-identity N/A

         \[\leadsto \frac{\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot {y}^{3} + \color{blue}{y}}{x} \cdot \sin x \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right), {y}^{3}, y\right)}}{x} \cdot \sin x \]

   7. Simplified 89.3%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)}}{x} \cdot \sin x \]

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{y + {y}^{3} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{y}^{3} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) + y} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot {y}^{3}} + y \]

      3. unpow3 N/A

         \[\leadsto \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot \color{blue}{\left(\left(y \cdot y\right) \cdot y\right)} + y \]

      4. unpow2 N/A

         \[\leadsto \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot \left(\color{blue}{{y}^{2}} \cdot y\right) + y \]

      5. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot {y}^{2}\right) \cdot y} + y \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)} \cdot y + y \]

      7. associate-*l* N/A

         \[\leadsto \color{blue}{{y}^{2} \cdot \left(\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot y\right)} + y \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({y}^{2}, \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot y, y\right)} \]

   10. Simplified 16.4%

       \[\leadsto \color{blue}{\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right) \cdot y, y\right)} \]

   11. Taylor expanded in y around inf

       \[\leadsto \color{blue}{\frac{1}{5040} \cdot {y}^{7}} \]
   12. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{5040} \cdot {y}^{7}} \]

       2. lower-pow.f64 49.5

          \[\leadsto 0.0001984126984126984 \cdot \color{blue}{{y}^{7}} \]

   13. Simplified 49.5%

       \[\leadsto \color{blue}{0.0001984126984126984 \cdot {y}^{7}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.6 \cdot 10^{-153}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, y \cdot \left(y \cdot \left(y \cdot \left(\left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right)\right), y\right)\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{+52}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right) \cdot \mathsf{fma}\left(y, -0.16666666666666666, \frac{y}{x \cdot x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.0001984126984126984 \cdot {y}^{7}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 35.7% accurate, 12.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x, x \cdot \left(y \cdot -0.16666666666666666\right), y\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (fma x (* x (* y -0.16666666666666666)) y))

C

double code(double x, double y) {
	return fma(x, (x * (y * -0.16666666666666666)), y);
}

Julia

function code(x, y)
	return fma(x, Float64(x * Float64(y * -0.16666666666666666)), y)
end

Wolfram

code[x_, y_] := N[(x * N[(x * N[(y * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision]

Derivation

1. Initial program 87.6%

   \[\frac{\sin x \cdot \sinh y}{x} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\frac{y \cdot \sin x}{x}} \]
4. Step-by-step derivation

   1. associate-/l* N/A

      \[\leadsto \color{blue}{y \cdot \frac{\sin x}{x}} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{y \cdot \frac{\sin x}{x}} \]

   3. lower-/.f64 N/A

      \[\leadsto y \cdot \color{blue}{\frac{\sin x}{x}} \]

   4. lower-sin.f64 57.1

      \[\leadsto y \cdot \frac{\color{blue}{\sin x}}{x} \]

5. Simplified 57.1%

   \[\leadsto \color{blue}{y \cdot \frac{\sin x}{x}} \]

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{y + {x}^{2} \cdot \left(\frac{-1}{6} \cdot y + \frac{1}{120} \cdot \left({x}^{2} \cdot y\right)\right)} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{-1}{6} \cdot y + \frac{1}{120} \cdot \left({x}^{2} \cdot y\right)\right) + y} \]

   2. unpow2 N/A

      \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{-1}{6} \cdot y + \frac{1}{120} \cdot \left({x}^{2} \cdot y\right)\right) + y \]

   3. associate-*l* N/A

      \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(\frac{-1}{6} \cdot y + \frac{1}{120} \cdot \left({x}^{2} \cdot y\right)\right)\right)} + y \]

   4. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot \left(\frac{-1}{6} \cdot y + \frac{1}{120} \cdot \left({x}^{2} \cdot y\right)\right), y\right)} \]

   5. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{-1}{6} \cdot y + \frac{1}{120} \cdot \left({x}^{2} \cdot y\right)\right)}, y\right) \]

   6. associate-*r* N/A

      \[\leadsto \mathsf{fma}\left(x, x \cdot \left(\frac{-1}{6} \cdot y + \color{blue}{\left(\frac{1}{120} \cdot {x}^{2}\right) \cdot y}\right), y\right) \]

   7. distribute-rgt-out N/A

      \[\leadsto \mathsf{fma}\left(x, x \cdot \color{blue}{\left(y \cdot \left(\frac{-1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right)}, y\right) \]

   8. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(x, x \cdot \left(y \cdot \color{blue}{\left(\frac{1}{120} \cdot {x}^{2} + \frac{-1}{6}\right)}\right), y\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(x, x \cdot \left(y \cdot \left(\frac{1}{120} \cdot {x}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}\right)\right), y\right) \]

   10. sub-neg N/A

       \[\leadsto \mathsf{fma}\left(x, x \cdot \left(y \cdot \color{blue}{\left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)}\right), y\right) \]

   11. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(x, x \cdot \color{blue}{\left(y \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right)}, y\right) \]

   12. sub-neg N/A

       \[\leadsto \mathsf{fma}\left(x, x \cdot \left(y \cdot \color{blue}{\left(\frac{1}{120} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)}\right), y\right) \]

   13. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(x, x \cdot \left(y \cdot \left(\frac{1}{120} \cdot {x}^{2} + \color{blue}{\frac{-1}{6}}\right)\right), y\right) \]

   14. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(x, x \cdot \left(y \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{120}, {x}^{2}, \frac{-1}{6}\right)}\right), y\right) \]

   15. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(x, x \cdot \left(y \cdot \mathsf{fma}\left(\frac{1}{120}, \color{blue}{x \cdot x}, \frac{-1}{6}\right)\right), y\right) \]

   16. lower-*.f64 39.5

       \[\leadsto \mathsf{fma}\left(x, x \cdot \left(y \cdot \mathsf{fma}\left(0.008333333333333333, \color{blue}{x \cdot x}, -0.16666666666666666\right)\right), y\right) \]

8. Simplified 39.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot \left(y \cdot \mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right)\right), y\right)} \]

9. Taylor expanded in x around 0

   \[\leadsto \mathsf{fma}\left(x, x \cdot \color{blue}{\left(\frac{-1}{6} \cdot y\right)}, y\right) \]
10. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(x, x \cdot \color{blue}{\left(y \cdot \frac{-1}{6}\right)}, y\right) \]

    2. lower-*.f64 37.4

       \[\leadsto \mathsf{fma}\left(x, x \cdot \color{blue}{\left(y \cdot -0.16666666666666666\right)}, y\right) \]

11. Simplified 37.4%

    \[\leadsto \mathsf{fma}\left(x, x \cdot \color{blue}{\left(y \cdot -0.16666666666666666\right)}, y\right) \]
12. Add Preprocessing

Alternative 17: 27.8% 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 87.6%

   \[\frac{\sin x \cdot \sinh y}{x} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\frac{y \cdot \sin x}{x}} \]
4. Step-by-step derivation

   1. associate-/l* N/A

      \[\leadsto \color{blue}{y \cdot \frac{\sin x}{x}} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{y \cdot \frac{\sin x}{x}} \]

   3. lower-/.f64 N/A

      \[\leadsto y \cdot \color{blue}{\frac{\sin x}{x}} \]

   4. lower-sin.f64 57.1

      \[\leadsto y \cdot \frac{\color{blue}{\sin x}}{x} \]

5. Simplified 57.1%

   \[\leadsto \color{blue}{y \cdot \frac{\sin x}{x}} \]

6. Taylor expanded in x around 0

   \[\leadsto y \cdot \color{blue}{1} \]
7. Step-by-step derivation

8. Simplified 31.7%

   \[\leadsto y \cdot \color{blue}{1} \]
9. Step-by-step derivation

   1. *-rgt-identity 31.7

      \[\leadsto \color{blue}{y} \]

10. Applied egg-rr 31.7%

    \[\leadsto \color{blue}{y} \]
11. Add Preprocessing

Developer Target 1: 99.8% 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 2024207 
(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))