Linear.Quaternion:$ccosh from linear-1.19.1.3

Percentage Accurate: 88.2% → 99.9%

Time: 15.1s

Alternatives: 22

Speedup: 1.0×

• 48.8% 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: 88.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.0%

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

   1. associate-*l/ N/A

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

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

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

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

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

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

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

   5. sinh-lowering-sinh.f64 99.9

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

4. Applied egg-rr 99.9%

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

Alternative 2: 86.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin x \cdot \sinh y}{x}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\frac{\sinh y \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right)\right)}{x}\\ \mathbf{elif}\;t\_0 \leq 10^{-114}:\\ \;\;\;\;\left(\sin x \cdot \frac{y}{x}\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sin x) (sinh y)) x)))
   (if (<= t_0 (- INFINITY))
     (/ (* (sinh y) (* x (fma x (* x -0.16666666666666666) 1.0))) x)
     (if (<= t_0 1e-114)
       (*
        (* (sin x) (/ y x))
        (fma
         (* y y)
         (fma y (* y 0.008333333333333333) 0.16666666666666666)
         1.0))
       (sinh y)))))

C

double code(double x, double y) {
	double t_0 = (sin(x) * sinh(y)) / x;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (sinh(y) * (x * fma(x, (x * -0.16666666666666666), 1.0))) / x;
	} else if (t_0 <= 1e-114) {
		tmp = (sin(x) * (y / x)) * fma((y * y), fma(y, (y * 0.008333333333333333), 0.16666666666666666), 1.0);
	} else {
		tmp = sinh(y);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(sin(x) * sinh(y)) / x)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(sinh(y) * Float64(x * fma(x, Float64(x * -0.16666666666666666), 1.0))) / x);
	elseif (t_0 <= 1e-114)
		tmp = Float64(Float64(sin(x) * Float64(y / x)) * fma(Float64(y * y), fma(y, Float64(y * 0.008333333333333333), 0.16666666666666666), 1.0));
	else
		tmp = sinh(y);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sin[x], $MachinePrecision] * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[Sinh[y], $MachinePrecision] * N[(x * N[(x * N[(x * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[t$95$0, 1e-114], N[(N[(N[Sin[x], $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[Sinh[y], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in 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. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

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

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

      7. *-lowering-*.f64 71.7

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

   5. Simplified 71.7%

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

   if -inf.0 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 1.0000000000000001e-114

   1. Initial program 81.3%

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

   3. Taylor expanded in y around 0

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

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

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

      2. +-commutative N/A

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

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

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

      4. unpow2 N/A

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

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

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 79.0

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

   5. Simplified 79.0%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      11. *-lowering-*.f64 96.0

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

   7. Applied egg-rr 96.0%

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

      1. associate-/r* N/A

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

      2. associate-/r/ N/A

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

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

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

      4. div-inv N/A

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

      5. clear-num N/A

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

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

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

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

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

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

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

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

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

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

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

      11. associate-*l* N/A

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

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

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

      13. *-lowering-*.f64 97.5

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

   9. Applied egg-rr 97.5%

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

   if 1.0000000000000001e-114 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 99.3%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 72.6%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-inverses N/A

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

      4. *-rgt-identity N/A

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

      5. sinh-lowering-sinh.f64 73.4

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

   7. Applied egg-rr 73.4%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin x \cdot \sinh y}{x} \leq -\infty:\\ \;\;\;\;\frac{\sinh y \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right)\right)}{x}\\ \mathbf{elif}\;\frac{\sin x \cdot \sinh y}{x} \leq 10^{-114}:\\ \;\;\;\;\left(\sin x \cdot \frac{y}{x}\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin x \cdot \sinh y}{x}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\frac{\sinh y \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right)\right)}{x}\\ \mathbf{elif}\;t\_0 \leq 10^{-114}:\\ \;\;\;\;\mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right) \cdot \left(\frac{\sin x}{x} \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sin x) (sinh y)) x)))
   (if (<= t_0 (- INFINITY))
     (/ (* (sinh y) (* x (fma x (* x -0.16666666666666666) 1.0))) x)
     (if (<= t_0 1e-114)
       (* (fma 0.16666666666666666 (* y y) 1.0) (* (/ (sin x) x) y))
       (sinh y)))))

C

double code(double x, double y) {
	double t_0 = (sin(x) * sinh(y)) / x;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (sinh(y) * (x * fma(x, (x * -0.16666666666666666), 1.0))) / x;
	} else if (t_0 <= 1e-114) {
		tmp = fma(0.16666666666666666, (y * y), 1.0) * ((sin(x) / x) * y);
	} else {
		tmp = sinh(y);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(sin(x) * sinh(y)) / x)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(sinh(y) * Float64(x * fma(x, Float64(x * -0.16666666666666666), 1.0))) / x);
	elseif (t_0 <= 1e-114)
		tmp = Float64(fma(0.16666666666666666, Float64(y * y), 1.0) * Float64(Float64(sin(x) / x) * y));
	else
		tmp = sinh(y);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sin[x], $MachinePrecision] * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[Sinh[y], $MachinePrecision] * N[(x * N[(x * N[(x * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[t$95$0, 1e-114], N[(N[(0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], N[Sinh[y], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in 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. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

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

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

      7. *-lowering-*.f64 71.7

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

   5. Simplified 71.7%

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

   if -inf.0 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 1.0000000000000001e-114

   1. Initial program 81.3%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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. distribute-lft-in N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. distribute-lft-in N/A

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

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

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. distribute-lft1-in N/A

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

      11. +-commutative N/A

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

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

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

   5. Simplified 97.0%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

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

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

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

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

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

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

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

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

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

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

      10. sin-lowering-sin.f64 97.0

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

   7. Applied egg-rr 97.0%

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

   if 1.0000000000000001e-114 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 99.3%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 72.6%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-inverses N/A

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

      4. *-rgt-identity N/A

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

      5. sinh-lowering-sinh.f64 73.4

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

   7. Applied egg-rr 73.4%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin x \cdot \sinh y}{x} \leq -\infty:\\ \;\;\;\;\frac{\sinh y \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right)\right)}{x}\\ \mathbf{elif}\;\frac{\sin x \cdot \sinh y}{x} \leq 10^{-114}:\\ \;\;\;\;\mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right) \cdot \left(\frac{\sin x}{x} \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin x \cdot \sinh y}{x}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right) \cdot \left(y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), 1\right)\right)\\ \mathbf{elif}\;t\_0 \leq 10^{-114}:\\ \;\;\;\;\mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right) \cdot \left(\frac{\sin x}{x} \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sin x) (sinh y)) x)))
   (if (<= t_0 (- INFINITY))
     (*
      (fma x (* x -0.16666666666666666) 1.0)
      (*
       y
       (fma
        y
        (* y (fma (* y y) 0.008333333333333333 0.16666666666666666))
        1.0)))
     (if (<= t_0 1e-114)
       (* (fma 0.16666666666666666 (* y y) 1.0) (* (/ (sin x) x) y))
       (sinh y)))))

C

double code(double x, double y) {
	double t_0 = (sin(x) * sinh(y)) / x;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma(x, (x * -0.16666666666666666), 1.0) * (y * fma(y, (y * fma((y * y), 0.008333333333333333, 0.16666666666666666)), 1.0));
	} else if (t_0 <= 1e-114) {
		tmp = fma(0.16666666666666666, (y * y), 1.0) * ((sin(x) / x) * y);
	} else {
		tmp = sinh(y);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(sin(x) * sinh(y)) / x)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(fma(x, Float64(x * -0.16666666666666666), 1.0) * Float64(y * fma(y, Float64(y * fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666)), 1.0)));
	elseif (t_0 <= 1e-114)
		tmp = Float64(fma(0.16666666666666666, Float64(y * y), 1.0) * Float64(Float64(sin(x) / x) * y));
	else
		tmp = sinh(y);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sin[x], $MachinePrecision] * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(x * N[(x * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * N[(y * N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e-114], N[(N[(0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], N[Sinh[y], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

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

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

      2. +-commutative N/A

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

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

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

      4. unpow2 N/A

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

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

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 85.7

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

   5. Simplified 85.7%

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

   6. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. distribute-lft1-in N/A

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

      3. +-commutative N/A

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

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

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

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

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

      11. *-commutative N/A

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

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

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

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

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

      14. +-commutative N/A

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

   8. Simplified 65.3%

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

   if -inf.0 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 1.0000000000000001e-114

   1. Initial program 81.3%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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. distribute-lft-in N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. distribute-lft-in N/A

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

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

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. distribute-lft1-in N/A

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

      11. +-commutative N/A

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

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

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

   5. Simplified 97.0%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

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

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

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

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

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

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

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

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

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

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

      10. sin-lowering-sin.f64 97.0

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

   7. Applied egg-rr 97.0%

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

   if 1.0000000000000001e-114 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 99.3%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 72.6%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-inverses N/A

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

      4. *-rgt-identity N/A

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

      5. sinh-lowering-sinh.f64 73.4

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

   7. Applied egg-rr 73.4%

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

Alternative 5: 83.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin x \cdot \sinh y}{x}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right) \cdot \left(y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), 1\right)\right)\\ \mathbf{elif}\;t\_0 \leq 10^{-114}:\\ \;\;\;\;y \cdot \left(\frac{\sin x}{x} \cdot \mathsf{fma}\left(y, y \cdot 0.16666666666666666, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sin x) (sinh y)) x)))
   (if (<= t_0 (- INFINITY))
     (*
      (fma x (* x -0.16666666666666666) 1.0)
      (*
       y
       (fma
        y
        (* y (fma (* y y) 0.008333333333333333 0.16666666666666666))
        1.0)))
     (if (<= t_0 1e-114)
       (* y (* (/ (sin x) x) (fma y (* y 0.16666666666666666) 1.0)))
       (sinh y)))))

C

double code(double x, double y) {
	double t_0 = (sin(x) * sinh(y)) / x;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma(x, (x * -0.16666666666666666), 1.0) * (y * fma(y, (y * fma((y * y), 0.008333333333333333, 0.16666666666666666)), 1.0));
	} else if (t_0 <= 1e-114) {
		tmp = y * ((sin(x) / x) * fma(y, (y * 0.16666666666666666), 1.0));
	} else {
		tmp = sinh(y);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(sin(x) * sinh(y)) / x)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(fma(x, Float64(x * -0.16666666666666666), 1.0) * Float64(y * fma(y, Float64(y * fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666)), 1.0)));
	elseif (t_0 <= 1e-114)
		tmp = Float64(y * Float64(Float64(sin(x) / x) * fma(y, Float64(y * 0.16666666666666666), 1.0)));
	else
		tmp = sinh(y);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sin[x], $MachinePrecision] * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(x * N[(x * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * N[(y * N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e-114], N[(y * N[(N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision] * N[(y * N[(y * 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sinh[y], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

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

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

      2. +-commutative N/A

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

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

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

      4. unpow2 N/A

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

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

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 85.7

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

   5. Simplified 85.7%

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

   6. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. distribute-lft1-in N/A

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

      3. +-commutative N/A

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

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

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

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

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

      11. *-commutative N/A

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

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

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

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

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

      14. +-commutative N/A

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

   8. Simplified 65.3%

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

   if -inf.0 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 1.0000000000000001e-114

   1. Initial program 81.3%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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. distribute-lft-in N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. distribute-lft-in N/A

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

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

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. distribute-lft1-in N/A

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

      11. +-commutative N/A

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

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

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

   5. Simplified 97.0%

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

   if 1.0000000000000001e-114 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 99.3%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 72.6%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-inverses N/A

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

      4. *-rgt-identity N/A

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

      5. sinh-lowering-sinh.f64 73.4

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

   7. Applied egg-rr 73.4%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin x \cdot \sinh y}{x} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right) \cdot \left(y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), 1\right)\right)\\ \mathbf{elif}\;\frac{\sin x \cdot \sinh y}{x} \leq 10^{-114}:\\ \;\;\;\;y \cdot \left(\frac{\sin x}{x} \cdot \mathsf{fma}\left(y, y \cdot 0.16666666666666666, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 83.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin x \cdot \sinh y}{x}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right) \cdot \left(y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), 1\right)\right)\\ \mathbf{elif}\;t\_0 \leq 10^{-114}:\\ \;\;\;\;\frac{\sin x}{x} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sin x) (sinh y)) x)))
   (if (<= t_0 (- INFINITY))
     (*
      (fma x (* x -0.16666666666666666) 1.0)
      (*
       y
       (fma
        y
        (* y (fma (* y y) 0.008333333333333333 0.16666666666666666))
        1.0)))
     (if (<= t_0 1e-114) (* (/ (sin x) x) y) (sinh y)))))

C

double code(double x, double y) {
	double t_0 = (sin(x) * sinh(y)) / x;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma(x, (x * -0.16666666666666666), 1.0) * (y * fma(y, (y * fma((y * y), 0.008333333333333333, 0.16666666666666666)), 1.0));
	} else if (t_0 <= 1e-114) {
		tmp = (sin(x) / x) * y;
	} else {
		tmp = sinh(y);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(sin(x) * sinh(y)) / x)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(fma(x, Float64(x * -0.16666666666666666), 1.0) * Float64(y * fma(y, Float64(y * fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666)), 1.0)));
	elseif (t_0 <= 1e-114)
		tmp = Float64(Float64(sin(x) / x) * y);
	else
		tmp = sinh(y);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sin[x], $MachinePrecision] * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(x * N[(x * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * N[(y * N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e-114], N[(N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision] * y), $MachinePrecision], N[Sinh[y], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

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

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

      2. +-commutative N/A

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

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

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

      4. unpow2 N/A

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

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

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 85.7

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

   5. Simplified 85.7%

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

   6. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. distribute-lft1-in N/A

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

      3. +-commutative N/A

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

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

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

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

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

      11. *-commutative N/A

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

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

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

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

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

      14. +-commutative N/A

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

   8. Simplified 65.3%

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

   if -inf.0 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 1.0000000000000001e-114

   1. Initial program 81.3%

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

      1. associate-*l/ N/A

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

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

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

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

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

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

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

      5. sinh-lowering-sinh.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in y around 0

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

   7. Simplified 96.3%

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

   if 1.0000000000000001e-114 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 99.3%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 72.6%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-inverses N/A

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

      4. *-rgt-identity N/A

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

      5. sinh-lowering-sinh.f64 73.4

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

   7. Applied egg-rr 73.4%

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

Alternative 7: 72.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin x \cdot \sinh y}{x}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-82}:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right) \cdot \left(y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), 1\right)\right)\\ \mathbf{elif}\;t\_0 \leq 10^{-114}:\\ \;\;\;\;y \cdot \frac{-1}{\mathsf{fma}\left(x, x \cdot -0.16666666666666666, -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sin x) (sinh y)) x)))
   (if (<= t_0 -2e-82)
     (*
      (fma x (* x -0.16666666666666666) 1.0)
      (*
       y
       (fma
        y
        (* y (fma (* y y) 0.008333333333333333 0.16666666666666666))
        1.0)))
     (if (<= t_0 1e-114)
       (* y (/ -1.0 (fma x (* x -0.16666666666666666) -1.0)))
       (sinh y)))))

C

double code(double x, double y) {
	double t_0 = (sin(x) * sinh(y)) / x;
	double tmp;
	if (t_0 <= -2e-82) {
		tmp = fma(x, (x * -0.16666666666666666), 1.0) * (y * fma(y, (y * fma((y * y), 0.008333333333333333, 0.16666666666666666)), 1.0));
	} else if (t_0 <= 1e-114) {
		tmp = y * (-1.0 / fma(x, (x * -0.16666666666666666), -1.0));
	} else {
		tmp = sinh(y);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(sin(x) * sinh(y)) / x)
	tmp = 0.0
	if (t_0 <= -2e-82)
		tmp = Float64(fma(x, Float64(x * -0.16666666666666666), 1.0) * Float64(y * fma(y, Float64(y * fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666)), 1.0)));
	elseif (t_0 <= 1e-114)
		tmp = Float64(y * Float64(-1.0 / fma(x, Float64(x * -0.16666666666666666), -1.0)));
	else
		tmp = sinh(y);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sin[x], $MachinePrecision] * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-82], N[(N[(x * N[(x * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * N[(y * N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e-114], N[(y * N[(-1.0 / N[(x * N[(x * -0.16666666666666666), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sinh[y], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -2e-82

   1. Initial program 99.0%

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

   3. Taylor expanded in y around 0

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

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

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

      2. +-commutative N/A

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

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

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

      4. unpow2 N/A

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

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

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 85.5

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

   5. Simplified 85.5%

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

   6. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. distribute-lft1-in N/A

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

      3. +-commutative N/A

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

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

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

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

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

      11. *-commutative N/A

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

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

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

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

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

      14. +-commutative N/A

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

   8. Simplified 68.3%

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

   if -2e-82 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 1.0000000000000001e-114

   1. Initial program 78.8%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

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

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

      7. *-lowering-*.f64 14.4

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

   5. Simplified 14.4%

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

   6. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

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

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

      10. *-commutative N/A

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

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

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 14.5

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

   8. Simplified 14.5%

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

      1. associate-/l* N/A

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

      2. flip-+ N/A

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

      3. associate-/l* N/A

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

      4. *-inverses N/A

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

      5. *-rgt-identity N/A

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

      6. associate-*l/ N/A

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

      7. div-inv N/A

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

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

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

   10. Applied egg-rr 34.8%

       \[\leadsto \color{blue}{\left(\mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot 0.027777777777777776, -1\right) \cdot y\right) \cdot \frac{1}{\mathsf{fma}\left(x, x \cdot -0.16666666666666666, -1\right)}} \]

   11. Taylor expanded in x around 0

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

       1. mul-1-neg N/A

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

       2. neg-sub0 N/A

          \[\leadsto \color{blue}{\left(0 - y\right)} \cdot \frac{1}{\mathsf{fma}\left(x, x \cdot \frac{-1}{6}, -1\right)} \]

       3. --lowering--.f64 71.3

          \[\leadsto \color{blue}{\left(0 - y\right)} \cdot \frac{1}{\mathsf{fma}\left(x, x \cdot -0.16666666666666666, -1\right)} \]

   13. Simplified 71.3%

       \[\leadsto \color{blue}{\left(0 - y\right)} \cdot \frac{1}{\mathsf{fma}\left(x, x \cdot -0.16666666666666666, -1\right)} \]

   if 1.0000000000000001e-114 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 99.3%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 72.6%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-inverses N/A

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

      4. *-rgt-identity N/A

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

      5. sinh-lowering-sinh.f64 73.4

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

   7. Applied egg-rr 73.4%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin x \cdot \sinh y}{x} \leq -2 \cdot 10^{-82}:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right) \cdot \left(y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), 1\right)\right)\\ \mathbf{elif}\;\frac{\sin x \cdot \sinh y}{x} \leq 10^{-114}:\\ \;\;\;\;y \cdot \frac{-1}{\mathsf{fma}\left(x, x \cdot -0.16666666666666666, -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 70.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin x \cdot \sinh y}{x}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-82}:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right) \cdot \left(y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), 1\right)\right)\\ \mathbf{elif}\;t\_0 \leq 10^{-114}:\\ \;\;\;\;y \cdot \frac{-1}{\mathsf{fma}\left(x, x \cdot -0.16666666666666666, -1\right)}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sin x) (sinh y)) x)))
   (if (<= t_0 -2e-82)
     (*
      (fma x (* x -0.16666666666666666) 1.0)
      (*
       y
       (fma
        y
        (* y (fma (* y y) 0.008333333333333333 0.16666666666666666))
        1.0)))
     (if (<= t_0 1e-114)
       (* y (/ -1.0 (fma x (* x -0.16666666666666666) -1.0)))
       (*
        y
        (fma
         (* y y)
         (fma
          (* y y)
          (fma (* y y) 0.0001984126984126984 0.008333333333333333)
          0.16666666666666666)
         1.0))))))

C

double code(double x, double y) {
	double t_0 = (sin(x) * sinh(y)) / x;
	double tmp;
	if (t_0 <= -2e-82) {
		tmp = fma(x, (x * -0.16666666666666666), 1.0) * (y * fma(y, (y * fma((y * y), 0.008333333333333333, 0.16666666666666666)), 1.0));
	} else if (t_0 <= 1e-114) {
		tmp = y * (-1.0 / fma(x, (x * -0.16666666666666666), -1.0));
	} else {
		tmp = y * fma((y * y), fma((y * y), fma((y * y), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(sin(x) * sinh(y)) / x)
	tmp = 0.0
	if (t_0 <= -2e-82)
		tmp = Float64(fma(x, Float64(x * -0.16666666666666666), 1.0) * Float64(y * fma(y, Float64(y * fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666)), 1.0)));
	elseif (t_0 <= 1e-114)
		tmp = Float64(y * Float64(-1.0 / fma(x, Float64(x * -0.16666666666666666), -1.0)));
	else
		tmp = Float64(y * fma(Float64(y * y), fma(Float64(y * y), fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sin[x], $MachinePrecision] * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-82], N[(N[(x * N[(x * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * N[(y * N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e-114], N[(y * N[(-1.0 / N[(x * N[(x * -0.16666666666666666), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -2e-82

   1. Initial program 99.0%

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

   3. Taylor expanded in y around 0

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

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

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

      2. +-commutative N/A

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

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

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

      4. unpow2 N/A

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

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

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 85.5

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

   5. Simplified 85.5%

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

   6. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. distribute-lft1-in N/A

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

      3. +-commutative N/A

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

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

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

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

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

      11. *-commutative N/A

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

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

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

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

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

      14. +-commutative N/A

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

   8. Simplified 68.3%

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

   if -2e-82 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 1.0000000000000001e-114

   1. Initial program 78.8%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

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

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

      7. *-lowering-*.f64 14.4

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

   5. Simplified 14.4%

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

   6. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

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

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

      10. *-commutative N/A

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

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

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 14.5

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

   8. Simplified 14.5%

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

      1. associate-/l* N/A

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

      2. flip-+ N/A

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

      3. associate-/l* N/A

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

      4. *-inverses N/A

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

      5. *-rgt-identity N/A

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

      6. associate-*l/ N/A

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

      7. div-inv N/A

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

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

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

   10. Applied egg-rr 34.8%

       \[\leadsto \color{blue}{\left(\mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot 0.027777777777777776, -1\right) \cdot y\right) \cdot \frac{1}{\mathsf{fma}\left(x, x \cdot -0.16666666666666666, -1\right)}} \]

   11. Taylor expanded in x around 0

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

       1. mul-1-neg N/A

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

       2. neg-sub0 N/A

          \[\leadsto \color{blue}{\left(0 - y\right)} \cdot \frac{1}{\mathsf{fma}\left(x, x \cdot \frac{-1}{6}, -1\right)} \]

       3. --lowering--.f64 71.3

          \[\leadsto \color{blue}{\left(0 - y\right)} \cdot \frac{1}{\mathsf{fma}\left(x, x \cdot -0.16666666666666666, -1\right)} \]

   13. Simplified 71.3%

       \[\leadsto \color{blue}{\left(0 - y\right)} \cdot \frac{1}{\mathsf{fma}\left(x, x \cdot -0.16666666666666666, -1\right)} \]

   if 1.0000000000000001e-114 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 99.3%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 72.6%

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

   6. Taylor expanded in y around 0

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

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

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

      2. +-commutative N/A

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

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

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

      4. unpow2 N/A

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

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

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

      6. +-commutative N/A

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

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

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

      8. unpow2 N/A

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

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

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 65.9

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

   8. Simplified 65.9%

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

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin x \cdot \sinh y}{x} \leq -2 \cdot 10^{-82}:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right) \cdot \left(y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), 1\right)\right)\\ \mathbf{elif}\;\frac{\sin x \cdot \sinh y}{x} \leq 10^{-114}:\\ \;\;\;\;y \cdot \frac{-1}{\mathsf{fma}\left(x, x \cdot -0.16666666666666666, -1\right)}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 68.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin x \cdot \sinh y}{x}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-82}:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right) \cdot \left(y \cdot \mathsf{fma}\left(y, y \cdot 0.16666666666666666, 1\right)\right)\\ \mathbf{elif}\;t\_0 \leq 10^{-114}:\\ \;\;\;\;y \cdot \frac{-1}{\mathsf{fma}\left(x, x \cdot -0.16666666666666666, -1\right)}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sin x) (sinh y)) x)))
   (if (<= t_0 -2e-82)
     (*
      (fma x (* x -0.16666666666666666) 1.0)
      (* y (fma y (* y 0.16666666666666666) 1.0)))
     (if (<= t_0 1e-114)
       (* y (/ -1.0 (fma x (* x -0.16666666666666666) -1.0)))
       (*
        y
        (fma
         (* y y)
         (fma
          (* y y)
          (fma (* y y) 0.0001984126984126984 0.008333333333333333)
          0.16666666666666666)
         1.0))))))

C

double code(double x, double y) {
	double t_0 = (sin(x) * sinh(y)) / x;
	double tmp;
	if (t_0 <= -2e-82) {
		tmp = fma(x, (x * -0.16666666666666666), 1.0) * (y * fma(y, (y * 0.16666666666666666), 1.0));
	} else if (t_0 <= 1e-114) {
		tmp = y * (-1.0 / fma(x, (x * -0.16666666666666666), -1.0));
	} else {
		tmp = y * fma((y * y), fma((y * y), fma((y * y), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(sin(x) * sinh(y)) / x)
	tmp = 0.0
	if (t_0 <= -2e-82)
		tmp = Float64(fma(x, Float64(x * -0.16666666666666666), 1.0) * Float64(y * fma(y, Float64(y * 0.16666666666666666), 1.0)));
	elseif (t_0 <= 1e-114)
		tmp = Float64(y * Float64(-1.0 / fma(x, Float64(x * -0.16666666666666666), -1.0)));
	else
		tmp = Float64(y * fma(Float64(y * y), fma(Float64(y * y), fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sin[x], $MachinePrecision] * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-82], N[(N[(x * N[(x * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * N[(y * N[(y * N[(y * 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e-114], N[(y * N[(-1.0 / N[(x * N[(x * -0.16666666666666666), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -2e-82

   1. Initial program 99.0%

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

   3. Taylor expanded in y around 0

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

      1. distribute-lft-in N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. distribute-lft-in N/A

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

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

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. distribute-lft1-in N/A

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

      11. +-commutative N/A

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

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

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

   5. Simplified 71.5%

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

   6. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. distribute-lft1-in N/A

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

      3. +-commutative N/A

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

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

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

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

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

      11. *-commutative N/A

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

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

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

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

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

      14. +-commutative N/A

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

      15. *-commutative N/A

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

      16. unpow2 N/A

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

      17. associate-*l* N/A

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

      18. *-commutative N/A

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

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

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

      20. *-commutative N/A

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

      21. *-lowering-*.f64 63.3

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

   8. Simplified 63.3%

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

   if -2e-82 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 1.0000000000000001e-114

   1. Initial program 78.8%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

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

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

      7. *-lowering-*.f64 14.4

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

   5. Simplified 14.4%

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

   6. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

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

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

      10. *-commutative N/A

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

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

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 14.5

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

   8. Simplified 14.5%

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

      1. associate-/l* N/A

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

      2. flip-+ N/A

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

      3. associate-/l* N/A

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

      4. *-inverses N/A

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

      5. *-rgt-identity N/A

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

      6. associate-*l/ N/A

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

      7. div-inv N/A

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

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

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

   10. Applied egg-rr 34.8%

       \[\leadsto \color{blue}{\left(\mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot 0.027777777777777776, -1\right) \cdot y\right) \cdot \frac{1}{\mathsf{fma}\left(x, x \cdot -0.16666666666666666, -1\right)}} \]

   11. Taylor expanded in x around 0

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

       1. mul-1-neg N/A

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

       2. neg-sub0 N/A

          \[\leadsto \color{blue}{\left(0 - y\right)} \cdot \frac{1}{\mathsf{fma}\left(x, x \cdot \frac{-1}{6}, -1\right)} \]

       3. --lowering--.f64 71.3

          \[\leadsto \color{blue}{\left(0 - y\right)} \cdot \frac{1}{\mathsf{fma}\left(x, x \cdot -0.16666666666666666, -1\right)} \]

   13. Simplified 71.3%

       \[\leadsto \color{blue}{\left(0 - y\right)} \cdot \frac{1}{\mathsf{fma}\left(x, x \cdot -0.16666666666666666, -1\right)} \]

   if 1.0000000000000001e-114 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 99.3%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 72.6%

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

   6. Taylor expanded in y around 0

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

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

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

      2. +-commutative N/A

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

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

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

      4. unpow2 N/A

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

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

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

      6. +-commutative N/A

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

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

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

      8. unpow2 N/A

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

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

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 65.9

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

   8. Simplified 65.9%

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

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin x \cdot \sinh y}{x} \leq -2 \cdot 10^{-82}:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right) \cdot \left(y \cdot \mathsf{fma}\left(y, y \cdot 0.16666666666666666, 1\right)\right)\\ \mathbf{elif}\;\frac{\sin x \cdot \sinh y}{x} \leq 10^{-114}:\\ \;\;\;\;y \cdot \frac{-1}{\mathsf{fma}\left(x, x \cdot -0.16666666666666666, -1\right)}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 67.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin x \cdot \sinh y}{x}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-82}:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right) \cdot \left(y \cdot \mathsf{fma}\left(y, y \cdot 0.16666666666666666, 1\right)\right)\\ \mathbf{elif}\;t\_0 \leq 10^{-114}:\\ \;\;\;\;y \cdot \frac{-1}{\mathsf{fma}\left(x, x \cdot -0.16666666666666666, -1\right)}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sin x) (sinh y)) x)))
   (if (<= t_0 -2e-82)
     (*
      (fma x (* x -0.16666666666666666) 1.0)
      (* y (fma y (* y 0.16666666666666666) 1.0)))
     (if (<= t_0 1e-114)
       (* y (/ -1.0 (fma x (* x -0.16666666666666666) -1.0)))
       (*
        y
        (fma
         y
         (* y (fma y (* y 0.008333333333333333) 0.16666666666666666))
         1.0))))))

C

double code(double x, double y) {
	double t_0 = (sin(x) * sinh(y)) / x;
	double tmp;
	if (t_0 <= -2e-82) {
		tmp = fma(x, (x * -0.16666666666666666), 1.0) * (y * fma(y, (y * 0.16666666666666666), 1.0));
	} else if (t_0 <= 1e-114) {
		tmp = y * (-1.0 / fma(x, (x * -0.16666666666666666), -1.0));
	} else {
		tmp = y * fma(y, (y * fma(y, (y * 0.008333333333333333), 0.16666666666666666)), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(sin(x) * sinh(y)) / x)
	tmp = 0.0
	if (t_0 <= -2e-82)
		tmp = Float64(fma(x, Float64(x * -0.16666666666666666), 1.0) * Float64(y * fma(y, Float64(y * 0.16666666666666666), 1.0)));
	elseif (t_0 <= 1e-114)
		tmp = Float64(y * Float64(-1.0 / fma(x, Float64(x * -0.16666666666666666), -1.0)));
	else
		tmp = Float64(y * fma(y, Float64(y * fma(y, Float64(y * 0.008333333333333333), 0.16666666666666666)), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sin[x], $MachinePrecision] * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-82], N[(N[(x * N[(x * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * N[(y * N[(y * N[(y * 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e-114], N[(y * N[(-1.0 / N[(x * N[(x * -0.16666666666666666), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(y * N[(y * N[(y * N[(y * 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -2e-82

   1. Initial program 99.0%

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

   3. Taylor expanded in y around 0

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

      1. distribute-lft-in N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. distribute-lft-in N/A

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

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

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. distribute-lft1-in N/A

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

      11. +-commutative N/A

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

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

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

   5. Simplified 71.5%

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

   6. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. distribute-lft1-in N/A

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

      3. +-commutative N/A

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

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

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

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

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

      11. *-commutative N/A

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

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

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

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

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

      14. +-commutative N/A

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

      15. *-commutative N/A

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

      16. unpow2 N/A

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

      17. associate-*l* N/A

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

      18. *-commutative N/A

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

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

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

      20. *-commutative N/A

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

      21. *-lowering-*.f64 63.3

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

   8. Simplified 63.3%

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

   if -2e-82 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 1.0000000000000001e-114

   1. Initial program 78.8%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

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

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

      7. *-lowering-*.f64 14.4

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

   5. Simplified 14.4%

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

   6. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

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

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

      10. *-commutative N/A

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

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

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 14.5

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

   8. Simplified 14.5%

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

      1. associate-/l* N/A

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

      2. flip-+ N/A

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

      3. associate-/l* N/A

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

      4. *-inverses N/A

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

      5. *-rgt-identity N/A

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

      6. associate-*l/ N/A

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

      7. div-inv N/A

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

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

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

   10. Applied egg-rr 34.8%

       \[\leadsto \color{blue}{\left(\mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot 0.027777777777777776, -1\right) \cdot y\right) \cdot \frac{1}{\mathsf{fma}\left(x, x \cdot -0.16666666666666666, -1\right)}} \]

   11. Taylor expanded in x around 0

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

       1. mul-1-neg N/A

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

       2. neg-sub0 N/A

          \[\leadsto \color{blue}{\left(0 - y\right)} \cdot \frac{1}{\mathsf{fma}\left(x, x \cdot \frac{-1}{6}, -1\right)} \]

       3. --lowering--.f64 71.3

          \[\leadsto \color{blue}{\left(0 - y\right)} \cdot \frac{1}{\mathsf{fma}\left(x, x \cdot -0.16666666666666666, -1\right)} \]

   13. Simplified 71.3%

       \[\leadsto \color{blue}{\left(0 - y\right)} \cdot \frac{1}{\mathsf{fma}\left(x, x \cdot -0.16666666666666666, -1\right)} \]

   if 1.0000000000000001e-114 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 99.3%

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

   3. Taylor expanded in y around 0

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

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

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

      2. +-commutative N/A

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

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

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

      4. unpow2 N/A

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

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

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 87.8

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

   5. Simplified 87.8%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      11. *-lowering-*.f64 93.6

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

   7. Applied egg-rr 93.6%

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

   8. Taylor expanded in x around 0

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

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

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

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

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

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

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

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

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

      13. *-commutative N/A

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

      14. *-lowering-*.f64 65.7

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

   10. Simplified 65.7%

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

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin x \cdot \sinh y}{x} \leq -2 \cdot 10^{-82}:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right) \cdot \left(y \cdot \mathsf{fma}\left(y, y \cdot 0.16666666666666666, 1\right)\right)\\ \mathbf{elif}\;\frac{\sin x \cdot \sinh y}{x} \leq 10^{-114}:\\ \;\;\;\;y \cdot \frac{-1}{\mathsf{fma}\left(x, x \cdot -0.16666666666666666, -1\right)}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 59.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin x \cdot \sinh y}{x}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-82}:\\ \;\;\;\;\left(0 - y\right) \cdot \mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot 0.027777777777777776, -1\right)\\ \mathbf{elif}\;t\_0 \leq 10^{-114}:\\ \;\;\;\;y \cdot \frac{-1}{\mathsf{fma}\left(x, x \cdot -0.16666666666666666, -1\right)}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sin x) (sinh y)) x)))
   (if (<= t_0 -2e-82)
     (* (- 0.0 y) (fma (* x x) (* (* x x) 0.027777777777777776) -1.0))
     (if (<= t_0 1e-114)
       (* y (/ -1.0 (fma x (* x -0.16666666666666666) -1.0)))
       (*
        y
        (fma
         y
         (* y (fma y (* y 0.008333333333333333) 0.16666666666666666))
         1.0))))))

C

double code(double x, double y) {
	double t_0 = (sin(x) * sinh(y)) / x;
	double tmp;
	if (t_0 <= -2e-82) {
		tmp = (0.0 - y) * fma((x * x), ((x * x) * 0.027777777777777776), -1.0);
	} else if (t_0 <= 1e-114) {
		tmp = y * (-1.0 / fma(x, (x * -0.16666666666666666), -1.0));
	} else {
		tmp = y * fma(y, (y * fma(y, (y * 0.008333333333333333), 0.16666666666666666)), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(sin(x) * sinh(y)) / x)
	tmp = 0.0
	if (t_0 <= -2e-82)
		tmp = Float64(Float64(0.0 - y) * fma(Float64(x * x), Float64(Float64(x * x) * 0.027777777777777776), -1.0));
	elseif (t_0 <= 1e-114)
		tmp = Float64(y * Float64(-1.0 / fma(x, Float64(x * -0.16666666666666666), -1.0)));
	else
		tmp = Float64(y * fma(y, Float64(y * fma(y, Float64(y * 0.008333333333333333), 0.16666666666666666)), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sin[x], $MachinePrecision] * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-82], N[(N[(0.0 - y), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.027777777777777776), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e-114], N[(y * N[(-1.0 / N[(x * N[(x * -0.16666666666666666), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(y * N[(y * N[(y * N[(y * 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -2e-82

   1. Initial program 99.0%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

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

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

      7. *-lowering-*.f64 73.7

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

   5. Simplified 73.7%

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

   6. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

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

      4. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{6} \cdot {x}^{2} + 1\right)} \cdot \left(x \cdot y\right)}{x} \]

      5. *-commutative N/A

         \[\leadsto \frac{\left(\color{blue}{{x}^{2} \cdot \frac{-1}{6}} + 1\right) \cdot \left(x \cdot y\right)}{x} \]

      6. unpow2 N/A

         \[\leadsto \frac{\left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{-1}{6} + 1\right) \cdot \left(x \cdot y\right)}{x} \]

      7. associate-*l* N/A

         \[\leadsto \frac{\left(\color{blue}{x \cdot \left(x \cdot \frac{-1}{6}\right)} + 1\right) \cdot \left(x \cdot y\right)}{x} \]

      8. *-commutative N/A

         \[\leadsto \frac{\left(x \cdot \color{blue}{\left(\frac{-1}{6} \cdot x\right)} + 1\right) \cdot \left(x \cdot y\right)}{x} \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, \frac{-1}{6} \cdot x, 1\right)} \cdot \left(x \cdot y\right)}{x} \]

      10. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, \color{blue}{x \cdot \frac{-1}{6}}, 1\right) \cdot \left(x \cdot y\right)}{x} \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, \color{blue}{x \cdot \frac{-1}{6}}, 1\right) \cdot \left(x \cdot y\right)}{x} \]

      12. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right) \cdot \color{blue}{\left(y \cdot x\right)}}{x} \]

      13. *-lowering-*.f64 38.9

          \[\leadsto \frac{\mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right) \cdot \color{blue}{\left(y \cdot x\right)}}{x} \]

   8. Simplified 38.9%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right) \cdot \left(y \cdot x\right)}}{x} \]
   9. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{\left(x \cdot \left(x \cdot \frac{-1}{6}\right) + 1\right) \cdot \frac{y \cdot x}{x}} \]

      2. flip-+ N/A

         \[\leadsto \color{blue}{\frac{\left(x \cdot \left(x \cdot \frac{-1}{6}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{-1}{6}\right)\right) - 1 \cdot 1}{x \cdot \left(x \cdot \frac{-1}{6}\right) - 1}} \cdot \frac{y \cdot x}{x} \]

      3. associate-/l* N/A

         \[\leadsto \frac{\left(x \cdot \left(x \cdot \frac{-1}{6}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{-1}{6}\right)\right) - 1 \cdot 1}{x \cdot \left(x \cdot \frac{-1}{6}\right) - 1} \cdot \color{blue}{\left(y \cdot \frac{x}{x}\right)} \]

      4. *-inverses N/A

         \[\leadsto \frac{\left(x \cdot \left(x \cdot \frac{-1}{6}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{-1}{6}\right)\right) - 1 \cdot 1}{x \cdot \left(x \cdot \frac{-1}{6}\right) - 1} \cdot \left(y \cdot \color{blue}{1}\right) \]

      5. *-rgt-identity N/A

         \[\leadsto \frac{\left(x \cdot \left(x \cdot \frac{-1}{6}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{-1}{6}\right)\right) - 1 \cdot 1}{x \cdot \left(x \cdot \frac{-1}{6}\right) - 1} \cdot \color{blue}{y} \]

      6. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\left(\left(x \cdot \left(x \cdot \frac{-1}{6}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{-1}{6}\right)\right) - 1 \cdot 1\right) \cdot y}{x \cdot \left(x \cdot \frac{-1}{6}\right) - 1}} \]

      7. div-inv N/A

         \[\leadsto \color{blue}{\left(\left(\left(x \cdot \left(x \cdot \frac{-1}{6}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{-1}{6}\right)\right) - 1 \cdot 1\right) \cdot y\right) \cdot \frac{1}{x \cdot \left(x \cdot \frac{-1}{6}\right) - 1}} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(x \cdot \left(x \cdot \frac{-1}{6}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{-1}{6}\right)\right) - 1 \cdot 1\right) \cdot y\right) \cdot \frac{1}{x \cdot \left(x \cdot \frac{-1}{6}\right) - 1}} \]

   10. Applied egg-rr 30.8%

       \[\leadsto \color{blue}{\left(\mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot 0.027777777777777776, -1\right) \cdot y\right) \cdot \frac{1}{\mathsf{fma}\left(x, x \cdot -0.16666666666666666, -1\right)}} \]

   11. Taylor expanded in x around 0

       \[\leadsto \left(\mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot \frac{1}{36}, -1\right) \cdot y\right) \cdot \color{blue}{-1} \]
   12. Step-by-step derivation

   13. Simplified 41.0%

       \[\leadsto \left(\mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot 0.027777777777777776, -1\right) \cdot y\right) \cdot \color{blue}{-1} \]

   if -2e-82 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 1.0000000000000001e-114

   1. Initial program 78.8%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)} \cdot \sinh y}{x} \]

      2. +-commutative N/A

         \[\leadsto \frac{\left(x \cdot \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2} + 1\right)}\right) \cdot \sinh y}{x} \]

      3. *-commutative N/A

         \[\leadsto \frac{\left(x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{-1}{6}} + 1\right)\right) \cdot \sinh y}{x} \]

      4. unpow2 N/A

         \[\leadsto \frac{\left(x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{-1}{6} + 1\right)\right) \cdot \sinh y}{x} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\left(x \cdot \left(\color{blue}{x \cdot \left(x \cdot \frac{-1}{6}\right)} + 1\right)\right) \cdot \sinh y}{x} \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\left(x \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)}\right) \cdot \sinh y}{x} \]

      7. *-lowering-*.f64 14.4

         \[\leadsto \frac{\left(x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot -0.16666666666666666}, 1\right)\right) \cdot \sinh y}{x} \]

   5. Simplified 14.4%

      \[\leadsto \frac{\color{blue}{\left(x \cdot \mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right)\right)} \cdot \sinh y}{x} \]

   6. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{x \cdot \left(y \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)}}{x} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot y\right) \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)}}{x} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{-1}{6} \cdot {x}^{2}\right) \cdot \left(x \cdot y\right)}}{x} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{-1}{6} \cdot {x}^{2}\right) \cdot \left(x \cdot y\right)}}{x} \]

      4. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{6} \cdot {x}^{2} + 1\right)} \cdot \left(x \cdot y\right)}{x} \]

      5. *-commutative N/A

         \[\leadsto \frac{\left(\color{blue}{{x}^{2} \cdot \frac{-1}{6}} + 1\right) \cdot \left(x \cdot y\right)}{x} \]

      6. unpow2 N/A

         \[\leadsto \frac{\left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{-1}{6} + 1\right) \cdot \left(x \cdot y\right)}{x} \]

      7. associate-*l* N/A

         \[\leadsto \frac{\left(\color{blue}{x \cdot \left(x \cdot \frac{-1}{6}\right)} + 1\right) \cdot \left(x \cdot y\right)}{x} \]

      8. *-commutative N/A

         \[\leadsto \frac{\left(x \cdot \color{blue}{\left(\frac{-1}{6} \cdot x\right)} + 1\right) \cdot \left(x \cdot y\right)}{x} \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, \frac{-1}{6} \cdot x, 1\right)} \cdot \left(x \cdot y\right)}{x} \]

      10. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, \color{blue}{x \cdot \frac{-1}{6}}, 1\right) \cdot \left(x \cdot y\right)}{x} \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, \color{blue}{x \cdot \frac{-1}{6}}, 1\right) \cdot \left(x \cdot y\right)}{x} \]

      12. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right) \cdot \color{blue}{\left(y \cdot x\right)}}{x} \]

      13. *-lowering-*.f64 14.5

          \[\leadsto \frac{\mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right) \cdot \color{blue}{\left(y \cdot x\right)}}{x} \]

   8. Simplified 14.5%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right) \cdot \left(y \cdot x\right)}}{x} \]
   9. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{\left(x \cdot \left(x \cdot \frac{-1}{6}\right) + 1\right) \cdot \frac{y \cdot x}{x}} \]

      2. flip-+ N/A

         \[\leadsto \color{blue}{\frac{\left(x \cdot \left(x \cdot \frac{-1}{6}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{-1}{6}\right)\right) - 1 \cdot 1}{x \cdot \left(x \cdot \frac{-1}{6}\right) - 1}} \cdot \frac{y \cdot x}{x} \]

      3. associate-/l* N/A

         \[\leadsto \frac{\left(x \cdot \left(x \cdot \frac{-1}{6}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{-1}{6}\right)\right) - 1 \cdot 1}{x \cdot \left(x \cdot \frac{-1}{6}\right) - 1} \cdot \color{blue}{\left(y \cdot \frac{x}{x}\right)} \]

      4. *-inverses N/A

         \[\leadsto \frac{\left(x \cdot \left(x \cdot \frac{-1}{6}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{-1}{6}\right)\right) - 1 \cdot 1}{x \cdot \left(x \cdot \frac{-1}{6}\right) - 1} \cdot \left(y \cdot \color{blue}{1}\right) \]

      5. *-rgt-identity N/A

         \[\leadsto \frac{\left(x \cdot \left(x \cdot \frac{-1}{6}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{-1}{6}\right)\right) - 1 \cdot 1}{x \cdot \left(x \cdot \frac{-1}{6}\right) - 1} \cdot \color{blue}{y} \]

      6. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\left(\left(x \cdot \left(x \cdot \frac{-1}{6}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{-1}{6}\right)\right) - 1 \cdot 1\right) \cdot y}{x \cdot \left(x \cdot \frac{-1}{6}\right) - 1}} \]

      7. div-inv N/A

         \[\leadsto \color{blue}{\left(\left(\left(x \cdot \left(x \cdot \frac{-1}{6}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{-1}{6}\right)\right) - 1 \cdot 1\right) \cdot y\right) \cdot \frac{1}{x \cdot \left(x \cdot \frac{-1}{6}\right) - 1}} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(x \cdot \left(x \cdot \frac{-1}{6}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{-1}{6}\right)\right) - 1 \cdot 1\right) \cdot y\right) \cdot \frac{1}{x \cdot \left(x \cdot \frac{-1}{6}\right) - 1}} \]

   10. Applied egg-rr 34.8%

       \[\leadsto \color{blue}{\left(\mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot 0.027777777777777776, -1\right) \cdot y\right) \cdot \frac{1}{\mathsf{fma}\left(x, x \cdot -0.16666666666666666, -1\right)}} \]

   11. Taylor expanded in x around 0

       \[\leadsto \color{blue}{\left(-1 \cdot y\right)} \cdot \frac{1}{\mathsf{fma}\left(x, x \cdot \frac{-1}{6}, -1\right)} \]
   12. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(y\right)\right)} \cdot \frac{1}{\mathsf{fma}\left(x, x \cdot \frac{-1}{6}, -1\right)} \]

       2. neg-sub0 N/A

          \[\leadsto \color{blue}{\left(0 - y\right)} \cdot \frac{1}{\mathsf{fma}\left(x, x \cdot \frac{-1}{6}, -1\right)} \]

       3. --lowering--.f64 71.3

          \[\leadsto \color{blue}{\left(0 - y\right)} \cdot \frac{1}{\mathsf{fma}\left(x, x \cdot -0.16666666666666666, -1\right)} \]

   13. Simplified 71.3%

       \[\leadsto \color{blue}{\left(0 - y\right)} \cdot \frac{1}{\mathsf{fma}\left(x, x \cdot -0.16666666666666666, -1\right)} \]

   if 1.0000000000000001e-114 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 99.3%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\sin x \cdot \color{blue}{\left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)}}{x} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \frac{\sin x \cdot \color{blue}{\left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)}}{x} \]

      2. +-commutative N/A

         \[\leadsto \frac{\sin x \cdot \left(y \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) + 1\right)}\right)}{x} \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\sin x \cdot \left(y \cdot \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{1}{6} + \frac{1}{120} \cdot {y}^{2}, 1\right)}\right)}{x} \]

      4. unpow2 N/A

         \[\leadsto \frac{\sin x \cdot \left(y \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{6} + \frac{1}{120} \cdot {y}^{2}, 1\right)\right)}{x} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{\sin x \cdot \left(y \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{6} + \frac{1}{120} \cdot {y}^{2}, 1\right)\right)}{x} \]

      6. +-commutative N/A

         \[\leadsto \frac{\sin x \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{\frac{1}{120} \cdot {y}^{2} + \frac{1}{6}}, 1\right)\right)}{x} \]

      7. *-commutative N/A

         \[\leadsto \frac{\sin x \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{{y}^{2} \cdot \frac{1}{120}} + \frac{1}{6}, 1\right)\right)}{x} \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\sin x \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{1}{120}, \frac{1}{6}\right)}, 1\right)\right)}{x} \]

      9. unpow2 N/A

         \[\leadsto \frac{\sin x \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{120}, \frac{1}{6}\right), 1\right)\right)}{x} \]

      10. *-lowering-*.f64 87.8

          \[\leadsto \frac{\sin x \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, 0.008333333333333333, 0.16666666666666666\right), 1\right)\right)}{x} \]

   5. Simplified 87.8%

      \[\leadsto \frac{\sin x \cdot \color{blue}{\left(y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), 1\right)\right)}}{x} \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{\sin x \cdot \frac{y \cdot \left(\left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{120} + \frac{1}{6}\right) + 1\right)}{x}} \]

      2. clear-num N/A

         \[\leadsto \sin x \cdot \color{blue}{\frac{1}{\frac{x}{y \cdot \left(\left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{120} + \frac{1}{6}\right) + 1\right)}}} \]

      3. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{\sin x}{\frac{x}{y \cdot \left(\left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{120} + \frac{1}{6}\right) + 1\right)}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin x}{\frac{x}{y \cdot \left(\left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{120} + \frac{1}{6}\right) + 1\right)}}} \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \frac{\color{blue}{\sin x}}{\frac{x}{y \cdot \left(\left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{120} + \frac{1}{6}\right) + 1\right)}} \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \frac{\sin x}{\color{blue}{\frac{x}{y \cdot \left(\left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{120} + \frac{1}{6}\right) + 1\right)}}} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{\sin x}{\frac{x}{\color{blue}{y \cdot \left(\left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{120} + \frac{1}{6}\right) + 1\right)}}} \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\sin x}{\frac{x}{y \cdot \color{blue}{\mathsf{fma}\left(y \cdot y, \left(y \cdot y\right) \cdot \frac{1}{120} + \frac{1}{6}, 1\right)}}} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \frac{\sin x}{\frac{x}{y \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, \left(y \cdot y\right) \cdot \frac{1}{120} + \frac{1}{6}, 1\right)}} \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\sin x}{\frac{x}{y \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{\mathsf{fma}\left(y \cdot y, \frac{1}{120}, \frac{1}{6}\right)}, 1\right)}} \]

      11. *-lowering-*.f64 93.6

          \[\leadsto \frac{\sin x}{\frac{x}{y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, 0.008333333333333333, 0.16666666666666666\right), 1\right)}} \]

   7. Applied egg-rr 93.6%

      \[\leadsto \color{blue}{\frac{\sin x}{\frac{x}{y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), 1\right)}}} \]

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto y \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) + 1\right)} \]

      3. unpow2 N/A

         \[\leadsto y \cdot \left(\color{blue}{\left(y \cdot y\right)} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) + 1\right) \]

      4. associate-*l* N/A

         \[\leadsto y \cdot \left(\color{blue}{y \cdot \left(y \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} + 1\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto y \cdot \color{blue}{\mathsf{fma}\left(y, y \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right), 1\right)} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto y \cdot \mathsf{fma}\left(y, \color{blue}{y \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)}, 1\right) \]

      7. +-commutative N/A

         \[\leadsto y \cdot \mathsf{fma}\left(y, y \cdot \color{blue}{\left(\frac{1}{120} \cdot {y}^{2} + \frac{1}{6}\right)}, 1\right) \]

      8. *-commutative N/A

         \[\leadsto y \cdot \mathsf{fma}\left(y, y \cdot \left(\color{blue}{{y}^{2} \cdot \frac{1}{120}} + \frac{1}{6}\right), 1\right) \]

      9. unpow2 N/A

         \[\leadsto y \cdot \mathsf{fma}\left(y, y \cdot \left(\color{blue}{\left(y \cdot y\right)} \cdot \frac{1}{120} + \frac{1}{6}\right), 1\right) \]

      10. associate-*l* N/A

          \[\leadsto y \cdot \mathsf{fma}\left(y, y \cdot \left(\color{blue}{y \cdot \left(y \cdot \frac{1}{120}\right)} + \frac{1}{6}\right), 1\right) \]

      11. *-commutative N/A

          \[\leadsto y \cdot \mathsf{fma}\left(y, y \cdot \left(y \cdot \color{blue}{\left(\frac{1}{120} \cdot y\right)} + \frac{1}{6}\right), 1\right) \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto y \cdot \mathsf{fma}\left(y, y \cdot \color{blue}{\mathsf{fma}\left(y, \frac{1}{120} \cdot y, \frac{1}{6}\right)}, 1\right) \]

      13. *-commutative N/A

          \[\leadsto y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, \color{blue}{y \cdot \frac{1}{120}}, \frac{1}{6}\right), 1\right) \]

      14. *-lowering-*.f64 65.7

          \[\leadsto y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, \color{blue}{y \cdot 0.008333333333333333}, 0.16666666666666666\right), 1\right) \]

   10. Simplified 65.7%

       \[\leadsto \color{blue}{y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 60.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin x \cdot \sinh y}{x} \leq -2 \cdot 10^{-82}:\\ \;\;\;\;\left(0 - y\right) \cdot \mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot 0.027777777777777776, -1\right)\\ \mathbf{elif}\;\frac{\sin x \cdot \sinh y}{x} \leq 10^{-114}:\\ \;\;\;\;y \cdot \frac{-1}{\mathsf{fma}\left(x, x \cdot -0.16666666666666666, -1\right)}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 58.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin x \cdot \sinh y}{x}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-82}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right)\\ \mathbf{elif}\;t\_0 \leq 10^{-114}:\\ \;\;\;\;y \cdot \frac{-1}{\mathsf{fma}\left(x, x \cdot -0.16666666666666666, -1\right)}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sin x) (sinh y)) x)))
   (if (<= t_0 -2e-82)
     (* y (fma x (* x -0.16666666666666666) 1.0))
     (if (<= t_0 1e-114)
       (* y (/ -1.0 (fma x (* x -0.16666666666666666) -1.0)))
       (*
        y
        (fma
         y
         (* y (fma y (* y 0.008333333333333333) 0.16666666666666666))
         1.0))))))

C

double code(double x, double y) {
	double t_0 = (sin(x) * sinh(y)) / x;
	double tmp;
	if (t_0 <= -2e-82) {
		tmp = y * fma(x, (x * -0.16666666666666666), 1.0);
	} else if (t_0 <= 1e-114) {
		tmp = y * (-1.0 / fma(x, (x * -0.16666666666666666), -1.0));
	} else {
		tmp = y * fma(y, (y * fma(y, (y * 0.008333333333333333), 0.16666666666666666)), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(sin(x) * sinh(y)) / x)
	tmp = 0.0
	if (t_0 <= -2e-82)
		tmp = Float64(y * fma(x, Float64(x * -0.16666666666666666), 1.0));
	elseif (t_0 <= 1e-114)
		tmp = Float64(y * Float64(-1.0 / fma(x, Float64(x * -0.16666666666666666), -1.0)));
	else
		tmp = Float64(y * fma(y, Float64(y * fma(y, Float64(y * 0.008333333333333333), 0.16666666666666666)), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sin[x], $MachinePrecision] * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-82], N[(y * N[(x * N[(x * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e-114], N[(y * N[(-1.0 / N[(x * N[(x * -0.16666666666666666), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(y * N[(y * N[(y * N[(y * 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -2e-82

   1. Initial program 99.0%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)} \cdot \sinh y}{x} \]

      2. +-commutative N/A

         \[\leadsto \frac{\left(x \cdot \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2} + 1\right)}\right) \cdot \sinh y}{x} \]

      3. *-commutative N/A

         \[\leadsto \frac{\left(x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{-1}{6}} + 1\right)\right) \cdot \sinh y}{x} \]

      4. unpow2 N/A

         \[\leadsto \frac{\left(x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{-1}{6} + 1\right)\right) \cdot \sinh y}{x} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\left(x \cdot \left(\color{blue}{x \cdot \left(x \cdot \frac{-1}{6}\right)} + 1\right)\right) \cdot \sinh y}{x} \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\left(x \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)}\right) \cdot \sinh y}{x} \]

      7. *-lowering-*.f64 73.7

         \[\leadsto \frac{\left(x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot -0.16666666666666666}, 1\right)\right) \cdot \sinh y}{x} \]

   5. Simplified 73.7%

      \[\leadsto \frac{\color{blue}{\left(x \cdot \mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right)\right)} \cdot \sinh y}{x} \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{y \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)} \]

      2. +-commutative N/A

         \[\leadsto y \cdot \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2} + 1\right)} \]

      3. *-commutative N/A

         \[\leadsto y \cdot \left(\color{blue}{{x}^{2} \cdot \frac{-1}{6}} + 1\right) \]

      4. unpow2 N/A

         \[\leadsto y \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{-1}{6} + 1\right) \]

      5. associate-*l* N/A

         \[\leadsto y \cdot \left(\color{blue}{x \cdot \left(x \cdot \frac{-1}{6}\right)} + 1\right) \]

      6. *-commutative N/A

         \[\leadsto y \cdot \left(x \cdot \color{blue}{\left(\frac{-1}{6} \cdot x\right)} + 1\right) \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto y \cdot \color{blue}{\mathsf{fma}\left(x, \frac{-1}{6} \cdot x, 1\right)} \]

      8. *-commutative N/A

         \[\leadsto y \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{-1}{6}}, 1\right) \]

      9. *-lowering-*.f64 38.7

         \[\leadsto y \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot -0.16666666666666666}, 1\right) \]

   8. Simplified 38.7%

      \[\leadsto \color{blue}{y \cdot \mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right)} \]

   if -2e-82 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 1.0000000000000001e-114

   1. Initial program 78.8%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)} \cdot \sinh y}{x} \]

      2. +-commutative N/A

         \[\leadsto \frac{\left(x \cdot \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2} + 1\right)}\right) \cdot \sinh y}{x} \]

      3. *-commutative N/A

         \[\leadsto \frac{\left(x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{-1}{6}} + 1\right)\right) \cdot \sinh y}{x} \]

      4. unpow2 N/A

         \[\leadsto \frac{\left(x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{-1}{6} + 1\right)\right) \cdot \sinh y}{x} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\left(x \cdot \left(\color{blue}{x \cdot \left(x \cdot \frac{-1}{6}\right)} + 1\right)\right) \cdot \sinh y}{x} \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\left(x \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)}\right) \cdot \sinh y}{x} \]

      7. *-lowering-*.f64 14.4

         \[\leadsto \frac{\left(x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot -0.16666666666666666}, 1\right)\right) \cdot \sinh y}{x} \]

   5. Simplified 14.4%

      \[\leadsto \frac{\color{blue}{\left(x \cdot \mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right)\right)} \cdot \sinh y}{x} \]

   6. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{x \cdot \left(y \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)}}{x} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot y\right) \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)}}{x} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{-1}{6} \cdot {x}^{2}\right) \cdot \left(x \cdot y\right)}}{x} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{-1}{6} \cdot {x}^{2}\right) \cdot \left(x \cdot y\right)}}{x} \]

      4. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{6} \cdot {x}^{2} + 1\right)} \cdot \left(x \cdot y\right)}{x} \]

      5. *-commutative N/A

         \[\leadsto \frac{\left(\color{blue}{{x}^{2} \cdot \frac{-1}{6}} + 1\right) \cdot \left(x \cdot y\right)}{x} \]

      6. unpow2 N/A

         \[\leadsto \frac{\left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{-1}{6} + 1\right) \cdot \left(x \cdot y\right)}{x} \]

      7. associate-*l* N/A

         \[\leadsto \frac{\left(\color{blue}{x \cdot \left(x \cdot \frac{-1}{6}\right)} + 1\right) \cdot \left(x \cdot y\right)}{x} \]

      8. *-commutative N/A

         \[\leadsto \frac{\left(x \cdot \color{blue}{\left(\frac{-1}{6} \cdot x\right)} + 1\right) \cdot \left(x \cdot y\right)}{x} \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, \frac{-1}{6} \cdot x, 1\right)} \cdot \left(x \cdot y\right)}{x} \]

      10. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, \color{blue}{x \cdot \frac{-1}{6}}, 1\right) \cdot \left(x \cdot y\right)}{x} \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, \color{blue}{x \cdot \frac{-1}{6}}, 1\right) \cdot \left(x \cdot y\right)}{x} \]

      12. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right) \cdot \color{blue}{\left(y \cdot x\right)}}{x} \]

      13. *-lowering-*.f64 14.5

          \[\leadsto \frac{\mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right) \cdot \color{blue}{\left(y \cdot x\right)}}{x} \]

   8. Simplified 14.5%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right) \cdot \left(y \cdot x\right)}}{x} \]
   9. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{\left(x \cdot \left(x \cdot \frac{-1}{6}\right) + 1\right) \cdot \frac{y \cdot x}{x}} \]

      2. flip-+ N/A

         \[\leadsto \color{blue}{\frac{\left(x \cdot \left(x \cdot \frac{-1}{6}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{-1}{6}\right)\right) - 1 \cdot 1}{x \cdot \left(x \cdot \frac{-1}{6}\right) - 1}} \cdot \frac{y \cdot x}{x} \]

      3. associate-/l* N/A

         \[\leadsto \frac{\left(x \cdot \left(x \cdot \frac{-1}{6}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{-1}{6}\right)\right) - 1 \cdot 1}{x \cdot \left(x \cdot \frac{-1}{6}\right) - 1} \cdot \color{blue}{\left(y \cdot \frac{x}{x}\right)} \]

      4. *-inverses N/A

         \[\leadsto \frac{\left(x \cdot \left(x \cdot \frac{-1}{6}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{-1}{6}\right)\right) - 1 \cdot 1}{x \cdot \left(x \cdot \frac{-1}{6}\right) - 1} \cdot \left(y \cdot \color{blue}{1}\right) \]

      5. *-rgt-identity N/A

         \[\leadsto \frac{\left(x \cdot \left(x \cdot \frac{-1}{6}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{-1}{6}\right)\right) - 1 \cdot 1}{x \cdot \left(x \cdot \frac{-1}{6}\right) - 1} \cdot \color{blue}{y} \]

      6. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\left(\left(x \cdot \left(x \cdot \frac{-1}{6}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{-1}{6}\right)\right) - 1 \cdot 1\right) \cdot y}{x \cdot \left(x \cdot \frac{-1}{6}\right) - 1}} \]

      7. div-inv N/A

         \[\leadsto \color{blue}{\left(\left(\left(x \cdot \left(x \cdot \frac{-1}{6}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{-1}{6}\right)\right) - 1 \cdot 1\right) \cdot y\right) \cdot \frac{1}{x \cdot \left(x \cdot \frac{-1}{6}\right) - 1}} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(x \cdot \left(x \cdot \frac{-1}{6}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{-1}{6}\right)\right) - 1 \cdot 1\right) \cdot y\right) \cdot \frac{1}{x \cdot \left(x \cdot \frac{-1}{6}\right) - 1}} \]

   10. Applied egg-rr 34.8%

       \[\leadsto \color{blue}{\left(\mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot 0.027777777777777776, -1\right) \cdot y\right) \cdot \frac{1}{\mathsf{fma}\left(x, x \cdot -0.16666666666666666, -1\right)}} \]

   11. Taylor expanded in x around 0

       \[\leadsto \color{blue}{\left(-1 \cdot y\right)} \cdot \frac{1}{\mathsf{fma}\left(x, x \cdot \frac{-1}{6}, -1\right)} \]
   12. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(y\right)\right)} \cdot \frac{1}{\mathsf{fma}\left(x, x \cdot \frac{-1}{6}, -1\right)} \]

       2. neg-sub0 N/A

          \[\leadsto \color{blue}{\left(0 - y\right)} \cdot \frac{1}{\mathsf{fma}\left(x, x \cdot \frac{-1}{6}, -1\right)} \]

       3. --lowering--.f64 71.3

          \[\leadsto \color{blue}{\left(0 - y\right)} \cdot \frac{1}{\mathsf{fma}\left(x, x \cdot -0.16666666666666666, -1\right)} \]

   13. Simplified 71.3%

       \[\leadsto \color{blue}{\left(0 - y\right)} \cdot \frac{1}{\mathsf{fma}\left(x, x \cdot -0.16666666666666666, -1\right)} \]

   if 1.0000000000000001e-114 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 99.3%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\sin x \cdot \color{blue}{\left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)}}{x} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \frac{\sin x \cdot \color{blue}{\left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)}}{x} \]

      2. +-commutative N/A

         \[\leadsto \frac{\sin x \cdot \left(y \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) + 1\right)}\right)}{x} \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\sin x \cdot \left(y \cdot \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{1}{6} + \frac{1}{120} \cdot {y}^{2}, 1\right)}\right)}{x} \]

      4. unpow2 N/A

         \[\leadsto \frac{\sin x \cdot \left(y \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{6} + \frac{1}{120} \cdot {y}^{2}, 1\right)\right)}{x} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{\sin x \cdot \left(y \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{6} + \frac{1}{120} \cdot {y}^{2}, 1\right)\right)}{x} \]

      6. +-commutative N/A

         \[\leadsto \frac{\sin x \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{\frac{1}{120} \cdot {y}^{2} + \frac{1}{6}}, 1\right)\right)}{x} \]

      7. *-commutative N/A

         \[\leadsto \frac{\sin x \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{{y}^{2} \cdot \frac{1}{120}} + \frac{1}{6}, 1\right)\right)}{x} \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\sin x \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{1}{120}, \frac{1}{6}\right)}, 1\right)\right)}{x} \]

      9. unpow2 N/A

         \[\leadsto \frac{\sin x \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{120}, \frac{1}{6}\right), 1\right)\right)}{x} \]

      10. *-lowering-*.f64 87.8

          \[\leadsto \frac{\sin x \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, 0.008333333333333333, 0.16666666666666666\right), 1\right)\right)}{x} \]

   5. Simplified 87.8%

      \[\leadsto \frac{\sin x \cdot \color{blue}{\left(y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), 1\right)\right)}}{x} \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{\sin x \cdot \frac{y \cdot \left(\left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{120} + \frac{1}{6}\right) + 1\right)}{x}} \]

      2. clear-num N/A

         \[\leadsto \sin x \cdot \color{blue}{\frac{1}{\frac{x}{y \cdot \left(\left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{120} + \frac{1}{6}\right) + 1\right)}}} \]

      3. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{\sin x}{\frac{x}{y \cdot \left(\left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{120} + \frac{1}{6}\right) + 1\right)}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin x}{\frac{x}{y \cdot \left(\left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{120} + \frac{1}{6}\right) + 1\right)}}} \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \frac{\color{blue}{\sin x}}{\frac{x}{y \cdot \left(\left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{120} + \frac{1}{6}\right) + 1\right)}} \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \frac{\sin x}{\color{blue}{\frac{x}{y \cdot \left(\left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{120} + \frac{1}{6}\right) + 1\right)}}} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{\sin x}{\frac{x}{\color{blue}{y \cdot \left(\left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{120} + \frac{1}{6}\right) + 1\right)}}} \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\sin x}{\frac{x}{y \cdot \color{blue}{\mathsf{fma}\left(y \cdot y, \left(y \cdot y\right) \cdot \frac{1}{120} + \frac{1}{6}, 1\right)}}} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \frac{\sin x}{\frac{x}{y \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, \left(y \cdot y\right) \cdot \frac{1}{120} + \frac{1}{6}, 1\right)}} \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\sin x}{\frac{x}{y \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{\mathsf{fma}\left(y \cdot y, \frac{1}{120}, \frac{1}{6}\right)}, 1\right)}} \]

      11. *-lowering-*.f64 93.6

          \[\leadsto \frac{\sin x}{\frac{x}{y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, 0.008333333333333333, 0.16666666666666666\right), 1\right)}} \]

   7. Applied egg-rr 93.6%

      \[\leadsto \color{blue}{\frac{\sin x}{\frac{x}{y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), 1\right)}}} \]

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto y \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) + 1\right)} \]

      3. unpow2 N/A

         \[\leadsto y \cdot \left(\color{blue}{\left(y \cdot y\right)} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) + 1\right) \]

      4. associate-*l* N/A

         \[\leadsto y \cdot \left(\color{blue}{y \cdot \left(y \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} + 1\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto y \cdot \color{blue}{\mathsf{fma}\left(y, y \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right), 1\right)} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto y \cdot \mathsf{fma}\left(y, \color{blue}{y \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)}, 1\right) \]

      7. +-commutative N/A

         \[\leadsto y \cdot \mathsf{fma}\left(y, y \cdot \color{blue}{\left(\frac{1}{120} \cdot {y}^{2} + \frac{1}{6}\right)}, 1\right) \]

      8. *-commutative N/A

         \[\leadsto y \cdot \mathsf{fma}\left(y, y \cdot \left(\color{blue}{{y}^{2} \cdot \frac{1}{120}} + \frac{1}{6}\right), 1\right) \]

      9. unpow2 N/A

         \[\leadsto y \cdot \mathsf{fma}\left(y, y \cdot \left(\color{blue}{\left(y \cdot y\right)} \cdot \frac{1}{120} + \frac{1}{6}\right), 1\right) \]

      10. associate-*l* N/A

          \[\leadsto y \cdot \mathsf{fma}\left(y, y \cdot \left(\color{blue}{y \cdot \left(y \cdot \frac{1}{120}\right)} + \frac{1}{6}\right), 1\right) \]

      11. *-commutative N/A

          \[\leadsto y \cdot \mathsf{fma}\left(y, y \cdot \left(y \cdot \color{blue}{\left(\frac{1}{120} \cdot y\right)} + \frac{1}{6}\right), 1\right) \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto y \cdot \mathsf{fma}\left(y, y \cdot \color{blue}{\mathsf{fma}\left(y, \frac{1}{120} \cdot y, \frac{1}{6}\right)}, 1\right) \]

      13. *-commutative N/A

          \[\leadsto y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, \color{blue}{y \cdot \frac{1}{120}}, \frac{1}{6}\right), 1\right) \]

      14. *-lowering-*.f64 65.7

          \[\leadsto y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, \color{blue}{y \cdot 0.008333333333333333}, 0.16666666666666666\right), 1\right) \]

   10. Simplified 65.7%

       \[\leadsto \color{blue}{y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 59.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin x \cdot \sinh y}{x} \leq -2 \cdot 10^{-82}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right)\\ \mathbf{elif}\;\frac{\sin x \cdot \sinh y}{x} \leq 10^{-114}:\\ \;\;\;\;y \cdot \frac{-1}{\mathsf{fma}\left(x, x \cdot -0.16666666666666666, -1\right)}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 57.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin x \cdot \sinh y}{x}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-202}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;x \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sin x) (sinh y)) x)))
   (if (<= t_0 -2e-202)
     (* y (fma x (* x -0.16666666666666666) 1.0))
     (if (<= t_0 0.0)
       (* x (/ y x))
       (*
        y
        (fma
         y
         (* y (fma y (* y 0.008333333333333333) 0.16666666666666666))
         1.0))))))

C

double code(double x, double y) {
	double t_0 = (sin(x) * sinh(y)) / x;
	double tmp;
	if (t_0 <= -2e-202) {
		tmp = y * fma(x, (x * -0.16666666666666666), 1.0);
	} else if (t_0 <= 0.0) {
		tmp = x * (y / x);
	} else {
		tmp = y * fma(y, (y * fma(y, (y * 0.008333333333333333), 0.16666666666666666)), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(sin(x) * sinh(y)) / x)
	tmp = 0.0
	if (t_0 <= -2e-202)
		tmp = Float64(y * fma(x, Float64(x * -0.16666666666666666), 1.0));
	elseif (t_0 <= 0.0)
		tmp = Float64(x * Float64(y / x));
	else
		tmp = Float64(y * fma(y, Float64(y * fma(y, Float64(y * 0.008333333333333333), 0.16666666666666666)), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sin[x], $MachinePrecision] * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-202], N[(y * N[(x * N[(x * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(x * N[(y / x), $MachinePrecision]), $MachinePrecision], N[(y * N[(y * N[(y * N[(y * N[(y * 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -2.0000000000000001e-202

   1. Initial program 99.1%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)} \cdot \sinh y}{x} \]

      2. +-commutative N/A

         \[\leadsto \frac{\left(x \cdot \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2} + 1\right)}\right) \cdot \sinh y}{x} \]

      3. *-commutative N/A

         \[\leadsto \frac{\left(x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{-1}{6}} + 1\right)\right) \cdot \sinh y}{x} \]

      4. unpow2 N/A

         \[\leadsto \frac{\left(x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{-1}{6} + 1\right)\right) \cdot \sinh y}{x} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\left(x \cdot \left(\color{blue}{x \cdot \left(x \cdot \frac{-1}{6}\right)} + 1\right)\right) \cdot \sinh y}{x} \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\left(x \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)}\right) \cdot \sinh y}{x} \]

      7. *-lowering-*.f64 72.3

         \[\leadsto \frac{\left(x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot -0.16666666666666666}, 1\right)\right) \cdot \sinh y}{x} \]

   5. Simplified 72.3%

      \[\leadsto \frac{\color{blue}{\left(x \cdot \mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right)\right)} \cdot \sinh y}{x} \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{y \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)} \]

      2. +-commutative N/A

         \[\leadsto y \cdot \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2} + 1\right)} \]

      3. *-commutative N/A

         \[\leadsto y \cdot \left(\color{blue}{{x}^{2} \cdot \frac{-1}{6}} + 1\right) \]

      4. unpow2 N/A

         \[\leadsto y \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{-1}{6} + 1\right) \]

      5. associate-*l* N/A

         \[\leadsto y \cdot \left(\color{blue}{x \cdot \left(x \cdot \frac{-1}{6}\right)} + 1\right) \]

      6. *-commutative N/A

         \[\leadsto y \cdot \left(x \cdot \color{blue}{\left(\frac{-1}{6} \cdot x\right)} + 1\right) \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto y \cdot \color{blue}{\mathsf{fma}\left(x, \frac{-1}{6} \cdot x, 1\right)} \]

      8. *-commutative N/A

         \[\leadsto y \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{-1}{6}}, 1\right) \]

      9. *-lowering-*.f64 42.6

         \[\leadsto y \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot -0.16666666666666666}, 1\right) \]

   8. Simplified 42.6%

      \[\leadsto \color{blue}{y \cdot \mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right)} \]

   if -2.0000000000000001e-202 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 0.0

   1. Initial program 69.8%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)} \cdot \sinh y}{x} \]

      2. +-commutative N/A

         \[\leadsto \frac{\left(x \cdot \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2} + 1\right)}\right) \cdot \sinh y}{x} \]

      3. *-commutative N/A

         \[\leadsto \frac{\left(x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{-1}{6}} + 1\right)\right) \cdot \sinh y}{x} \]

      4. unpow2 N/A

         \[\leadsto \frac{\left(x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{-1}{6} + 1\right)\right) \cdot \sinh y}{x} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\left(x \cdot \left(\color{blue}{x \cdot \left(x \cdot \frac{-1}{6}\right)} + 1\right)\right) \cdot \sinh y}{x} \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\left(x \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)}\right) \cdot \sinh y}{x} \]

      7. *-lowering-*.f64 6.1

         \[\leadsto \frac{\left(x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot -0.16666666666666666}, 1\right)\right) \cdot \sinh y}{x} \]

   5. Simplified 6.1%

      \[\leadsto \frac{\color{blue}{\left(x \cdot \mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right)\right)} \cdot \sinh y}{x} \]

   6. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{x \cdot \left(y \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)}}{x} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot y\right) \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)}}{x} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{-1}{6} \cdot {x}^{2}\right) \cdot \left(x \cdot y\right)}}{x} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{-1}{6} \cdot {x}^{2}\right) \cdot \left(x \cdot y\right)}}{x} \]

      4. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{6} \cdot {x}^{2} + 1\right)} \cdot \left(x \cdot y\right)}{x} \]

      5. *-commutative N/A

         \[\leadsto \frac{\left(\color{blue}{{x}^{2} \cdot \frac{-1}{6}} + 1\right) \cdot \left(x \cdot y\right)}{x} \]

      6. unpow2 N/A

         \[\leadsto \frac{\left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{-1}{6} + 1\right) \cdot \left(x \cdot y\right)}{x} \]

      7. associate-*l* N/A

         \[\leadsto \frac{\left(\color{blue}{x \cdot \left(x \cdot \frac{-1}{6}\right)} + 1\right) \cdot \left(x \cdot y\right)}{x} \]

      8. *-commutative N/A

         \[\leadsto \frac{\left(x \cdot \color{blue}{\left(\frac{-1}{6} \cdot x\right)} + 1\right) \cdot \left(x \cdot y\right)}{x} \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, \frac{-1}{6} \cdot x, 1\right)} \cdot \left(x \cdot y\right)}{x} \]

      10. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, \color{blue}{x \cdot \frac{-1}{6}}, 1\right) \cdot \left(x \cdot y\right)}{x} \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, \color{blue}{x \cdot \frac{-1}{6}}, 1\right) \cdot \left(x \cdot y\right)}{x} \]

      12. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right) \cdot \color{blue}{\left(y \cdot x\right)}}{x} \]

      13. *-lowering-*.f64 6.2

          \[\leadsto \frac{\mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right) \cdot \color{blue}{\left(y \cdot x\right)}}{x} \]

   8. Simplified 6.2%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right) \cdot \left(y \cdot x\right)}}{x} \]
   9. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot \left(x \cdot \frac{-1}{6}\right) + 1\right) \cdot y\right) \cdot x}}{x} \]

      2. *-rgt-identity N/A

         \[\leadsto \frac{\left(\left(x \cdot \left(x \cdot \frac{-1}{6}\right) + 1\right) \cdot y\right) \cdot x}{\color{blue}{x \cdot 1}} \]

      3. times-frac N/A

         \[\leadsto \color{blue}{\frac{\left(x \cdot \left(x \cdot \frac{-1}{6}\right) + 1\right) \cdot y}{x} \cdot \frac{x}{1}} \]

      4. /-rgt-identity N/A

         \[\leadsto \frac{\left(x \cdot \left(x \cdot \frac{-1}{6}\right) + 1\right) \cdot y}{x} \cdot \color{blue}{x} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(x \cdot \left(x \cdot \frac{-1}{6}\right) + 1\right) \cdot y}{x} \cdot x} \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(x \cdot \left(x \cdot \frac{-1}{6}\right) + 1\right) \cdot y}{x}} \cdot x \]

      7. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(x \cdot \left(x \cdot \frac{-1}{6}\right) + 1\right)}}{x} \cdot x \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(x \cdot \left(x \cdot \frac{-1}{6}\right) + 1\right)}}{x} \cdot x \]

      9. associate-*r* N/A

         \[\leadsto \frac{y \cdot \left(\color{blue}{\left(x \cdot x\right) \cdot \frac{-1}{6}} + 1\right)}{x} \cdot x \]

      10. *-commutative N/A

          \[\leadsto \frac{y \cdot \left(\color{blue}{\frac{-1}{6} \cdot \left(x \cdot x\right)} + 1\right)}{x} \cdot x \]

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{y \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{6}, x \cdot x, 1\right)}}{x} \cdot x \]

      12. *-lowering-*.f64 36.2

          \[\leadsto \frac{y \cdot \mathsf{fma}\left(-0.16666666666666666, \color{blue}{x \cdot x}, 1\right)}{x} \cdot x \]

   10. Applied egg-rr 36.2%

       \[\leadsto \color{blue}{\frac{y \cdot \mathsf{fma}\left(-0.16666666666666666, x \cdot x, 1\right)}{x} \cdot x} \]

   11. Taylor expanded in x around 0

       \[\leadsto \color{blue}{\frac{y}{x}} \cdot x \]
   12. Step-by-step derivation

       1. /-lowering-/.f64 81.3

          \[\leadsto \color{blue}{\frac{y}{x}} \cdot x \]

   13. Simplified 81.3%

       \[\leadsto \color{blue}{\frac{y}{x}} \cdot x \]

   if 0.0 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 99.4%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\sin x \cdot \color{blue}{\left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)}}{x} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \frac{\sin x \cdot \color{blue}{\left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)}}{x} \]

      2. +-commutative N/A

         \[\leadsto \frac{\sin x \cdot \left(y \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) + 1\right)}\right)}{x} \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\sin x \cdot \left(y \cdot \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{1}{6} + \frac{1}{120} \cdot {y}^{2}, 1\right)}\right)}{x} \]

      4. unpow2 N/A

         \[\leadsto \frac{\sin x \cdot \left(y \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{6} + \frac{1}{120} \cdot {y}^{2}, 1\right)\right)}{x} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{\sin x \cdot \left(y \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{6} + \frac{1}{120} \cdot {y}^{2}, 1\right)\right)}{x} \]

      6. +-commutative N/A

         \[\leadsto \frac{\sin x \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{\frac{1}{120} \cdot {y}^{2} + \frac{1}{6}}, 1\right)\right)}{x} \]

      7. *-commutative N/A

         \[\leadsto \frac{\sin x \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{{y}^{2} \cdot \frac{1}{120}} + \frac{1}{6}, 1\right)\right)}{x} \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\sin x \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{1}{120}, \frac{1}{6}\right)}, 1\right)\right)}{x} \]

      9. unpow2 N/A

         \[\leadsto \frac{\sin x \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{120}, \frac{1}{6}\right), 1\right)\right)}{x} \]

      10. *-lowering-*.f64 89.2

          \[\leadsto \frac{\sin x \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, 0.008333333333333333, 0.16666666666666666\right), 1\right)\right)}{x} \]

   5. Simplified 89.2%

      \[\leadsto \frac{\sin x \cdot \color{blue}{\left(y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), 1\right)\right)}}{x} \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{\sin x \cdot \frac{y \cdot \left(\left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{120} + \frac{1}{6}\right) + 1\right)}{x}} \]

      2. clear-num N/A

         \[\leadsto \sin x \cdot \color{blue}{\frac{1}{\frac{x}{y \cdot \left(\left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{120} + \frac{1}{6}\right) + 1\right)}}} \]

      3. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{\sin x}{\frac{x}{y \cdot \left(\left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{120} + \frac{1}{6}\right) + 1\right)}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin x}{\frac{x}{y \cdot \left(\left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{120} + \frac{1}{6}\right) + 1\right)}}} \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \frac{\color{blue}{\sin x}}{\frac{x}{y \cdot \left(\left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{120} + \frac{1}{6}\right) + 1\right)}} \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \frac{\sin x}{\color{blue}{\frac{x}{y \cdot \left(\left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{120} + \frac{1}{6}\right) + 1\right)}}} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{\sin x}{\frac{x}{\color{blue}{y \cdot \left(\left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{120} + \frac{1}{6}\right) + 1\right)}}} \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\sin x}{\frac{x}{y \cdot \color{blue}{\mathsf{fma}\left(y \cdot y, \left(y \cdot y\right) \cdot \frac{1}{120} + \frac{1}{6}, 1\right)}}} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \frac{\sin x}{\frac{x}{y \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, \left(y \cdot y\right) \cdot \frac{1}{120} + \frac{1}{6}, 1\right)}} \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\sin x}{\frac{x}{y \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{\mathsf{fma}\left(y \cdot y, \frac{1}{120}, \frac{1}{6}\right)}, 1\right)}} \]

      11. *-lowering-*.f64 92.4

          \[\leadsto \frac{\sin x}{\frac{x}{y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, 0.008333333333333333, 0.16666666666666666\right), 1\right)}} \]

   7. Applied egg-rr 92.4%

      \[\leadsto \color{blue}{\frac{\sin x}{\frac{x}{y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), 1\right)}}} \]

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto y \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) + 1\right)} \]

      3. unpow2 N/A

         \[\leadsto y \cdot \left(\color{blue}{\left(y \cdot y\right)} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) + 1\right) \]

      4. associate-*l* N/A

         \[\leadsto y \cdot \left(\color{blue}{y \cdot \left(y \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} + 1\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto y \cdot \color{blue}{\mathsf{fma}\left(y, y \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right), 1\right)} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto y \cdot \mathsf{fma}\left(y, \color{blue}{y \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)}, 1\right) \]

      7. +-commutative N/A

         \[\leadsto y \cdot \mathsf{fma}\left(y, y \cdot \color{blue}{\left(\frac{1}{120} \cdot {y}^{2} + \frac{1}{6}\right)}, 1\right) \]

      8. *-commutative N/A

         \[\leadsto y \cdot \mathsf{fma}\left(y, y \cdot \left(\color{blue}{{y}^{2} \cdot \frac{1}{120}} + \frac{1}{6}\right), 1\right) \]

      9. unpow2 N/A

         \[\leadsto y \cdot \mathsf{fma}\left(y, y \cdot \left(\color{blue}{\left(y \cdot y\right)} \cdot \frac{1}{120} + \frac{1}{6}\right), 1\right) \]

      10. associate-*l* N/A

          \[\leadsto y \cdot \mathsf{fma}\left(y, y \cdot \left(\color{blue}{y \cdot \left(y \cdot \frac{1}{120}\right)} + \frac{1}{6}\right), 1\right) \]

      11. *-commutative N/A

          \[\leadsto y \cdot \mathsf{fma}\left(y, y \cdot \left(y \cdot \color{blue}{\left(\frac{1}{120} \cdot y\right)} + \frac{1}{6}\right), 1\right) \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto y \cdot \mathsf{fma}\left(y, y \cdot \color{blue}{\mathsf{fma}\left(y, \frac{1}{120} \cdot y, \frac{1}{6}\right)}, 1\right) \]

      13. *-commutative N/A

          \[\leadsto y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, \color{blue}{y \cdot \frac{1}{120}}, \frac{1}{6}\right), 1\right) \]

      14. *-lowering-*.f64 55.7

          \[\leadsto y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, \color{blue}{y \cdot 0.008333333333333333}, 0.16666666666666666\right), 1\right) \]

   10. Simplified 55.7%

       \[\leadsto \color{blue}{y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 58.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin x \cdot \sinh y}{x} \leq -2 \cdot 10^{-202}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right)\\ \mathbf{elif}\;\frac{\sin x \cdot \sinh y}{x} \leq 0:\\ \;\;\;\;x \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 55.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin x \cdot \sinh y}{x}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-202}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;x \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(y, y \cdot 0.16666666666666666, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sin x) (sinh y)) x)))
   (if (<= t_0 -2e-202)
     (* y (fma x (* x -0.16666666666666666) 1.0))
     (if (<= t_0 0.0)
       (* x (/ y x))
       (* y (fma y (* y 0.16666666666666666) 1.0))))))

C

double code(double x, double y) {
	double t_0 = (sin(x) * sinh(y)) / x;
	double tmp;
	if (t_0 <= -2e-202) {
		tmp = y * fma(x, (x * -0.16666666666666666), 1.0);
	} else if (t_0 <= 0.0) {
		tmp = x * (y / x);
	} else {
		tmp = y * fma(y, (y * 0.16666666666666666), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(sin(x) * sinh(y)) / x)
	tmp = 0.0
	if (t_0 <= -2e-202)
		tmp = Float64(y * fma(x, Float64(x * -0.16666666666666666), 1.0));
	elseif (t_0 <= 0.0)
		tmp = Float64(x * Float64(y / x));
	else
		tmp = Float64(y * fma(y, Float64(y * 0.16666666666666666), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sin[x], $MachinePrecision] * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-202], N[(y * N[(x * N[(x * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(x * N[(y / x), $MachinePrecision]), $MachinePrecision], N[(y * N[(y * N[(y * 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -2.0000000000000001e-202

   1. Initial program 99.1%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)} \cdot \sinh y}{x} \]

      2. +-commutative N/A

         \[\leadsto \frac{\left(x \cdot \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2} + 1\right)}\right) \cdot \sinh y}{x} \]

      3. *-commutative N/A

         \[\leadsto \frac{\left(x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{-1}{6}} + 1\right)\right) \cdot \sinh y}{x} \]

      4. unpow2 N/A

         \[\leadsto \frac{\left(x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{-1}{6} + 1\right)\right) \cdot \sinh y}{x} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\left(x \cdot \left(\color{blue}{x \cdot \left(x \cdot \frac{-1}{6}\right)} + 1\right)\right) \cdot \sinh y}{x} \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\left(x \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)}\right) \cdot \sinh y}{x} \]

      7. *-lowering-*.f64 72.3

         \[\leadsto \frac{\left(x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot -0.16666666666666666}, 1\right)\right) \cdot \sinh y}{x} \]

   5. Simplified 72.3%

      \[\leadsto \frac{\color{blue}{\left(x \cdot \mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right)\right)} \cdot \sinh y}{x} \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{y \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)} \]

      2. +-commutative N/A

         \[\leadsto y \cdot \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2} + 1\right)} \]

      3. *-commutative N/A

         \[\leadsto y \cdot \left(\color{blue}{{x}^{2} \cdot \frac{-1}{6}} + 1\right) \]

      4. unpow2 N/A

         \[\leadsto y \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{-1}{6} + 1\right) \]

      5. associate-*l* N/A

         \[\leadsto y \cdot \left(\color{blue}{x \cdot \left(x \cdot \frac{-1}{6}\right)} + 1\right) \]

      6. *-commutative N/A

         \[\leadsto y \cdot \left(x \cdot \color{blue}{\left(\frac{-1}{6} \cdot x\right)} + 1\right) \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto y \cdot \color{blue}{\mathsf{fma}\left(x, \frac{-1}{6} \cdot x, 1\right)} \]

      8. *-commutative N/A

         \[\leadsto y \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{-1}{6}}, 1\right) \]

      9. *-lowering-*.f64 42.6

         \[\leadsto y \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot -0.16666666666666666}, 1\right) \]

   8. Simplified 42.6%

      \[\leadsto \color{blue}{y \cdot \mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right)} \]

   if -2.0000000000000001e-202 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 0.0

   1. Initial program 69.8%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)} \cdot \sinh y}{x} \]

      2. +-commutative N/A

         \[\leadsto \frac{\left(x \cdot \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2} + 1\right)}\right) \cdot \sinh y}{x} \]

      3. *-commutative N/A

         \[\leadsto \frac{\left(x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{-1}{6}} + 1\right)\right) \cdot \sinh y}{x} \]

      4. unpow2 N/A

         \[\leadsto \frac{\left(x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{-1}{6} + 1\right)\right) \cdot \sinh y}{x} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\left(x \cdot \left(\color{blue}{x \cdot \left(x \cdot \frac{-1}{6}\right)} + 1\right)\right) \cdot \sinh y}{x} \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\left(x \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)}\right) \cdot \sinh y}{x} \]

      7. *-lowering-*.f64 6.1

         \[\leadsto \frac{\left(x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot -0.16666666666666666}, 1\right)\right) \cdot \sinh y}{x} \]

   5. Simplified 6.1%

      \[\leadsto \frac{\color{blue}{\left(x \cdot \mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right)\right)} \cdot \sinh y}{x} \]

   6. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{x \cdot \left(y \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)}}{x} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot y\right) \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)}}{x} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{-1}{6} \cdot {x}^{2}\right) \cdot \left(x \cdot y\right)}}{x} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{-1}{6} \cdot {x}^{2}\right) \cdot \left(x \cdot y\right)}}{x} \]

      4. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{6} \cdot {x}^{2} + 1\right)} \cdot \left(x \cdot y\right)}{x} \]

      5. *-commutative N/A

         \[\leadsto \frac{\left(\color{blue}{{x}^{2} \cdot \frac{-1}{6}} + 1\right) \cdot \left(x \cdot y\right)}{x} \]

      6. unpow2 N/A

         \[\leadsto \frac{\left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{-1}{6} + 1\right) \cdot \left(x \cdot y\right)}{x} \]

      7. associate-*l* N/A

         \[\leadsto \frac{\left(\color{blue}{x \cdot \left(x \cdot \frac{-1}{6}\right)} + 1\right) \cdot \left(x \cdot y\right)}{x} \]

      8. *-commutative N/A

         \[\leadsto \frac{\left(x \cdot \color{blue}{\left(\frac{-1}{6} \cdot x\right)} + 1\right) \cdot \left(x \cdot y\right)}{x} \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, \frac{-1}{6} \cdot x, 1\right)} \cdot \left(x \cdot y\right)}{x} \]

      10. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, \color{blue}{x \cdot \frac{-1}{6}}, 1\right) \cdot \left(x \cdot y\right)}{x} \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, \color{blue}{x \cdot \frac{-1}{6}}, 1\right) \cdot \left(x \cdot y\right)}{x} \]

      12. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right) \cdot \color{blue}{\left(y \cdot x\right)}}{x} \]

      13. *-lowering-*.f64 6.2

          \[\leadsto \frac{\mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right) \cdot \color{blue}{\left(y \cdot x\right)}}{x} \]

   8. Simplified 6.2%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right) \cdot \left(y \cdot x\right)}}{x} \]
   9. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot \left(x \cdot \frac{-1}{6}\right) + 1\right) \cdot y\right) \cdot x}}{x} \]

      2. *-rgt-identity N/A

         \[\leadsto \frac{\left(\left(x \cdot \left(x \cdot \frac{-1}{6}\right) + 1\right) \cdot y\right) \cdot x}{\color{blue}{x \cdot 1}} \]

      3. times-frac N/A

         \[\leadsto \color{blue}{\frac{\left(x \cdot \left(x \cdot \frac{-1}{6}\right) + 1\right) \cdot y}{x} \cdot \frac{x}{1}} \]

      4. /-rgt-identity N/A

         \[\leadsto \frac{\left(x \cdot \left(x \cdot \frac{-1}{6}\right) + 1\right) \cdot y}{x} \cdot \color{blue}{x} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(x \cdot \left(x \cdot \frac{-1}{6}\right) + 1\right) \cdot y}{x} \cdot x} \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(x \cdot \left(x \cdot \frac{-1}{6}\right) + 1\right) \cdot y}{x}} \cdot x \]

      7. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(x \cdot \left(x \cdot \frac{-1}{6}\right) + 1\right)}}{x} \cdot x \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(x \cdot \left(x \cdot \frac{-1}{6}\right) + 1\right)}}{x} \cdot x \]

      9. associate-*r* N/A

         \[\leadsto \frac{y \cdot \left(\color{blue}{\left(x \cdot x\right) \cdot \frac{-1}{6}} + 1\right)}{x} \cdot x \]

      10. *-commutative N/A

          \[\leadsto \frac{y \cdot \left(\color{blue}{\frac{-1}{6} \cdot \left(x \cdot x\right)} + 1\right)}{x} \cdot x \]

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{y \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{6}, x \cdot x, 1\right)}}{x} \cdot x \]

      12. *-lowering-*.f64 36.2

          \[\leadsto \frac{y \cdot \mathsf{fma}\left(-0.16666666666666666, \color{blue}{x \cdot x}, 1\right)}{x} \cdot x \]

   10. Applied egg-rr 36.2%

       \[\leadsto \color{blue}{\frac{y \cdot \mathsf{fma}\left(-0.16666666666666666, x \cdot x, 1\right)}{x} \cdot x} \]

   11. Taylor expanded in x around 0

       \[\leadsto \color{blue}{\frac{y}{x}} \cdot x \]
   12. Step-by-step derivation

       1. /-lowering-/.f64 81.3

          \[\leadsto \color{blue}{\frac{y}{x}} \cdot x \]

   13. Simplified 81.3%

       \[\leadsto \color{blue}{\frac{y}{x}} \cdot x \]

   if 0.0 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 99.4%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{x} \cdot \sinh y}{x} \]
   4. Step-by-step derivation

   5. Simplified 61.3%

      \[\leadsto \frac{\color{blue}{x} \cdot \sinh y}{x} \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{y \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)} \]

      2. +-commutative N/A

         \[\leadsto y \cdot \color{blue}{\left(\frac{1}{6} \cdot {y}^{2} + 1\right)} \]

      3. *-commutative N/A

         \[\leadsto y \cdot \left(\color{blue}{{y}^{2} \cdot \frac{1}{6}} + 1\right) \]

      4. unpow2 N/A

         \[\leadsto y \cdot \left(\color{blue}{\left(y \cdot y\right)} \cdot \frac{1}{6} + 1\right) \]

      5. associate-*l* N/A

         \[\leadsto y \cdot \left(\color{blue}{y \cdot \left(y \cdot \frac{1}{6}\right)} + 1\right) \]

      6. *-commutative N/A

         \[\leadsto y \cdot \left(y \cdot \color{blue}{\left(\frac{1}{6} \cdot y\right)} + 1\right) \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto y \cdot \color{blue}{\mathsf{fma}\left(y, \frac{1}{6} \cdot y, 1\right)} \]

      8. *-commutative N/A

         \[\leadsto y \cdot \mathsf{fma}\left(y, \color{blue}{y \cdot \frac{1}{6}}, 1\right) \]

      9. *-lowering-*.f64 50.5

         \[\leadsto y \cdot \mathsf{fma}\left(y, \color{blue}{y \cdot 0.16666666666666666}, 1\right) \]

   8. Simplified 50.5%

      \[\leadsto \color{blue}{y \cdot \mathsf{fma}\left(y, y \cdot 0.16666666666666666, 1\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 56.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin x \cdot \sinh y}{x} \leq -2 \cdot 10^{-202}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right)\\ \mathbf{elif}\;\frac{\sin x \cdot \sinh y}{x} \leq 0:\\ \;\;\;\;x \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(y, y \cdot 0.16666666666666666, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 89.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin x \cdot \sinh y}{x} \leq 10^{-114}:\\ \;\;\;\;\frac{\sin x}{x} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (sin x) (sinh y)) x) 1e-114)
   (*
    (/ (sin x) x)
    (fma
     (fma
      y
      (* y (fma (* y y) 0.0001984126984126984 0.008333333333333333))
      0.16666666666666666)
     (* y (* y y))
     y))
   (sinh y)))

C

double code(double x, double y) {
	double tmp;
	if (((sin(x) * sinh(y)) / x) <= 1e-114) {
		tmp = (sin(x) / x) * fma(fma(y, (y * fma((y * y), 0.0001984126984126984, 0.008333333333333333)), 0.16666666666666666), (y * (y * y)), y);
	} else {
		tmp = sinh(y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(sin(x) * sinh(y)) / x) <= 1e-114)
		tmp = Float64(Float64(sin(x) / x) * fma(fma(y, Float64(y * fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333)), 0.16666666666666666), Float64(y * Float64(y * y)), y));
	else
		tmp = sinh(y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[Sin[x], $MachinePrecision] * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], 1e-114], N[(N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision] * N[(N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision], N[Sinh[y], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 1.0000000000000001e-114

   1. Initial program 87.5%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot \sinh y} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot \sinh y} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin x}{x}} \cdot \sinh y \]

      4. sin-lowering-sin.f64 N/A

         \[\leadsto \frac{\color{blue}{\sin x}}{x} \cdot \sinh y \]

      5. sinh-lowering-sinh.f64 99.9

         \[\leadsto \frac{\sin x}{x} \cdot \color{blue}{\sinh y} \]

   4. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot \sinh y} \]

   5. Taylor expanded in y around 0

      \[\leadsto \frac{\sin x}{x} \cdot \color{blue}{\left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \frac{\sin x}{x} \cdot \color{blue}{\left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \frac{\sin x}{x} \cdot \left(y \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) + 1\right)}\right) \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\sin x}{x} \cdot \left(y \cdot \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right), 1\right)}\right) \]

      4. unpow2 N/A

         \[\leadsto \frac{\sin x}{x} \cdot \left(y \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right), 1\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{\sin x}{x} \cdot \left(y \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right), 1\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \frac{\sin x}{x} \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{{y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right) + \frac{1}{6}}, 1\right)\right) \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\sin x}{x} \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}, \frac{1}{6}\right)}, 1\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \frac{\sin x}{x} \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}, \frac{1}{6}\right), 1\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \frac{\sin x}{x} \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}, \frac{1}{6}\right), 1\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \frac{\sin x}{x} \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \color{blue}{\frac{1}{5040} \cdot {y}^{2} + \frac{1}{120}}, \frac{1}{6}\right), 1\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \frac{\sin x}{x} \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \color{blue}{{y}^{2} \cdot \frac{1}{5040}} + \frac{1}{120}, \frac{1}{6}\right), 1\right)\right) \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\sin x}{x} \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{1}{5040}, \frac{1}{120}\right)}, \frac{1}{6}\right), 1\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \frac{\sin x}{x} \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right)\right) \]

      14. *-lowering-*.f64 95.9

          \[\leadsto \frac{\sin x}{x} \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)\right) \]

   7. Simplified 95.9%

      \[\leadsto \frac{\sin x}{x} \cdot \color{blue}{\left(y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)\right)} \]
   8. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \frac{\sin x}{x} \cdot \color{blue}{\left(\left(\left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{5040} + \frac{1}{120}\right) + \frac{1}{6}\right)\right) \cdot y + 1 \cdot y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\sin x}{x} \cdot \left(\color{blue}{\left(\left(\left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{5040} + \frac{1}{120}\right) + \frac{1}{6}\right) \cdot \left(y \cdot y\right)\right)} \cdot y + 1 \cdot y\right) \]

      3. associate-*l* N/A

         \[\leadsto \frac{\sin x}{x} \cdot \left(\color{blue}{\left(\left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{5040} + \frac{1}{120}\right) + \frac{1}{6}\right) \cdot \left(\left(y \cdot y\right) \cdot y\right)} + 1 \cdot y\right) \]

      4. pow3 N/A

         \[\leadsto \frac{\sin x}{x} \cdot \left(\left(\left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{5040} + \frac{1}{120}\right) + \frac{1}{6}\right) \cdot \color{blue}{{y}^{3}} + 1 \cdot y\right) \]

      5. *-lft-identity N/A

         \[\leadsto \frac{\sin x}{x} \cdot \left(\left(\left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{5040} + \frac{1}{120}\right) + \frac{1}{6}\right) \cdot {y}^{3} + \color{blue}{y}\right) \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\sin x}{x} \cdot \color{blue}{\mathsf{fma}\left(\left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{5040} + \frac{1}{120}\right) + \frac{1}{6}, {y}^{3}, y\right)} \]

      7. associate-*l* N/A

         \[\leadsto \frac{\sin x}{x} \cdot \mathsf{fma}\left(\color{blue}{y \cdot \left(y \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{5040} + \frac{1}{120}\right)\right)} + \frac{1}{6}, {y}^{3}, y\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\sin x}{x} \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(y, y \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{5040} + \frac{1}{120}\right), \frac{1}{6}\right)}, {y}^{3}, y\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \frac{\sin x}{x} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y, \color{blue}{y \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{5040} + \frac{1}{120}\right)}, \frac{1}{6}\right), {y}^{3}, y\right) \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\sin x}{x} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \color{blue}{\mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right)}, \frac{1}{6}\right), {y}^{3}, y\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \frac{\sin x}{x} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), {y}^{3}, y\right) \]

      12. cube-unmult N/A

          \[\leadsto \frac{\sin x}{x} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), \color{blue}{y \cdot \left(y \cdot y\right)}, y\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{\sin x}{x} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), \color{blue}{y \cdot \left(y \cdot y\right)}, y\right) \]

      14. *-lowering-*.f64 95.9

          \[\leadsto \frac{\sin x}{x} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y \cdot \color{blue}{\left(y \cdot y\right)}, y\right) \]

   9. Applied egg-rr 95.9%

      \[\leadsto \frac{\sin x}{x} \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)} \]

   if 1.0000000000000001e-114 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 99.3%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{x} \cdot \sinh y}{x} \]
   4. Step-by-step derivation

   5. Simplified 72.6%

      \[\leadsto \frac{\color{blue}{x} \cdot \sinh y}{x} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sinh y \cdot x}}{x} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\sinh y \cdot \frac{x}{x}} \]

      3. *-inverses N/A

         \[\leadsto \sinh y \cdot \color{blue}{1} \]

      4. *-rgt-identity N/A

         \[\leadsto \color{blue}{\sinh y} \]

      5. sinh-lowering-sinh.f64 73.4

         \[\leadsto \color{blue}{\sinh y} \]

   7. Applied egg-rr 73.4%

      \[\leadsto \color{blue}{\sinh y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 89.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin x \cdot \sinh y}{x} \leq 10^{-114}:\\ \;\;\;\;\frac{\sin x}{x} \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (sin x) (sinh y)) x) 1e-114)
   (*
    (/ (sin x) x)
    (*
     y
     (fma
      (* y y)
      (fma
       (* y y)
       (fma (* y y) 0.0001984126984126984 0.008333333333333333)
       0.16666666666666666)
      1.0)))
   (sinh y)))

C

double code(double x, double y) {
	double tmp;
	if (((sin(x) * sinh(y)) / x) <= 1e-114) {
		tmp = (sin(x) / x) * (y * fma((y * y), fma((y * y), fma((y * y), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666), 1.0));
	} else {
		tmp = sinh(y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(sin(x) * sinh(y)) / x) <= 1e-114)
		tmp = Float64(Float64(sin(x) / x) * Float64(y * fma(Float64(y * y), fma(Float64(y * y), fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666), 1.0)));
	else
		tmp = sinh(y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[Sin[x], $MachinePrecision] * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], 1e-114], N[(N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision] * N[(y * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sinh[y], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 1.0000000000000001e-114

   1. Initial program 87.5%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot \sinh y} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot \sinh y} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin x}{x}} \cdot \sinh y \]

      4. sin-lowering-sin.f64 N/A

         \[\leadsto \frac{\color{blue}{\sin x}}{x} \cdot \sinh y \]

      5. sinh-lowering-sinh.f64 99.9

         \[\leadsto \frac{\sin x}{x} \cdot \color{blue}{\sinh y} \]

   4. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot \sinh y} \]

   5. Taylor expanded in y around 0

      \[\leadsto \frac{\sin x}{x} \cdot \color{blue}{\left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \frac{\sin x}{x} \cdot \color{blue}{\left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \frac{\sin x}{x} \cdot \left(y \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) + 1\right)}\right) \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\sin x}{x} \cdot \left(y \cdot \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right), 1\right)}\right) \]

      4. unpow2 N/A

         \[\leadsto \frac{\sin x}{x} \cdot \left(y \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right), 1\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{\sin x}{x} \cdot \left(y \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right), 1\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \frac{\sin x}{x} \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{{y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right) + \frac{1}{6}}, 1\right)\right) \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\sin x}{x} \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}, \frac{1}{6}\right)}, 1\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \frac{\sin x}{x} \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}, \frac{1}{6}\right), 1\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \frac{\sin x}{x} \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}, \frac{1}{6}\right), 1\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \frac{\sin x}{x} \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \color{blue}{\frac{1}{5040} \cdot {y}^{2} + \frac{1}{120}}, \frac{1}{6}\right), 1\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \frac{\sin x}{x} \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \color{blue}{{y}^{2} \cdot \frac{1}{5040}} + \frac{1}{120}, \frac{1}{6}\right), 1\right)\right) \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\sin x}{x} \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{1}{5040}, \frac{1}{120}\right)}, \frac{1}{6}\right), 1\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \frac{\sin x}{x} \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right)\right) \]

      14. *-lowering-*.f64 95.9

          \[\leadsto \frac{\sin x}{x} \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)\right) \]

   7. Simplified 95.9%

      \[\leadsto \frac{\sin x}{x} \cdot \color{blue}{\left(y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)\right)} \]

   if 1.0000000000000001e-114 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 99.3%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{x} \cdot \sinh y}{x} \]
   4. Step-by-step derivation

   5. Simplified 72.6%

      \[\leadsto \frac{\color{blue}{x} \cdot \sinh y}{x} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sinh y \cdot x}}{x} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\sinh y \cdot \frac{x}{x}} \]

      3. *-inverses N/A

         \[\leadsto \sinh y \cdot \color{blue}{1} \]

      4. *-rgt-identity N/A

         \[\leadsto \color{blue}{\sinh y} \]

      5. sinh-lowering-sinh.f64 73.4

         \[\leadsto \color{blue}{\sinh y} \]

   7. Applied egg-rr 73.4%

      \[\leadsto \color{blue}{\sinh y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 89.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin x \cdot \sinh y}{x} \leq 10^{-114}:\\ \;\;\;\;y \cdot \left(\frac{\sin x}{x} \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (sin x) (sinh y)) x) 1e-114)
   (*
    y
    (*
     (/ (sin x) x)
     (fma
      (* y y)
      (fma
       y
       (* y (fma (* y y) 0.0001984126984126984 0.008333333333333333))
       0.16666666666666666)
      1.0)))
   (sinh y)))

C

double code(double x, double y) {
	double tmp;
	if (((sin(x) * sinh(y)) / x) <= 1e-114) {
		tmp = y * ((sin(x) / x) * fma((y * y), fma(y, (y * fma((y * y), 0.0001984126984126984, 0.008333333333333333)), 0.16666666666666666), 1.0));
	} else {
		tmp = sinh(y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(sin(x) * sinh(y)) / x) <= 1e-114)
		tmp = Float64(y * Float64(Float64(sin(x) / x) * fma(Float64(y * y), fma(y, Float64(y * fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333)), 0.16666666666666666), 1.0)));
	else
		tmp = sinh(y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[Sin[x], $MachinePrecision] * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], 1e-114], N[(y * N[(N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sinh[y], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 1.0000000000000001e-114

   1. Initial program 87.5%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot \sinh y} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot \sinh y} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin x}{x}} \cdot \sinh y \]

      4. sin-lowering-sin.f64 N/A

         \[\leadsto \frac{\color{blue}{\sin x}}{x} \cdot \sinh y \]

      5. sinh-lowering-sinh.f64 99.9

         \[\leadsto \frac{\sin x}{x} \cdot \color{blue}{\sinh y} \]

   4. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot \sinh y} \]

   5. Taylor expanded in y around 0

      \[\leadsto \frac{\sin x}{x} \cdot \color{blue}{\left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \frac{\sin x}{x} \cdot \color{blue}{\left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \frac{\sin x}{x} \cdot \left(y \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) + 1\right)}\right) \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\sin x}{x} \cdot \left(y \cdot \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right), 1\right)}\right) \]

      4. unpow2 N/A

         \[\leadsto \frac{\sin x}{x} \cdot \left(y \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right), 1\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{\sin x}{x} \cdot \left(y \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right), 1\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \frac{\sin x}{x} \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{{y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right) + \frac{1}{6}}, 1\right)\right) \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\sin x}{x} \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}, \frac{1}{6}\right)}, 1\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \frac{\sin x}{x} \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}, \frac{1}{6}\right), 1\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \frac{\sin x}{x} \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}, \frac{1}{6}\right), 1\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \frac{\sin x}{x} \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \color{blue}{\frac{1}{5040} \cdot {y}^{2} + \frac{1}{120}}, \frac{1}{6}\right), 1\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \frac{\sin x}{x} \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \color{blue}{{y}^{2} \cdot \frac{1}{5040}} + \frac{1}{120}, \frac{1}{6}\right), 1\right)\right) \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\sin x}{x} \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{1}{5040}, \frac{1}{120}\right)}, \frac{1}{6}\right), 1\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \frac{\sin x}{x} \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right)\right) \]

      14. *-lowering-*.f64 95.9

          \[\leadsto \frac{\sin x}{x} \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)\right) \]

   7. Simplified 95.9%

      \[\leadsto \frac{\sin x}{x} \cdot \color{blue}{\left(y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)\right)} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin x}{x} \cdot \color{blue}{\left(\left(\left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{5040} + \frac{1}{120}\right) + \frac{1}{6}\right) + 1\right) \cdot y\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{\sin x}{x} \cdot \left(\left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{5040} + \frac{1}{120}\right) + \frac{1}{6}\right) + 1\right)\right) \cdot y} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{\sin x}{x} \cdot \left(\left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{5040} + \frac{1}{120}\right) + \frac{1}{6}\right) + 1\right)\right) \cdot y} \]

   9. Applied egg-rr 95.9%

      \[\leadsto \color{blue}{\left(\frac{\sin x}{x} \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)\right) \cdot y} \]

   if 1.0000000000000001e-114 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 99.3%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{x} \cdot \sinh y}{x} \]
   4. Step-by-step derivation

   5. Simplified 72.6%

      \[\leadsto \frac{\color{blue}{x} \cdot \sinh y}{x} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sinh y \cdot x}}{x} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\sinh y \cdot \frac{x}{x}} \]

      3. *-inverses N/A

         \[\leadsto \sinh y \cdot \color{blue}{1} \]

      4. *-rgt-identity N/A

         \[\leadsto \color{blue}{\sinh y} \]

      5. sinh-lowering-sinh.f64 73.4

         \[\leadsto \color{blue}{\sinh y} \]

   7. Applied egg-rr 73.4%

      \[\leadsto \color{blue}{\sinh y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin x \cdot \sinh y}{x} \leq 10^{-114}:\\ \;\;\;\;y \cdot \left(\frac{\sin x}{x} \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 70.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right)\\ \mathbf{if}\;\frac{\sin x \cdot \sinh y}{x} \leq -2 \cdot 10^{-202}:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right) \cdot \left(y \cdot \mathsf{fma}\left(y, y \cdot t\_0, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{x}{y \cdot \mathsf{fma}\left(y \cdot y, t\_0, 1\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma (* y y) 0.008333333333333333 0.16666666666666666)))
   (if (<= (/ (* (sin x) (sinh y)) x) -2e-202)
     (* (fma x (* x -0.16666666666666666) 1.0) (* y (fma y (* y t_0) 1.0)))
     (/ x (/ x (* y (fma (* y y) t_0 1.0)))))))

C

double code(double x, double y) {
	double t_0 = fma((y * y), 0.008333333333333333, 0.16666666666666666);
	double tmp;
	if (((sin(x) * sinh(y)) / x) <= -2e-202) {
		tmp = fma(x, (x * -0.16666666666666666), 1.0) * (y * fma(y, (y * t_0), 1.0));
	} else {
		tmp = x / (x / (y * fma((y * y), t_0, 1.0)));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666)
	tmp = 0.0
	if (Float64(Float64(sin(x) * sinh(y)) / x) <= -2e-202)
		tmp = Float64(fma(x, Float64(x * -0.16666666666666666), 1.0) * Float64(y * fma(y, Float64(y * t_0), 1.0)));
	else
		tmp = Float64(x / Float64(x / Float64(y * fma(Float64(y * y), t_0, 1.0))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sin[x], $MachinePrecision] * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], -2e-202], N[(N[(x * N[(x * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * N[(y * N[(y * N[(y * t$95$0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(x / N[(y * N[(N[(y * y), $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -2.0000000000000001e-202

   1. Initial program 99.1%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\sin x \cdot \color{blue}{\left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)}}{x} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \frac{\sin x \cdot \color{blue}{\left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)}}{x} \]

      2. +-commutative N/A

         \[\leadsto \frac{\sin x \cdot \left(y \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) + 1\right)}\right)}{x} \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\sin x \cdot \left(y \cdot \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{1}{6} + \frac{1}{120} \cdot {y}^{2}, 1\right)}\right)}{x} \]

      4. unpow2 N/A

         \[\leadsto \frac{\sin x \cdot \left(y \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{6} + \frac{1}{120} \cdot {y}^{2}, 1\right)\right)}{x} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{\sin x \cdot \left(y \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{6} + \frac{1}{120} \cdot {y}^{2}, 1\right)\right)}{x} \]

      6. +-commutative N/A

         \[\leadsto \frac{\sin x \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{\frac{1}{120} \cdot {y}^{2} + \frac{1}{6}}, 1\right)\right)}{x} \]

      7. *-commutative N/A

         \[\leadsto \frac{\sin x \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{{y}^{2} \cdot \frac{1}{120}} + \frac{1}{6}, 1\right)\right)}{x} \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\sin x \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{1}{120}, \frac{1}{6}\right)}, 1\right)\right)}{x} \]

      9. unpow2 N/A

         \[\leadsto \frac{\sin x \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{120}, \frac{1}{6}\right), 1\right)\right)}{x} \]

      10. *-lowering-*.f64 87.6

          \[\leadsto \frac{\sin x \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, 0.008333333333333333, 0.16666666666666666\right), 1\right)\right)}{x} \]

   5. Simplified 87.6%

      \[\leadsto \frac{\sin x \cdot \color{blue}{\left(y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), 1\right)\right)}}{x} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-1}{6} \cdot \left({x}^{2} \cdot \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right) + y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2}\right) \cdot \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)} + y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) \]

      2. distribute-lft1-in N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2} + 1\right) \cdot \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(1 + \frac{-1}{6} \cdot {x}^{2}\right)} \cdot \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(1 + \frac{-1}{6} \cdot {x}^{2}\right) \cdot \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)} \]

      5. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2} + 1\right)} \cdot \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \left(\color{blue}{{x}^{2} \cdot \frac{-1}{6}} + 1\right) \cdot \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{-1}{6} + 1\right) \cdot \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \left(\color{blue}{x \cdot \left(x \cdot \frac{-1}{6}\right)} + 1\right) \cdot \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \left(x \cdot \color{blue}{\left(\frac{-1}{6} \cdot x\right)} + 1\right) \cdot \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{-1}{6} \cdot x, 1\right)} \cdot \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{-1}{6}}, 1\right) \cdot \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{-1}{6}}, 1\right) \cdot \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right) \cdot \color{blue}{\left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)} \]

      14. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right) \cdot \left(y \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) + 1\right)}\right) \]

   8. Simplified 67.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right) \cdot \left(y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), 1\right)\right)} \]

   if -2.0000000000000001e-202 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 86.4%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\sin x \cdot \color{blue}{\left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)}}{x} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \frac{\sin x \cdot \color{blue}{\left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)}}{x} \]

      2. +-commutative N/A

         \[\leadsto \frac{\sin x \cdot \left(y \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) + 1\right)}\right)}{x} \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\sin x \cdot \left(y \cdot \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{1}{6} + \frac{1}{120} \cdot {y}^{2}, 1\right)}\right)}{x} \]

      4. unpow2 N/A

         \[\leadsto \frac{\sin x \cdot \left(y \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{6} + \frac{1}{120} \cdot {y}^{2}, 1\right)\right)}{x} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{\sin x \cdot \left(y \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{6} + \frac{1}{120} \cdot {y}^{2}, 1\right)\right)}{x} \]

      6. +-commutative N/A

         \[\leadsto \frac{\sin x \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{\frac{1}{120} \cdot {y}^{2} + \frac{1}{6}}, 1\right)\right)}{x} \]

      7. *-commutative N/A

         \[\leadsto \frac{\sin x \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{{y}^{2} \cdot \frac{1}{120}} + \frac{1}{6}, 1\right)\right)}{x} \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\sin x \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{1}{120}, \frac{1}{6}\right)}, 1\right)\right)}{x} \]

      9. unpow2 N/A

         \[\leadsto \frac{\sin x \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{120}, \frac{1}{6}\right), 1\right)\right)}{x} \]

      10. *-lowering-*.f64 80.7

          \[\leadsto \frac{\sin x \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, 0.008333333333333333, 0.16666666666666666\right), 1\right)\right)}{x} \]

   5. Simplified 80.7%

      \[\leadsto \frac{\sin x \cdot \color{blue}{\left(y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), 1\right)\right)}}{x} \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{\sin x \cdot \frac{y \cdot \left(\left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{120} + \frac{1}{6}\right) + 1\right)}{x}} \]

      2. clear-num N/A

         \[\leadsto \sin x \cdot \color{blue}{\frac{1}{\frac{x}{y \cdot \left(\left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{120} + \frac{1}{6}\right) + 1\right)}}} \]

      3. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{\sin x}{\frac{x}{y \cdot \left(\left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{120} + \frac{1}{6}\right) + 1\right)}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin x}{\frac{x}{y \cdot \left(\left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{120} + \frac{1}{6}\right) + 1\right)}}} \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \frac{\color{blue}{\sin x}}{\frac{x}{y \cdot \left(\left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{120} + \frac{1}{6}\right) + 1\right)}} \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \frac{\sin x}{\color{blue}{\frac{x}{y \cdot \left(\left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{120} + \frac{1}{6}\right) + 1\right)}}} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{\sin x}{\frac{x}{\color{blue}{y \cdot \left(\left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{120} + \frac{1}{6}\right) + 1\right)}}} \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\sin x}{\frac{x}{y \cdot \color{blue}{\mathsf{fma}\left(y \cdot y, \left(y \cdot y\right) \cdot \frac{1}{120} + \frac{1}{6}, 1\right)}}} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \frac{\sin x}{\frac{x}{y \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, \left(y \cdot y\right) \cdot \frac{1}{120} + \frac{1}{6}, 1\right)}} \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\sin x}{\frac{x}{y \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{\mathsf{fma}\left(y \cdot y, \frac{1}{120}, \frac{1}{6}\right)}, 1\right)}} \]

      11. *-lowering-*.f64 95.4

          \[\leadsto \frac{\sin x}{\frac{x}{y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, 0.008333333333333333, 0.16666666666666666\right), 1\right)}} \]

   7. Applied egg-rr 95.4%

      \[\leadsto \color{blue}{\frac{\sin x}{\frac{x}{y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), 1\right)}}} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{x}}{\frac{x}{y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \frac{1}{120}, \frac{1}{6}\right), 1\right)}} \]
   9. Step-by-step derivation

   10. Simplified 71.1%

       \[\leadsto \frac{\color{blue}{x}}{\frac{x}{y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), 1\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 44.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin x \cdot \sinh y}{x} \leq -2 \cdot 10^{-202}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(y, y \cdot 0.16666666666666666, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (sin x) (sinh y)) x) -2e-202)
   (* y (fma x (* x -0.16666666666666666) 1.0))
   (* y (fma y (* y 0.16666666666666666) 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (((sin(x) * sinh(y)) / x) <= -2e-202) {
		tmp = y * fma(x, (x * -0.16666666666666666), 1.0);
	} else {
		tmp = y * fma(y, (y * 0.16666666666666666), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(sin(x) * sinh(y)) / x) <= -2e-202)
		tmp = Float64(y * fma(x, Float64(x * -0.16666666666666666), 1.0));
	else
		tmp = Float64(y * fma(y, Float64(y * 0.16666666666666666), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[Sin[x], $MachinePrecision] * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], -2e-202], N[(y * N[(x * N[(x * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(y * N[(y * N[(y * 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -2.0000000000000001e-202

   1. Initial program 99.1%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)} \cdot \sinh y}{x} \]

      2. +-commutative N/A

         \[\leadsto \frac{\left(x \cdot \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2} + 1\right)}\right) \cdot \sinh y}{x} \]

      3. *-commutative N/A

         \[\leadsto \frac{\left(x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{-1}{6}} + 1\right)\right) \cdot \sinh y}{x} \]

      4. unpow2 N/A

         \[\leadsto \frac{\left(x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{-1}{6} + 1\right)\right) \cdot \sinh y}{x} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\left(x \cdot \left(\color{blue}{x \cdot \left(x \cdot \frac{-1}{6}\right)} + 1\right)\right) \cdot \sinh y}{x} \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\left(x \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)}\right) \cdot \sinh y}{x} \]

      7. *-lowering-*.f64 72.3

         \[\leadsto \frac{\left(x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot -0.16666666666666666}, 1\right)\right) \cdot \sinh y}{x} \]

   5. Simplified 72.3%

      \[\leadsto \frac{\color{blue}{\left(x \cdot \mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right)\right)} \cdot \sinh y}{x} \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{y \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)} \]

      2. +-commutative N/A

         \[\leadsto y \cdot \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2} + 1\right)} \]

      3. *-commutative N/A

         \[\leadsto y \cdot \left(\color{blue}{{x}^{2} \cdot \frac{-1}{6}} + 1\right) \]

      4. unpow2 N/A

         \[\leadsto y \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{-1}{6} + 1\right) \]

      5. associate-*l* N/A

         \[\leadsto y \cdot \left(\color{blue}{x \cdot \left(x \cdot \frac{-1}{6}\right)} + 1\right) \]

      6. *-commutative N/A

         \[\leadsto y \cdot \left(x \cdot \color{blue}{\left(\frac{-1}{6} \cdot x\right)} + 1\right) \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto y \cdot \color{blue}{\mathsf{fma}\left(x, \frac{-1}{6} \cdot x, 1\right)} \]

      8. *-commutative N/A

         \[\leadsto y \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{-1}{6}}, 1\right) \]

      9. *-lowering-*.f64 42.6

         \[\leadsto y \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot -0.16666666666666666}, 1\right) \]

   8. Simplified 42.6%

      \[\leadsto \color{blue}{y \cdot \mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right)} \]

   if -2.0000000000000001e-202 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 86.4%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{x} \cdot \sinh y}{x} \]
   4. Step-by-step derivation

   5. Simplified 38.2%

      \[\leadsto \frac{\color{blue}{x} \cdot \sinh y}{x} \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{y \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)} \]

      2. +-commutative N/A

         \[\leadsto y \cdot \color{blue}{\left(\frac{1}{6} \cdot {y}^{2} + 1\right)} \]

      3. *-commutative N/A

         \[\leadsto y \cdot \left(\color{blue}{{y}^{2} \cdot \frac{1}{6}} + 1\right) \]

      4. unpow2 N/A

         \[\leadsto y \cdot \left(\color{blue}{\left(y \cdot y\right)} \cdot \frac{1}{6} + 1\right) \]

      5. associate-*l* N/A

         \[\leadsto y \cdot \left(\color{blue}{y \cdot \left(y \cdot \frac{1}{6}\right)} + 1\right) \]

      6. *-commutative N/A

         \[\leadsto y \cdot \left(y \cdot \color{blue}{\left(\frac{1}{6} \cdot y\right)} + 1\right) \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto y \cdot \color{blue}{\mathsf{fma}\left(y, \frac{1}{6} \cdot y, 1\right)} \]

      8. *-commutative N/A

         \[\leadsto y \cdot \mathsf{fma}\left(y, \color{blue}{y \cdot \frac{1}{6}}, 1\right) \]

      9. *-lowering-*.f64 45.4

         \[\leadsto y \cdot \mathsf{fma}\left(y, \color{blue}{y \cdot 0.16666666666666666}, 1\right) \]

   8. Simplified 45.4%

      \[\leadsto \color{blue}{y \cdot \mathsf{fma}\left(y, y \cdot 0.16666666666666666, 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 32.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin x \cdot \sinh y}{x} \leq -2 \cdot 10^{-202}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right)\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (sin x) (sinh y)) x) -2e-202)
   (* y (fma x (* x -0.16666666666666666) 1.0))
   y))

C

double code(double x, double y) {
	double tmp;
	if (((sin(x) * sinh(y)) / x) <= -2e-202) {
		tmp = y * fma(x, (x * -0.16666666666666666), 1.0);
	} else {
		tmp = y;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(sin(x) * sinh(y)) / x) <= -2e-202)
		tmp = Float64(y * fma(x, Float64(x * -0.16666666666666666), 1.0));
	else
		tmp = y;
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[Sin[x], $MachinePrecision] * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], -2e-202], N[(y * N[(x * N[(x * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], y]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -2.0000000000000001e-202

   1. Initial program 99.1%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)} \cdot \sinh y}{x} \]

      2. +-commutative N/A

         \[\leadsto \frac{\left(x \cdot \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2} + 1\right)}\right) \cdot \sinh y}{x} \]

      3. *-commutative N/A

         \[\leadsto \frac{\left(x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{-1}{6}} + 1\right)\right) \cdot \sinh y}{x} \]

      4. unpow2 N/A

         \[\leadsto \frac{\left(x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{-1}{6} + 1\right)\right) \cdot \sinh y}{x} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\left(x \cdot \left(\color{blue}{x \cdot \left(x \cdot \frac{-1}{6}\right)} + 1\right)\right) \cdot \sinh y}{x} \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\left(x \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)}\right) \cdot \sinh y}{x} \]

      7. *-lowering-*.f64 72.3

         \[\leadsto \frac{\left(x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot -0.16666666666666666}, 1\right)\right) \cdot \sinh y}{x} \]

   5. Simplified 72.3%

      \[\leadsto \frac{\color{blue}{\left(x \cdot \mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right)\right)} \cdot \sinh y}{x} \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{y \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)} \]

      2. +-commutative N/A

         \[\leadsto y \cdot \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2} + 1\right)} \]

      3. *-commutative N/A

         \[\leadsto y \cdot \left(\color{blue}{{x}^{2} \cdot \frac{-1}{6}} + 1\right) \]

      4. unpow2 N/A

         \[\leadsto y \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{-1}{6} + 1\right) \]

      5. associate-*l* N/A

         \[\leadsto y \cdot \left(\color{blue}{x \cdot \left(x \cdot \frac{-1}{6}\right)} + 1\right) \]

      6. *-commutative N/A

         \[\leadsto y \cdot \left(x \cdot \color{blue}{\left(\frac{-1}{6} \cdot x\right)} + 1\right) \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto y \cdot \color{blue}{\mathsf{fma}\left(x, \frac{-1}{6} \cdot x, 1\right)} \]

      8. *-commutative N/A

         \[\leadsto y \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{-1}{6}}, 1\right) \]

      9. *-lowering-*.f64 42.6

         \[\leadsto y \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot -0.16666666666666666}, 1\right) \]

   8. Simplified 42.6%

      \[\leadsto \color{blue}{y \cdot \mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right)} \]

   if -2.0000000000000001e-202 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 86.4%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{x} \cdot \sinh y}{x} \]
   4. Step-by-step derivation

   5. Simplified 38.2%

      \[\leadsto \frac{\color{blue}{x} \cdot \sinh y}{x} \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y} \]
   7. Step-by-step derivation

   8. Simplified 24.6%

      \[\leadsto \color{blue}{y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 26.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin x \cdot \sinh y}{x} \leq -2 \cdot 10^{-202}:\\ \;\;\;\;y \cdot \left(-0.16666666666666666 \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (sin x) (sinh y)) x) -2e-202)
   (* y (* -0.16666666666666666 (* x x)))
   y))

C

double code(double x, double y) {
	double tmp;
	if (((sin(x) * sinh(y)) / x) <= -2e-202) {
		tmp = y * (-0.16666666666666666 * (x * x));
	} else {
		tmp = y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (((sin(x) * sinh(y)) / x) <= (-2d-202)) then
        tmp = y * ((-0.16666666666666666d0) * (x * x))
    else
        tmp = y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (((Math.sin(x) * Math.sinh(y)) / x) <= -2e-202) {
		tmp = y * (-0.16666666666666666 * (x * x));
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((math.sin(x) * math.sinh(y)) / x) <= -2e-202:
		tmp = y * (-0.16666666666666666 * (x * x))
	else:
		tmp = y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(sin(x) * sinh(y)) / x) <= -2e-202)
		tmp = Float64(y * Float64(-0.16666666666666666 * Float64(x * x)));
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((sin(x) * sinh(y)) / x) <= -2e-202)
		tmp = y * (-0.16666666666666666 * (x * x));
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[Sin[x], $MachinePrecision] * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], -2e-202], N[(y * N[(-0.16666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], y]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -2.0000000000000001e-202

   1. Initial program 99.1%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)} \cdot \sinh y}{x} \]

      2. +-commutative N/A

         \[\leadsto \frac{\left(x \cdot \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2} + 1\right)}\right) \cdot \sinh y}{x} \]

      3. *-commutative N/A

         \[\leadsto \frac{\left(x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{-1}{6}} + 1\right)\right) \cdot \sinh y}{x} \]

      4. unpow2 N/A

         \[\leadsto \frac{\left(x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{-1}{6} + 1\right)\right) \cdot \sinh y}{x} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\left(x \cdot \left(\color{blue}{x \cdot \left(x \cdot \frac{-1}{6}\right)} + 1\right)\right) \cdot \sinh y}{x} \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\left(x \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)}\right) \cdot \sinh y}{x} \]

      7. *-lowering-*.f64 72.3

         \[\leadsto \frac{\left(x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot -0.16666666666666666}, 1\right)\right) \cdot \sinh y}{x} \]

   5. Simplified 72.3%

      \[\leadsto \frac{\color{blue}{\left(x \cdot \mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right)\right)} \cdot \sinh y}{x} \]

   6. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{x \cdot \left(y \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)}}{x} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot y\right) \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)}}{x} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{-1}{6} \cdot {x}^{2}\right) \cdot \left(x \cdot y\right)}}{x} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{-1}{6} \cdot {x}^{2}\right) \cdot \left(x \cdot y\right)}}{x} \]

      4. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{6} \cdot {x}^{2} + 1\right)} \cdot \left(x \cdot y\right)}{x} \]

      5. *-commutative N/A

         \[\leadsto \frac{\left(\color{blue}{{x}^{2} \cdot \frac{-1}{6}} + 1\right) \cdot \left(x \cdot y\right)}{x} \]

      6. unpow2 N/A

         \[\leadsto \frac{\left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{-1}{6} + 1\right) \cdot \left(x \cdot y\right)}{x} \]

      7. associate-*l* N/A

         \[\leadsto \frac{\left(\color{blue}{x \cdot \left(x \cdot \frac{-1}{6}\right)} + 1\right) \cdot \left(x \cdot y\right)}{x} \]

      8. *-commutative N/A

         \[\leadsto \frac{\left(x \cdot \color{blue}{\left(\frac{-1}{6} \cdot x\right)} + 1\right) \cdot \left(x \cdot y\right)}{x} \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, \frac{-1}{6} \cdot x, 1\right)} \cdot \left(x \cdot y\right)}{x} \]

      10. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, \color{blue}{x \cdot \frac{-1}{6}}, 1\right) \cdot \left(x \cdot y\right)}{x} \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, \color{blue}{x \cdot \frac{-1}{6}}, 1\right) \cdot \left(x \cdot y\right)}{x} \]

      12. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right) \cdot \color{blue}{\left(y \cdot x\right)}}{x} \]

      13. *-lowering-*.f64 42.8

          \[\leadsto \frac{\mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right) \cdot \color{blue}{\left(y \cdot x\right)}}{x} \]

   8. Simplified 42.8%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right) \cdot \left(y \cdot x\right)}}{x} \]

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{-1}{6} \cdot \left({x}^{2} \cdot y\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2}\right) \cdot y} \]

       2. *-commutative N/A

          \[\leadsto \color{blue}{y \cdot \left(\frac{-1}{6} \cdot {x}^{2}\right)} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{y \cdot \left(\frac{-1}{6} \cdot {x}^{2}\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto y \cdot \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2}\right)} \]

       5. unpow2 N/A

          \[\leadsto y \cdot \left(\frac{-1}{6} \cdot \color{blue}{\left(x \cdot x\right)}\right) \]

       6. *-lowering-*.f64 18.1

          \[\leadsto y \cdot \left(-0.16666666666666666 \cdot \color{blue}{\left(x \cdot x\right)}\right) \]

   11. Simplified 18.1%

       \[\leadsto \color{blue}{y \cdot \left(-0.16666666666666666 \cdot \left(x \cdot x\right)\right)} \]

   if -2.0000000000000001e-202 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 86.4%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{x} \cdot \sinh y}{x} \]
   4. Step-by-step derivation

   5. Simplified 38.2%

      \[\leadsto \frac{\color{blue}{x} \cdot \sinh y}{x} \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y} \]
   7. Step-by-step derivation

   8. Simplified 24.6%

      \[\leadsto \color{blue}{y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 22: 28.8% 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 91.0%

   \[\frac{\sin x \cdot \sinh y}{x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \frac{\color{blue}{x} \cdot \sinh y}{x} \]
4. Step-by-step derivation

5. Simplified 49.7%

   \[\leadsto \frac{\color{blue}{x} \cdot \sinh y}{x} \]

6. Taylor expanded in y around 0

   \[\leadsto \color{blue}{y} \]
7. Step-by-step derivation

8. Simplified 25.6%

   \[\leadsto \color{blue}{y} \]
9. 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 2024195 
(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))