Linear.Quaternion:$ccos from linear-1.19.1.3

Percentage Accurate: 100.0% → 100.0%

Time: 15.1s

Alternatives: 19

Speedup: 1.0×

• 49.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

Alternative 2: 87.2% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sinh y) y)) (t_1 (* (sin x) t_0)))
   (if (<= t_1 (- INFINITY))
     (* t_0 (* x (fma x (* x -0.16666666666666666) 1.0)))
     (if (<= t_1 1.0)
       (*
        (sin x)
        (fma
         (* y y)
         (fma 0.008333333333333333 (* y y) 0.16666666666666666)
         1.0))
       (* x t_0)))))

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

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

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

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

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

      7. *-lowering-*.f64 85.0

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

   5. Simplified 85.0%

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

   if -inf.0 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

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

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

      3. unpow2 N/A

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

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

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

      5. +-commutative N/A

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 98.6

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

   5. Simplified 98.6%

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

   if 1 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 76.3%

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

4. Final simplification 88.8%

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

Alternative 3: 84.0% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sinh y) y)) (t_1 (* (sin x) t_0)))
   (if (<= t_1 (- INFINITY))
     (*
      (* (* y y) (* y y))
      (* (* x (fma x (* x -0.16666666666666666) 1.0)) 0.008333333333333333))
     (if (<= t_1 1.0)
       (*
        (sin x)
        (fma
         (* y y)
         (fma 0.008333333333333333 (* y y) 0.16666666666666666)
         1.0))
       (* x t_0)))))

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

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

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

      3. unpow2 N/A

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

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

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

      5. +-commutative N/A

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 77.7

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

   5. Simplified 77.7%

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

   6. Taylor expanded in x around 0

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

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

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

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

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

      7. *-lowering-*.f64 73.8

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

   8. Simplified 73.8%

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

   9. Taylor expanded in y around inf

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. associate-*r* N/A

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

       6. +-commutative N/A

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

       7. distribute-rgt-in N/A

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

       8. *-lft-identity N/A

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

       9. +-commutative N/A

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

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

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

       11. metadata-eval N/A

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

       12. pow-sqr N/A

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

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

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

       14. unpow2 N/A

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

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

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

       16. unpow2 N/A

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

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

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

   11. Simplified 73.8%

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

   if -inf.0 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

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

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

      3. unpow2 N/A

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

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

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

      5. +-commutative N/A

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 98.6

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

   5. Simplified 98.6%

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

   if 1 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 76.3%

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

4. Final simplification 86.2%

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

Alternative 4: 83.9% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sinh y) y)) (t_1 (* (sin x) t_0)))
   (if (<= t_1 (- INFINITY))
     (*
      (* (* y y) (* y y))
      (* (* x (fma x (* x -0.16666666666666666) 1.0)) 0.008333333333333333))
     (if (<= t_1 1.0)
       (* (sin x) (fma 0.16666666666666666 (* y y) 1.0))
       (* x t_0)))))

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

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

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

      3. unpow2 N/A

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

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

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

      5. +-commutative N/A

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 77.7

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

   5. Simplified 77.7%

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

   6. Taylor expanded in x around 0

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

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

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

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

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

      7. *-lowering-*.f64 73.8

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

   8. Simplified 73.8%

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

   9. Taylor expanded in y around inf

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. associate-*r* N/A

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

       6. +-commutative N/A

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

       7. distribute-rgt-in N/A

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

       8. *-lft-identity N/A

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

       9. +-commutative N/A

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

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

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

       11. metadata-eval N/A

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

       12. pow-sqr N/A

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

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

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

       14. unpow2 N/A

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

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

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

       16. unpow2 N/A

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

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

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

   11. Simplified 73.8%

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

   if -inf.0 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 98.6

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

   5. Simplified 98.6%

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

   if 1 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 76.3%

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

4. Final simplification 86.1%

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

Alternative 5: 83.7% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sinh y) y)) (t_1 (* (sin x) t_0)))
   (if (<= t_1 (- INFINITY))
     (*
      (* (* y y) (* y y))
      (* (* x (fma x (* x -0.16666666666666666) 1.0)) 0.008333333333333333))
     (if (<= t_1 1.0) (sin x) (* x t_0)))))

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

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

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

      3. unpow2 N/A

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

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

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

      5. +-commutative N/A

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 77.7

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

   5. Simplified 77.7%

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

   6. Taylor expanded in x around 0

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

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

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

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

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

      7. *-lowering-*.f64 73.8

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

   8. Simplified 73.8%

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

   9. Taylor expanded in y around inf

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. associate-*r* N/A

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

       6. +-commutative N/A

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

       7. distribute-rgt-in N/A

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

       8. *-lft-identity N/A

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

       9. +-commutative N/A

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

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

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

       11. metadata-eval N/A

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

       12. pow-sqr N/A

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

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

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

       14. unpow2 N/A

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

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

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

       16. unpow2 N/A

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

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

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

   11. Simplified 73.8%

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

   if -inf.0 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. sin-lowering-sin.f64 98.5

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

   5. Simplified 98.5%

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

   if 1 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 76.3%

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

4. Final simplification 86.1%

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

Alternative 6: 81.6% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sin x) (/ (sinh y) y))))
   (if (<= t_0 (- INFINITY))
     (*
      (* (* y y) (* y y))
      (* (* x (fma x (* x -0.16666666666666666) 1.0)) 0.008333333333333333))
     (if (<= t_0 1.0)
       (sin x)
       (/
        1.0
        (/
         y
         (*
          x
          (*
           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) / y);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = ((y * y) * (y * y)) * ((x * fma(x, (x * -0.16666666666666666), 1.0)) * 0.008333333333333333);
	} else if (t_0 <= 1.0) {
		tmp = sin(x);
	} else {
		tmp = 1.0 / (y / (x * (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(sin(x) * Float64(sinh(y) / y))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(y * y) * Float64(y * y)) * Float64(Float64(x * fma(x, Float64(x * -0.16666666666666666), 1.0)) * 0.008333333333333333));
	elseif (t_0 <= 1.0)
		tmp = sin(x);
	else
		tmp = Float64(1.0 / Float64(y / Float64(x * Float64(y * fma(y, Float64(y * fma(Float64(y * y), fma(y, Float64(y * 0.0001984126984126984), 0.008333333333333333), 0.16666666666666666)), 1.0)))));
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

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

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

      3. unpow2 N/A

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

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

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

      5. +-commutative N/A

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 77.7

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

   5. Simplified 77.7%

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

   6. Taylor expanded in x around 0

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

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

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

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

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

      7. *-lowering-*.f64 73.8

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

   8. Simplified 73.8%

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

   9. Taylor expanded in y around inf

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. associate-*r* N/A

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

       6. +-commutative N/A

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

       7. distribute-rgt-in N/A

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

       8. *-lft-identity N/A

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

       9. +-commutative N/A

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

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

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

       11. metadata-eval N/A

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

       12. pow-sqr N/A

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

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

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

       14. unpow2 N/A

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

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

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

       16. unpow2 N/A

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

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

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

   11. Simplified 73.8%

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

   if -inf.0 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. sin-lowering-sin.f64 98.5

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

   5. Simplified 98.5%

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

   if 1 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 76.3%

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

   6. Taylor expanded in y around 0

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

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

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

      2. +-commutative N/A

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

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

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

      4. unpow2 N/A

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

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

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

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

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

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

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

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

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 67.5

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

   8. Simplified 67.5%

      \[\leadsto x \cdot \frac{\color{blue}{y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)}}{y} \]
   9. Step-by-step derivation

      1. *-commutative N/A

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

      2. frac-2neg N/A

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

      3. associate-*l/ N/A

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

      4. clear-num N/A

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

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

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

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

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

      7. neg-sub0 N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{0 - y}}{\left(\mathsf{neg}\left(y \cdot \left(\left(y \cdot y\right) \cdot \left(y \cdot \left(y \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{5040} + \frac{1}{120}\right)\right) + \frac{1}{6}\right) + 1\right)\right)\right) \cdot x}} \]

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

         \[\leadsto \frac{1}{\frac{\color{blue}{0 - y}}{\left(\mathsf{neg}\left(y \cdot \left(\left(y \cdot y\right) \cdot \left(y \cdot \left(y \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{5040} + \frac{1}{120}\right)\right) + \frac{1}{6}\right) + 1\right)\right)\right) \cdot x}} \]

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

         \[\leadsto \frac{1}{\frac{0 - y}{\color{blue}{\left(\mathsf{neg}\left(y \cdot \left(\left(y \cdot y\right) \cdot \left(y \cdot \left(y \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{5040} + \frac{1}{120}\right)\right) + \frac{1}{6}\right) + 1\right)\right)\right) \cdot x}}} \]

   10. Applied egg-rr 67.5%

       \[\leadsto \color{blue}{\frac{1}{\frac{0 - y}{\left(0 - y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)\right) \cdot x}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.5%

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

Alternative 7: 57.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin x \cdot \frac{\sinh y}{y} \leq -0.05:\\ \;\;\;\;\left(x \cdot \mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right)\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (sin x) (/ (sinh y) y)) -0.05)
   (*
    (* x (fma x (* x -0.16666666666666666) 1.0))
    (fma (* y y) (fma 0.008333333333333333 (* y y) 0.16666666666666666) 1.0))
   (*
    x
    (/
     (*
      y
      (fma
       (* y y)
       (fma
        y
        (* y (fma (* y y) 0.0001984126984126984 0.008333333333333333))
        0.16666666666666666)
       1.0))
     y))))

C

double code(double x, double y) {
	double tmp;
	if ((sin(x) * (sinh(y) / y)) <= -0.05) {
		tmp = (x * fma(x, (x * -0.16666666666666666), 1.0)) * fma((y * y), fma(0.008333333333333333, (y * y), 0.16666666666666666), 1.0);
	} else {
		tmp = x * ((y * fma((y * y), fma(y, (y * fma((y * y), 0.0001984126984126984, 0.008333333333333333)), 0.16666666666666666), 1.0)) / y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(sin(x) * Float64(sinh(y) / y)) <= -0.05)
		tmp = Float64(Float64(x * fma(x, Float64(x * -0.16666666666666666), 1.0)) * fma(Float64(y * y), fma(0.008333333333333333, Float64(y * y), 0.16666666666666666), 1.0));
	else
		tmp = Float64(x * Float64(Float64(y * fma(Float64(y * y), fma(y, Float64(y * fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333)), 0.16666666666666666), 1.0)) / y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Sin[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], -0.05], N[(N[(x * N[(x * N[(x * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(0.008333333333333333 * N[(y * y), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(y * N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < -0.050000000000000003

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

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

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

      3. unpow2 N/A

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

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

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

      5. +-commutative N/A

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 84.3

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

   5. Simplified 84.3%

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

   6. Taylor expanded in x around 0

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

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

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

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

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

      7. *-lowering-*.f64 52.6

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

   8. Simplified 52.6%

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

   if -0.050000000000000003 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 68.4%

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

   6. Taylor expanded in y around 0

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

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

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

      2. +-commutative N/A

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

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

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

      4. unpow2 N/A

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

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

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

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

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

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

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

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

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 63.5

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

   8. Simplified 63.5%

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

Alternative 8: 57.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin x \cdot \frac{\sinh y}{y} \leq -0.05:\\ \;\;\;\;\left(x \cdot \mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right)\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \left(\left(y \cdot y\right) \cdot 0.0001984126984126984\right), 0.16666666666666666\right), 1\right)}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (sin x) (/ (sinh y) y)) -0.05)
   (*
    (* x (fma x (* x -0.16666666666666666) 1.0))
    (fma (* y y) (fma 0.008333333333333333 (* y y) 0.16666666666666666) 1.0))
   (*
    x
    (/
     (*
      y
      (fma
       (* y y)
       (fma y (* y (* (* y y) 0.0001984126984126984)) 0.16666666666666666)
       1.0))
     y))))

C

double code(double x, double y) {
	double tmp;
	if ((sin(x) * (sinh(y) / y)) <= -0.05) {
		tmp = (x * fma(x, (x * -0.16666666666666666), 1.0)) * fma((y * y), fma(0.008333333333333333, (y * y), 0.16666666666666666), 1.0);
	} else {
		tmp = x * ((y * fma((y * y), fma(y, (y * ((y * y) * 0.0001984126984126984)), 0.16666666666666666), 1.0)) / y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(sin(x) * Float64(sinh(y) / y)) <= -0.05)
		tmp = Float64(Float64(x * fma(x, Float64(x * -0.16666666666666666), 1.0)) * fma(Float64(y * y), fma(0.008333333333333333, Float64(y * y), 0.16666666666666666), 1.0));
	else
		tmp = Float64(x * Float64(Float64(y * fma(Float64(y * y), fma(y, Float64(y * Float64(Float64(y * y) * 0.0001984126984126984)), 0.16666666666666666), 1.0)) / y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Sin[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], -0.05], N[(N[(x * N[(x * N[(x * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(0.008333333333333333 * N[(y * y), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(y * N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < -0.050000000000000003

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

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

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

      3. unpow2 N/A

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

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

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

      5. +-commutative N/A

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 84.3

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

   5. Simplified 84.3%

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

   6. Taylor expanded in x around 0

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

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

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

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

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

      7. *-lowering-*.f64 52.6

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

   8. Simplified 52.6%

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

   if -0.050000000000000003 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 68.4%

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

   6. Taylor expanded in y around 0

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

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

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

      2. +-commutative N/A

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

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

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

      4. unpow2 N/A

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

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

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

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

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

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

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

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

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 63.5

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

   8. Simplified 63.5%

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

   9. Taylor expanded in y around inf

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

       1. *-commutative N/A

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

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

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

       3. unpow2 N/A

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

       4. *-lowering-*.f64 63.5

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

   11. Simplified 63.5%

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

Alternative 9: 57.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin x \cdot \frac{\sinh y}{y} \leq -0.05:\\ \;\;\;\;\left(x \cdot \mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right)\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (sin x) (/ (sinh y) y)) -0.05)
   (*
    (* x (fma x (* x -0.16666666666666666) 1.0))
    (fma (* y y) (fma 0.008333333333333333 (* y y) 0.16666666666666666) 1.0))
   (*
    x
    (fma
     (* y y)
     (fma
      y
      (* y (fma (* y y) 0.0001984126984126984 0.008333333333333333))
      0.16666666666666666)
     1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((sin(x) * (sinh(y) / y)) <= -0.05) {
		tmp = (x * fma(x, (x * -0.16666666666666666), 1.0)) * fma((y * y), fma(0.008333333333333333, (y * y), 0.16666666666666666), 1.0);
	} else {
		tmp = x * fma((y * y), fma(y, (y * fma((y * y), 0.0001984126984126984, 0.008333333333333333)), 0.16666666666666666), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(sin(x) * Float64(sinh(y) / y)) <= -0.05)
		tmp = Float64(Float64(x * fma(x, Float64(x * -0.16666666666666666), 1.0)) * fma(Float64(y * y), fma(0.008333333333333333, Float64(y * y), 0.16666666666666666), 1.0));
	else
		tmp = Float64(x * fma(Float64(y * y), fma(y, Float64(y * fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333)), 0.16666666666666666), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Sin[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], -0.05], N[(N[(x * N[(x * N[(x * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(0.008333333333333333 * N[(y * y), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < -0.050000000000000003

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

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

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

      3. unpow2 N/A

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

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

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

      5. +-commutative N/A

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 84.3

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

   5. Simplified 84.3%

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

   6. Taylor expanded in x around 0

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

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

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

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

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

      7. *-lowering-*.f64 52.6

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

   8. Simplified 52.6%

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

   if -0.050000000000000003 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 68.4%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

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

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

      3. unpow2 N/A

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

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

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

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

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

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

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 63.5

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

   8. Simplified 63.5%

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

Alternative 10: 57.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin x \cdot \frac{\sinh y}{y} \leq -0.1:\\ \;\;\;\;\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \left(\left(x \cdot \mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right)\right) \cdot 0.008333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (sin x) (/ (sinh y) y)) -0.1)
   (*
    (* (* y y) (* y y))
    (* (* x (fma x (* x -0.16666666666666666) 1.0)) 0.008333333333333333))
   (*
    x
    (fma
     (* y y)
     (fma
      y
      (* y (fma (* y y) 0.0001984126984126984 0.008333333333333333))
      0.16666666666666666)
     1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((sin(x) * (sinh(y) / y)) <= -0.1) {
		tmp = ((y * y) * (y * y)) * ((x * fma(x, (x * -0.16666666666666666), 1.0)) * 0.008333333333333333);
	} else {
		tmp = x * fma((y * y), fma(y, (y * fma((y * y), 0.0001984126984126984, 0.008333333333333333)), 0.16666666666666666), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(sin(x) * Float64(sinh(y) / y)) <= -0.1)
		tmp = Float64(Float64(Float64(y * y) * Float64(y * y)) * Float64(Float64(x * fma(x, Float64(x * -0.16666666666666666), 1.0)) * 0.008333333333333333));
	else
		tmp = Float64(x * fma(Float64(y * y), fma(y, Float64(y * fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333)), 0.16666666666666666), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Sin[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], -0.1], N[(N[(N[(y * y), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(N[(x * N[(x * N[(x * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 0.008333333333333333), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < -0.10000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

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

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

      3. unpow2 N/A

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

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

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

      5. +-commutative N/A

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 84.1

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

   5. Simplified 84.1%

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

   6. Taylor expanded in x around 0

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

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

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

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

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

      7. *-lowering-*.f64 53.2

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

   8. Simplified 53.2%

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

   9. Taylor expanded in y around inf

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. associate-*r* N/A

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

       6. +-commutative N/A

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

       7. distribute-rgt-in N/A

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

       8. *-lft-identity N/A

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

       9. +-commutative N/A

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

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

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

       11. metadata-eval N/A

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

       12. pow-sqr N/A

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

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

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

       14. unpow2 N/A

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

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

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

       16. unpow2 N/A

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

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

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

   11. Simplified 53.1%

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

   if -0.10000000000000001 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 68.0%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

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

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

      3. unpow2 N/A

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

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

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

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

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

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

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 63.1

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

   8. Simplified 63.1%

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

4. Final simplification 59.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin x \cdot \frac{\sinh y}{y} \leq -0.1:\\ \;\;\;\;\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \left(\left(x \cdot \mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right)\right) \cdot 0.008333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 47.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin x \cdot \frac{\sinh y}{y} \leq -0.05:\\ \;\;\;\;\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(y \cdot y, -0.027777777777777776, -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (sin x) (/ (sinh y) y)) -0.05)
   (* (* x (* x x)) (fma (* y y) -0.027777777777777776 -0.16666666666666666))
   (*
    x
    (fma
     (* y y)
     (fma
      y
      (* y (fma (* y y) 0.0001984126984126984 0.008333333333333333))
      0.16666666666666666)
     1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((sin(x) * (sinh(y) / y)) <= -0.05) {
		tmp = (x * (x * x)) * fma((y * y), -0.027777777777777776, -0.16666666666666666);
	} else {
		tmp = x * fma((y * y), fma(y, (y * fma((y * y), 0.0001984126984126984, 0.008333333333333333)), 0.16666666666666666), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(sin(x) * Float64(sinh(y) / y)) <= -0.05)
		tmp = Float64(Float64(x * Float64(x * x)) * fma(Float64(y * y), -0.027777777777777776, -0.16666666666666666));
	else
		tmp = Float64(x * fma(Float64(y * y), fma(y, Float64(y * fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333)), 0.16666666666666666), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Sin[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], -0.05], N[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * -0.027777777777777776 + -0.16666666666666666), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < -0.050000000000000003

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 63.1

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

   5. Simplified 63.1%

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

   6. Taylor expanded in x around 0

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

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

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

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

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

      7. *-lowering-*.f64 40.1

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

   8. Simplified 40.1%

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

   9. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

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

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

       5. cube-mult N/A

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

       6. unpow2 N/A

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

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

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

       8. unpow2 N/A

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

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

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

       10. +-commutative N/A

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

       11. distribute-rgt-in N/A

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

       12. *-commutative N/A

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

       13. associate-*l* N/A

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

       14. metadata-eval N/A

           \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left({y}^{2} \cdot \color{blue}{\frac{-1}{36}} + 1 \cdot \frac{-1}{6}\right) \]

       15. metadata-eval N/A

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

       16. metadata-eval N/A

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

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

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

       18. unpow2 N/A

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

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

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

       20. metadata-eval 18.6

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

   11. Simplified 18.6%

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

   if -0.050000000000000003 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 68.4%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

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

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

      3. unpow2 N/A

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

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

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

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

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

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

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 63.5

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

   8. Simplified 63.5%

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

Alternative 12: 47.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin x \cdot \frac{\sinh y}{y} \leq -0.05:\\ \;\;\;\;\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(y \cdot y, -0.027777777777777776, -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, 0.0001984126984126984 \cdot \left(y \cdot \left(y \cdot y\right)\right), 0.16666666666666666\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (sin x) (/ (sinh y) y)) -0.05)
   (* (* x (* x x)) (fma (* y y) -0.027777777777777776 -0.16666666666666666))
   (*
    x
    (fma
     y
     (* y (fma y (* 0.0001984126984126984 (* y (* y y))) 0.16666666666666666))
     1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((sin(x) * (sinh(y) / y)) <= -0.05) {
		tmp = (x * (x * x)) * fma((y * y), -0.027777777777777776, -0.16666666666666666);
	} else {
		tmp = x * fma(y, (y * fma(y, (0.0001984126984126984 * (y * (y * y))), 0.16666666666666666)), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(sin(x) * Float64(sinh(y) / y)) <= -0.05)
		tmp = Float64(Float64(x * Float64(x * x)) * fma(Float64(y * y), -0.027777777777777776, -0.16666666666666666));
	else
		tmp = Float64(x * fma(y, Float64(y * fma(y, Float64(0.0001984126984126984 * Float64(y * Float64(y * y))), 0.16666666666666666)), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Sin[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], -0.05], N[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * -0.027777777777777776 + -0.16666666666666666), $MachinePrecision]), $MachinePrecision], N[(x * N[(y * N[(y * N[(y * N[(0.0001984126984126984 * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < -0.050000000000000003

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 63.1

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

   5. Simplified 63.1%

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

   6. Taylor expanded in x around 0

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

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

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

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

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

      7. *-lowering-*.f64 40.1

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

   8. Simplified 40.1%

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

   9. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

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

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

       5. cube-mult N/A

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

       6. unpow2 N/A

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

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

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

       8. unpow2 N/A

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

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

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

       10. +-commutative N/A

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

       11. distribute-rgt-in N/A

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

       12. *-commutative N/A

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

       13. associate-*l* N/A

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

       14. metadata-eval N/A

           \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left({y}^{2} \cdot \color{blue}{\frac{-1}{36}} + 1 \cdot \frac{-1}{6}\right) \]

       15. metadata-eval N/A

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

       16. metadata-eval N/A

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

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

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

       18. unpow2 N/A

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

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

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

       20. metadata-eval 18.6

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

   11. Simplified 18.6%

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

   if -0.050000000000000003 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 68.4%

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

   6. Taylor expanded in y around 0

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

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

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

      2. +-commutative N/A

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

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

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

      4. unpow2 N/A

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

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

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

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

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

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

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

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

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 63.5

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

   8. Simplified 63.5%

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

   9. Taylor expanded in y around inf

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

       1. *-commutative N/A

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

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

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

       3. unpow2 N/A

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

       4. *-lowering-*.f64 63.5

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

   11. Simplified 63.5%

       \[\leadsto x \cdot \frac{y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \color{blue}{\left(\left(y \cdot y\right) \cdot 0.0001984126984126984\right)}, 0.16666666666666666\right), 1\right)}{y} \]
   12. Step-by-step derivation

       1. *-commutative N/A

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

   13. Applied egg-rr 63.5%

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

4. Final simplification 48.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin x \cdot \frac{\sinh y}{y} \leq -0.05:\\ \;\;\;\;\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(y \cdot y, -0.027777777777777776, -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, 0.0001984126984126984 \cdot \left(y \cdot \left(y \cdot y\right)\right), 0.16666666666666666\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 38.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin x \cdot \frac{\sinh y}{y} \leq 0.95:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\left(y \cdot y\right) \cdot 0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (sin x) (/ (sinh y) y)) 0.95)
   x
   (* x (* (* y y) 0.16666666666666666))))

C

double code(double x, double y) {
	double tmp;
	if ((sin(x) * (sinh(y) / y)) <= 0.95) {
		tmp = x;
	} else {
		tmp = x * ((y * y) * 0.16666666666666666);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((sin(x) * (sinh(y) / y)) <= 0.95d0) then
        tmp = x
    else
        tmp = x * ((y * y) * 0.16666666666666666d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((Math.sin(x) * (Math.sinh(y) / y)) <= 0.95) {
		tmp = x;
	} else {
		tmp = x * ((y * y) * 0.16666666666666666);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (math.sin(x) * (math.sinh(y) / y)) <= 0.95:
		tmp = x
	else:
		tmp = x * ((y * y) * 0.16666666666666666)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(sin(x) * Float64(sinh(y) / y)) <= 0.95)
		tmp = x;
	else
		tmp = Float64(x * Float64(Float64(y * y) * 0.16666666666666666));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((sin(x) * (sinh(y) / y)) <= 0.95)
		tmp = x;
	else
		tmp = x * ((y * y) * 0.16666666666666666);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Sin[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], 0.95], x, N[(x * N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < 0.94999999999999996

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. sin-lowering-sin.f64 65.8

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

   5. Simplified 65.8%

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

   6. Taylor expanded in x around 0

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

   8. Simplified 33.6%

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

   if 0.94999999999999996 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 72.6%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

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

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

      6. *-lowering-*.f64 47.4

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

   8. Simplified 47.4%

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

   9. Taylor expanded in y around inf

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

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

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

       2. unpow2 N/A

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

       3. *-lowering-*.f64 47.5

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

   11. Simplified 47.5%

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

4. Final simplification 37.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin x \cdot \frac{\sinh y}{y} \leq 0.95:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\left(y \cdot y\right) \cdot 0.16666666666666666\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 35.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin x \cdot \frac{\sinh y}{y} \leq 0.934:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot \left(y \cdot \left(x \cdot y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (sin x) (/ (sinh y) y)) 0.934)
   x
   (* 0.16666666666666666 (* y (* x y)))))

C

double code(double x, double y) {
	double tmp;
	if ((sin(x) * (sinh(y) / y)) <= 0.934) {
		tmp = x;
	} else {
		tmp = 0.16666666666666666 * (y * (x * 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) / y)) <= 0.934d0) then
        tmp = x
    else
        tmp = 0.16666666666666666d0 * (y * (x * 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) / y)) <= 0.934) {
		tmp = x;
	} else {
		tmp = 0.16666666666666666 * (y * (x * y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (math.sin(x) * (math.sinh(y) / y)) <= 0.934:
		tmp = x
	else:
		tmp = 0.16666666666666666 * (y * (x * y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(sin(x) * Float64(sinh(y) / y)) <= 0.934)
		tmp = x;
	else
		tmp = Float64(0.16666666666666666 * Float64(y * Float64(x * y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((sin(x) * (sinh(y) / y)) <= 0.934)
		tmp = x;
	else
		tmp = 0.16666666666666666 * (y * (x * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Sin[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], 0.934], x, N[(0.16666666666666666 * N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < 0.934000000000000052

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. sin-lowering-sin.f64 65.6

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

   5. Simplified 65.6%

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

   6. Taylor expanded in x around 0

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

   8. Simplified 33.8%

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

   if 0.934000000000000052 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 71.8%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

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

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

      6. *-lowering-*.f64 46.9

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

   8. Simplified 46.9%

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

   9. Taylor expanded in y around inf

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

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

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

       2. *-commutative N/A

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

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

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

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

       6. *-lowering-*.f64 32.0

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

   11. Simplified 32.0%

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

4. Final simplification 33.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin x \cdot \frac{\sinh y}{y} \leq 0.934:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot \left(y \cdot \left(x \cdot y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 55.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin x \leq -0.01:\\ \;\;\;\;\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(y \cdot y, -0.027777777777777776, -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sin x) -0.01)
   (* (* x (* x x)) (fma (* y y) -0.027777777777777776 -0.16666666666666666))
   (*
    x
    (fma y (* y (fma y (* y 0.008333333333333333) 0.16666666666666666)) 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (sin(x) <= -0.01) {
		tmp = (x * (x * x)) * fma((y * y), -0.027777777777777776, -0.16666666666666666);
	} else {
		tmp = x * fma(y, (y * fma(y, (y * 0.008333333333333333), 0.16666666666666666)), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (sin(x) <= -0.01)
		tmp = Float64(Float64(x * Float64(x * x)) * fma(Float64(y * y), -0.027777777777777776, -0.16666666666666666));
	else
		tmp = Float64(x * fma(y, Float64(y * fma(y, Float64(y * 0.008333333333333333), 0.16666666666666666)), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[Sin[x], $MachinePrecision], -0.01], N[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * -0.027777777777777776 + -0.16666666666666666), $MachinePrecision]), $MachinePrecision], N[(x * N[(y * N[(y * N[(y * N[(y * 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 69.3

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

   5. Simplified 69.3%

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

   6. Taylor expanded in x around 0

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

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

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

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

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

      7. *-lowering-*.f64 29.9

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

   8. Simplified 29.9%

      \[\leadsto \color{blue}{\left(x \cdot \mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right)\right)} \cdot \mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right) \]

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{-1}{6} \cdot \left({x}^{3} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \color{blue}{\left({x}^{3} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right) \cdot \frac{-1}{6}} \]

       2. associate-*r* N/A

          \[\leadsto \color{blue}{{x}^{3} \cdot \left(\left(1 + \frac{1}{6} \cdot {y}^{2}\right) \cdot \frac{-1}{6}\right)} \]

       3. *-commutative N/A

          \[\leadsto {x}^{3} \cdot \color{blue}{\left(\frac{-1}{6} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{{x}^{3} \cdot \left(\frac{-1}{6} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)} \]

       5. cube-mult N/A

          \[\leadsto \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot \left(\frac{-1}{6} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \left(x \cdot \color{blue}{{x}^{2}}\right) \cdot \left(\frac{-1}{6} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\left(x \cdot {x}^{2}\right)} \cdot \left(\frac{-1}{6} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(\frac{-1}{6} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(\frac{-1}{6} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right) \]

       10. +-commutative N/A

           \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{-1}{6} \cdot \color{blue}{\left(\frac{1}{6} \cdot {y}^{2} + 1\right)}\right) \]

       11. distribute-rgt-in N/A

           \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(\left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \frac{-1}{6} + 1 \cdot \frac{-1}{6}\right)} \]

       12. *-commutative N/A

           \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\color{blue}{\left({y}^{2} \cdot \frac{1}{6}\right)} \cdot \frac{-1}{6} + 1 \cdot \frac{-1}{6}\right) \]

       13. associate-*l* N/A

           \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\color{blue}{{y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{-1}{6}\right)} + 1 \cdot \frac{-1}{6}\right) \]

       14. metadata-eval N/A

           \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left({y}^{2} \cdot \color{blue}{\frac{-1}{36}} + 1 \cdot \frac{-1}{6}\right) \]

       15. metadata-eval N/A

           \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left({y}^{2} \cdot \color{blue}{\left(\frac{-1}{6} \cdot \frac{1}{6}\right)} + 1 \cdot \frac{-1}{6}\right) \]

       16. metadata-eval N/A

           \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left({y}^{2} \cdot \left(\frac{-1}{6} \cdot \frac{1}{6}\right) + \color{blue}{\frac{-1}{6}}\right) \]

       17. accelerator-lowering-fma.f64 N/A

           \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{-1}{6} \cdot \frac{1}{6}, \frac{-1}{6}\right)} \]

       18. unpow2 N/A

           \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{-1}{6} \cdot \frac{1}{6}, \frac{-1}{6}\right) \]

       19. *-lowering-*.f64 N/A

           \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{-1}{6} \cdot \frac{1}{6}, \frac{-1}{6}\right) \]

       20. metadata-eval 29.9

           \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{-0.027777777777777776}, -0.16666666666666666\right) \]

   11. Simplified 29.9%

       \[\leadsto \color{blue}{\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(y \cdot y, -0.027777777777777776, -0.16666666666666666\right)} \]

   if -0.0100000000000000002 < (sin.f64 x)

   1. Initial program 100.0%

      \[\sin x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x} \cdot \frac{\sinh y}{y} \]
   4. Step-by-step derivation

   5. Simplified 74.0%

      \[\leadsto \color{blue}{x} \cdot \frac{\sinh y}{y} \]

   6. Taylor expanded in y around 0

      \[\leadsto x \cdot \frac{\color{blue}{y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)}}{y} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \frac{\color{blue}{y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)}}{y} \]

      2. +-commutative N/A

         \[\leadsto x \cdot \frac{y \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) + 1\right)}}{y} \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \frac{y \cdot \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right), 1\right)}}{y} \]

      4. unpow2 N/A

         \[\leadsto x \cdot \frac{y \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right), 1\right)}{y} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \frac{y \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right), 1\right)}{y} \]

      6. +-commutative N/A

         \[\leadsto x \cdot \frac{y \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{{y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right) + \frac{1}{6}}, 1\right)}{y} \]

      7. unpow2 N/A

         \[\leadsto x \cdot \frac{y \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{\left(y \cdot y\right)} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right) + \frac{1}{6}, 1\right)}{y} \]

      8. associate-*l* N/A

         \[\leadsto x \cdot \frac{y \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{y \cdot \left(y \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)} + \frac{1}{6}, 1\right)}{y} \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \frac{y \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{\mathsf{fma}\left(y, y \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right), \frac{1}{6}\right)}, 1\right)}{y} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \frac{y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, \color{blue}{y \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)}, \frac{1}{6}\right), 1\right)}{y} \]

      11. +-commutative N/A

          \[\leadsto x \cdot \frac{y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \color{blue}{\left(\frac{1}{5040} \cdot {y}^{2} + \frac{1}{120}\right)}, \frac{1}{6}\right), 1\right)}{y} \]

      12. *-commutative N/A

          \[\leadsto x \cdot \frac{y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \left(\color{blue}{{y}^{2} \cdot \frac{1}{5040}} + \frac{1}{120}\right), \frac{1}{6}\right), 1\right)}{y} \]

      13. accelerator-lowering-fma.f64 N/A

          \[\leadsto x \cdot \frac{y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{1}{5040}, \frac{1}{120}\right)}, \frac{1}{6}\right), 1\right)}{y} \]

      14. unpow2 N/A

          \[\leadsto x \cdot \frac{y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right)}{y} \]

      15. *-lowering-*.f64 68.4

          \[\leadsto x \cdot \frac{y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)}{y} \]

   8. Simplified 68.4%

      \[\leadsto x \cdot \frac{\color{blue}{y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)}}{y} \]
   9. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto x \cdot \color{blue}{\left(y \cdot \frac{\left(y \cdot y\right) \cdot \left(y \cdot \left(y \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{5040} + \frac{1}{120}\right)\right) + \frac{1}{6}\right) + 1}{y}\right)} \]

      2. clear-num N/A

         \[\leadsto x \cdot \left(y \cdot \color{blue}{\frac{1}{\frac{y}{\left(y \cdot y\right) \cdot \left(y \cdot \left(y \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{5040} + \frac{1}{120}\right)\right) + \frac{1}{6}\right) + 1}}}\right) \]

      3. un-div-inv N/A

         \[\leadsto x \cdot \color{blue}{\frac{y}{\frac{y}{\left(y \cdot y\right) \cdot \left(y \cdot \left(y \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{5040} + \frac{1}{120}\right)\right) + \frac{1}{6}\right) + 1}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto x \cdot \color{blue}{\frac{y}{\frac{y}{\left(y \cdot y\right) \cdot \left(y \cdot \left(y \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{5040} + \frac{1}{120}\right)\right) + \frac{1}{6}\right) + 1}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto x \cdot \frac{y}{\color{blue}{\frac{y}{\left(y \cdot y\right) \cdot \left(y \cdot \left(y \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{5040} + \frac{1}{120}\right)\right) + \frac{1}{6}\right) + 1}}} \]

      6. associate-*l* N/A

         \[\leadsto x \cdot \frac{y}{\frac{y}{\color{blue}{y \cdot \left(y \cdot \left(y \cdot \left(y \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{5040} + \frac{1}{120}\right)\right) + \frac{1}{6}\right)\right)} + 1}} \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \frac{y}{\frac{y}{\color{blue}{\mathsf{fma}\left(y, y \cdot \left(y \cdot \left(y \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{5040} + \frac{1}{120}\right)\right) + \frac{1}{6}\right), 1\right)}}} \]

   10. Applied egg-rr 68.0%

       \[\leadsto x \cdot \color{blue}{\frac{y}{\frac{y}{\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)}}} \]

   11. Taylor expanded in y around 0

       \[\leadsto x \cdot \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} \]
   12. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto x \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) + 1\right)} \]

       2. unpow2 N/A

          \[\leadsto x \cdot \left(\color{blue}{\left(y \cdot y\right)} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) + 1\right) \]

       3. associate-*l* N/A

          \[\leadsto x \cdot \left(\color{blue}{y \cdot \left(y \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} + 1\right) \]

       4. accelerator-lowering-fma.f64 N/A

          \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(y, y \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right), 1\right)} \]

       5. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(y, \color{blue}{y \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)}, 1\right) \]

       6. +-commutative N/A

          \[\leadsto x \cdot \mathsf{fma}\left(y, y \cdot \color{blue}{\left(\frac{1}{120} \cdot {y}^{2} + \frac{1}{6}\right)}, 1\right) \]

       7. *-commutative N/A

          \[\leadsto x \cdot \mathsf{fma}\left(y, y \cdot \left(\color{blue}{{y}^{2} \cdot \frac{1}{120}} + \frac{1}{6}\right), 1\right) \]

       8. unpow2 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(y, y \cdot \left(\color{blue}{\left(y \cdot y\right)} \cdot \frac{1}{120} + \frac{1}{6}\right), 1\right) \]

       9. associate-*l* N/A

          \[\leadsto x \cdot \mathsf{fma}\left(y, y \cdot \left(\color{blue}{y \cdot \left(y \cdot \frac{1}{120}\right)} + \frac{1}{6}\right), 1\right) \]

       10. accelerator-lowering-fma.f64 N/A

           \[\leadsto x \cdot \mathsf{fma}\left(y, y \cdot \color{blue}{\mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right)}, 1\right) \]

       11. *-lowering-*.f64 64.7

           \[\leadsto x \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, \color{blue}{y \cdot 0.008333333333333333}, 0.16666666666666666\right), 1\right) \]

   13. Simplified 64.7%

       \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 49.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin x \leq -0.01:\\ \;\;\;\;\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(y \cdot y, -0.027777777777777776, -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sin x) -0.01)
   (* (* x (* x x)) (fma (* y y) -0.027777777777777776 -0.16666666666666666))
   (* x (fma 0.16666666666666666 (* y y) 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (sin(x) <= -0.01) {
		tmp = (x * (x * x)) * fma((y * y), -0.027777777777777776, -0.16666666666666666);
	} else {
		tmp = x * fma(0.16666666666666666, (y * y), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (sin(x) <= -0.01)
		tmp = Float64(Float64(x * Float64(x * x)) * fma(Float64(y * y), -0.027777777777777776, -0.16666666666666666));
	else
		tmp = Float64(x * fma(0.16666666666666666, Float64(y * y), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[Sin[x], $MachinePrecision], -0.01], N[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * -0.027777777777777776 + -0.16666666666666666), $MachinePrecision]), $MachinePrecision], N[(x * N[(0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 x) < -0.0100000000000000002

   1. Initial program 100.0%

      \[\sin x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \sin x \cdot \color{blue}{\left(1 + \frac{1}{6} \cdot {y}^{2}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \sin x \cdot \color{blue}{\left(\frac{1}{6} \cdot {y}^{2} + 1\right)} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \sin x \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{6}, {y}^{2}, 1\right)} \]

      3. unpow2 N/A

         \[\leadsto \sin x \cdot \mathsf{fma}\left(\frac{1}{6}, \color{blue}{y \cdot y}, 1\right) \]

      4. *-lowering-*.f64 69.3

         \[\leadsto \sin x \cdot \mathsf{fma}\left(0.16666666666666666, \color{blue}{y \cdot y}, 1\right) \]

   5. Simplified 69.3%

      \[\leadsto \sin x \cdot \color{blue}{\mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)} \cdot \mathsf{fma}\left(\frac{1}{6}, y \cdot y, 1\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)} \cdot \mathsf{fma}\left(\frac{1}{6}, y \cdot y, 1\right) \]

      2. +-commutative N/A

         \[\leadsto \left(x \cdot \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2} + 1\right)}\right) \cdot \mathsf{fma}\left(\frac{1}{6}, y \cdot y, 1\right) \]

      3. *-commutative N/A

         \[\leadsto \left(x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{-1}{6}} + 1\right)\right) \cdot \mathsf{fma}\left(\frac{1}{6}, y \cdot y, 1\right) \]

      4. unpow2 N/A

         \[\leadsto \left(x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{-1}{6} + 1\right)\right) \cdot \mathsf{fma}\left(\frac{1}{6}, y \cdot y, 1\right) \]

      5. associate-*l* N/A

         \[\leadsto \left(x \cdot \left(\color{blue}{x \cdot \left(x \cdot \frac{-1}{6}\right)} + 1\right)\right) \cdot \mathsf{fma}\left(\frac{1}{6}, y \cdot y, 1\right) \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(x \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)}\right) \cdot \mathsf{fma}\left(\frac{1}{6}, y \cdot y, 1\right) \]

      7. *-lowering-*.f64 29.9

         \[\leadsto \left(x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot -0.16666666666666666}, 1\right)\right) \cdot \mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right) \]

   8. Simplified 29.9%

      \[\leadsto \color{blue}{\left(x \cdot \mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right)\right)} \cdot \mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right) \]

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{-1}{6} \cdot \left({x}^{3} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \color{blue}{\left({x}^{3} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right) \cdot \frac{-1}{6}} \]

       2. associate-*r* N/A

          \[\leadsto \color{blue}{{x}^{3} \cdot \left(\left(1 + \frac{1}{6} \cdot {y}^{2}\right) \cdot \frac{-1}{6}\right)} \]

       3. *-commutative N/A

          \[\leadsto {x}^{3} \cdot \color{blue}{\left(\frac{-1}{6} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{{x}^{3} \cdot \left(\frac{-1}{6} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)} \]

       5. cube-mult N/A

          \[\leadsto \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot \left(\frac{-1}{6} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \left(x \cdot \color{blue}{{x}^{2}}\right) \cdot \left(\frac{-1}{6} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\left(x \cdot {x}^{2}\right)} \cdot \left(\frac{-1}{6} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(\frac{-1}{6} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(\frac{-1}{6} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right) \]

       10. +-commutative N/A

           \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{-1}{6} \cdot \color{blue}{\left(\frac{1}{6} \cdot {y}^{2} + 1\right)}\right) \]

       11. distribute-rgt-in N/A

           \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(\left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \frac{-1}{6} + 1 \cdot \frac{-1}{6}\right)} \]

       12. *-commutative N/A

           \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\color{blue}{\left({y}^{2} \cdot \frac{1}{6}\right)} \cdot \frac{-1}{6} + 1 \cdot \frac{-1}{6}\right) \]

       13. associate-*l* N/A

           \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\color{blue}{{y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{-1}{6}\right)} + 1 \cdot \frac{-1}{6}\right) \]

       14. metadata-eval N/A

           \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left({y}^{2} \cdot \color{blue}{\frac{-1}{36}} + 1 \cdot \frac{-1}{6}\right) \]

       15. metadata-eval N/A

           \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left({y}^{2} \cdot \color{blue}{\left(\frac{-1}{6} \cdot \frac{1}{6}\right)} + 1 \cdot \frac{-1}{6}\right) \]

       16. metadata-eval N/A

           \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left({y}^{2} \cdot \left(\frac{-1}{6} \cdot \frac{1}{6}\right) + \color{blue}{\frac{-1}{6}}\right) \]

       17. accelerator-lowering-fma.f64 N/A

           \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{-1}{6} \cdot \frac{1}{6}, \frac{-1}{6}\right)} \]

       18. unpow2 N/A

           \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{-1}{6} \cdot \frac{1}{6}, \frac{-1}{6}\right) \]

       19. *-lowering-*.f64 N/A

           \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{-1}{6} \cdot \frac{1}{6}, \frac{-1}{6}\right) \]

       20. metadata-eval 29.9

           \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{-0.027777777777777776}, -0.16666666666666666\right) \]

   11. Simplified 29.9%

       \[\leadsto \color{blue}{\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(y \cdot y, -0.027777777777777776, -0.16666666666666666\right)} \]

   if -0.0100000000000000002 < (sin.f64 x)

   1. Initial program 100.0%

      \[\sin x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \sin x \cdot \color{blue}{\left(1 + \frac{1}{6} \cdot {y}^{2}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \sin x \cdot \color{blue}{\left(\frac{1}{6} \cdot {y}^{2} + 1\right)} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \sin x \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{6}, {y}^{2}, 1\right)} \]

      3. unpow2 N/A

         \[\leadsto \sin x \cdot \mathsf{fma}\left(\frac{1}{6}, \color{blue}{y \cdot y}, 1\right) \]

      4. *-lowering-*.f64 77.0

         \[\leadsto \sin x \cdot \mathsf{fma}\left(0.16666666666666666, \color{blue}{y \cdot y}, 1\right) \]

   5. Simplified 77.0%

      \[\leadsto \sin x \cdot \color{blue}{\mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x} \cdot \mathsf{fma}\left(\frac{1}{6}, y \cdot y, 1\right) \]
   7. Step-by-step derivation

   8. Simplified 55.7%

      \[\leadsto \color{blue}{x} \cdot \mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 47.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin x \leq -0.01:\\ \;\;\;\;x \cdot \mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sin x) -0.01)
   (* x (fma x (* x -0.16666666666666666) 1.0))
   (* x (fma 0.16666666666666666 (* y y) 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (sin(x) <= -0.01) {
		tmp = x * fma(x, (x * -0.16666666666666666), 1.0);
	} else {
		tmp = x * fma(0.16666666666666666, (y * y), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (sin(x) <= -0.01)
		tmp = Float64(x * fma(x, Float64(x * -0.16666666666666666), 1.0));
	else
		tmp = Float64(x * fma(0.16666666666666666, Float64(y * y), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[Sin[x], $MachinePrecision], -0.01], N[(x * N[(x * N[(x * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(x * N[(0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 x) < -0.0100000000000000002

   1. Initial program 100.0%

      \[\sin x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\sin x} \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 51.6

         \[\leadsto \color{blue}{\sin x} \]

   5. Simplified 51.6%

      \[\leadsto \color{blue}{\sin x} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)} \]

      2. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2} + 1\right)} \]

      3. *-commutative N/A

         \[\leadsto x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{-1}{6}} + 1\right) \]

      4. unpow2 N/A

         \[\leadsto x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{-1}{6} + 1\right) \]

      5. associate-*l* N/A

         \[\leadsto x \cdot \left(\color{blue}{x \cdot \left(x \cdot \frac{-1}{6}\right)} + 1\right) \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)} \]

      7. *-lowering-*.f64 29.9

         \[\leadsto x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot -0.16666666666666666}, 1\right) \]

   8. Simplified 29.9%

      \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right)} \]

   if -0.0100000000000000002 < (sin.f64 x)

   1. Initial program 100.0%

      \[\sin x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \sin x \cdot \color{blue}{\left(1 + \frac{1}{6} \cdot {y}^{2}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \sin x \cdot \color{blue}{\left(\frac{1}{6} \cdot {y}^{2} + 1\right)} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \sin x \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{6}, {y}^{2}, 1\right)} \]

      3. unpow2 N/A

         \[\leadsto \sin x \cdot \mathsf{fma}\left(\frac{1}{6}, \color{blue}{y \cdot y}, 1\right) \]

      4. *-lowering-*.f64 77.0

         \[\leadsto \sin x \cdot \mathsf{fma}\left(0.16666666666666666, \color{blue}{y \cdot y}, 1\right) \]

   5. Simplified 77.0%

      \[\leadsto \sin x \cdot \color{blue}{\mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x} \cdot \mathsf{fma}\left(\frac{1}{6}, y \cdot y, 1\right) \]
   7. Step-by-step derivation

   8. Simplified 55.7%

      \[\leadsto \color{blue}{x} \cdot \mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 48.9% accurate, 12.8× speedup

Math

\[\begin{array}{l} \\ x \cdot \mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* x (fma 0.16666666666666666 (* y y) 1.0)))

C

double code(double x, double y) {
	return x * fma(0.16666666666666666, (y * y), 1.0);
}

Julia

function code(x, y)
	return Float64(x * fma(0.16666666666666666, Float64(y * y), 1.0))
end

Wolfram

code[x_, y_] := N[(x * N[(0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\sin x \cdot \frac{\sinh y}{y} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \sin x \cdot \color{blue}{\left(1 + \frac{1}{6} \cdot {y}^{2}\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \sin x \cdot \color{blue}{\left(\frac{1}{6} \cdot {y}^{2} + 1\right)} \]

   2. accelerator-lowering-fma.f64 N/A

      \[\leadsto \sin x \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{6}, {y}^{2}, 1\right)} \]

   3. unpow2 N/A

      \[\leadsto \sin x \cdot \mathsf{fma}\left(\frac{1}{6}, \color{blue}{y \cdot y}, 1\right) \]

   4. *-lowering-*.f64 75.5

      \[\leadsto \sin x \cdot \mathsf{fma}\left(0.16666666666666666, \color{blue}{y \cdot y}, 1\right) \]

5. Simplified 75.5%

   \[\leadsto \sin x \cdot \color{blue}{\mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right)} \]

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x} \cdot \mathsf{fma}\left(\frac{1}{6}, y \cdot y, 1\right) \]
7. Step-by-step derivation

8. Simplified 47.5%

   \[\leadsto \color{blue}{x} \cdot \mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right) \]
9. Add Preprocessing

Alternative 19: 27.5% accurate, 217.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

(FPCore (x y) :precision binary64 x)

C

double code(double x, double y) {
	return x;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x
end function

Java

public static double code(double x, double y) {
	return x;
}

Python

def code(x, y):
	return x

Julia

function code(x, y)
	return x
end

MATLAB

function tmp = code(x, y)
	tmp = x;
end

Wolfram

code[x_, y_] := x

Derivation

1. Initial program 100.0%

   \[\sin x \cdot \frac{\sinh y}{y} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\sin x} \]
4. Step-by-step derivation

   1. sin-lowering-sin.f64 47.6

      \[\leadsto \color{blue}{\sin x} \]

5. Simplified 47.6%

   \[\leadsto \color{blue}{\sin x} \]

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x} \]
7. Step-by-step derivation

8. Simplified 24.0%

   \[\leadsto \color{blue}{x} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024195 
(FPCore (x y)
  :name "Linear.Quaternion:$ccos from linear-1.19.1.3"
  :precision binary64
  (* (sin x) (/ (sinh y) y)))