Linear.Quaternion:$ccos from linear-1.19.1.3

Percentage Accurate: 100.0% → 100.0%

Time: 12.2s

Alternatives: 21

Speedup: 1.0×

• Could not converge on a ground truth

  1. x = 1.9280807727414027e-291
  2. y = 3.776807843739643e+86

• 49.0% 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: 86.7% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sinh y) y)) (t_1 (* (sin x) t_0)))
   (if (<= t_1 (- INFINITY))
     (* t_0 (fma -0.16666666666666666 (* x (* x x)) x))
     (if (<= t_1 1.0)
       (*
        (sin x)
        (fma
         (* y y)
         (fma y (* y 0.008333333333333333) 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 * fma(-0.16666666666666666, (x * (x * x)), x);
	} else if (t_1 <= 1.0) {
		tmp = sin(x) * fma((y * y), fma(y, (y * 0.008333333333333333), 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 * fma(-0.16666666666666666, Float64(x * Float64(x * x)), x));
	elseif (t_1 <= 1.0)
		tmp = Float64(sin(x) * fma(Float64(y * y), fma(y, Float64(y * 0.008333333333333333), 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[(-0.16666666666666666 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1.0], N[(N[Sin[x], $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * 0.008333333333333333), $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. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-rgt-identity N/A

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

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

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 69.2

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

   5. Simplified 69.2%

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

   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

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

      1. distribute-rgt-in N/A

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

      2. *-rgt-identity N/A

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

      3. distribute-rgt-in N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out N/A

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

      7. +-commutative N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. distribute-lft-in N/A

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

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

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

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

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

      13. +-commutative N/A

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

   5. Simplified 99.4%

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

   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 67.3%

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

4. Final simplification 85.2%

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

Alternative 3: 83.7% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sinh y) y))
        (t_1 (* (sin x) t_0))
        (t_2
         (fma
          (* y y)
          (fma y (* y 0.008333333333333333) 0.16666666666666666)
          1.0)))
   (if (<= t_1 (- INFINITY))
     (* t_2 (fma x (* x (* x -0.16666666666666666)) x))
     (if (<= t_1 1.0) (* (sin x) t_2) (* x t_0)))))

C

double code(double x, double y) {
	double t_0 = sinh(y) / y;
	double t_1 = sin(x) * t_0;
	double t_2 = fma((y * y), fma(y, (y * 0.008333333333333333), 0.16666666666666666), 1.0);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_2 * fma(x, (x * (x * -0.16666666666666666)), x);
	} else if (t_1 <= 1.0) {
		tmp = sin(x) * t_2;
	} 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)
	t_2 = fma(Float64(y * y), fma(y, Float64(y * 0.008333333333333333), 0.16666666666666666), 1.0)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(t_2 * fma(x, Float64(x * Float64(x * -0.16666666666666666)), x));
	elseif (t_1 <= 1.0)
		tmp = Float64(sin(x) * t_2);
	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]}, Block[{t$95$2 = N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(t$95$2 * N[(x * N[(x * N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1.0], N[(N[Sin[x], $MachinePrecision] * t$95$2), $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

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

      1. distribute-rgt-in N/A

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

      2. *-rgt-identity N/A

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

      3. distribute-rgt-in N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out N/A

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

      7. +-commutative N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. distribute-lft-in N/A

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

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

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

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

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

      13. +-commutative N/A

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

   5. Simplified 69.3%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

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

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

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

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

      9. *-lowering-*.f64 54.5

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

   8. Simplified 54.5%

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

   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

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

      1. distribute-rgt-in N/A

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

      2. *-rgt-identity N/A

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

      3. distribute-rgt-in N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out N/A

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

      7. +-commutative N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. distribute-lft-in N/A

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

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

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

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

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

      13. +-commutative N/A

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

   5. Simplified 99.4%

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

   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 67.3%

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

4. Final simplification 81.4%

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

Alternative 4: 83.6% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sinh y) y)) (t_1 (* (sin x) t_0)))
   (if (<= t_1 (- INFINITY))
     (*
      (fma (* y y) (fma y (* y 0.008333333333333333) 0.16666666666666666) 1.0)
      (fma x (* x (* x -0.16666666666666666)) x))
     (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 = fma((y * y), fma(y, (y * 0.008333333333333333), 0.16666666666666666), 1.0) * fma(x, (x * (x * -0.16666666666666666)), x);
	} 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(fma(Float64(y * y), fma(y, Float64(y * 0.008333333333333333), 0.16666666666666666), 1.0) * fma(x, Float64(x * Float64(x * -0.16666666666666666)), x));
	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 * N[(y * 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x * N[(x * N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + x), $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

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

      1. distribute-rgt-in N/A

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

      2. *-rgt-identity N/A

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

      3. distribute-rgt-in N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out N/A

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

      7. +-commutative N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. distribute-lft-in N/A

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

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

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

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

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

      13. +-commutative N/A

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

   5. Simplified 69.3%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

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

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

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

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

      9. *-lowering-*.f64 54.5

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

   8. Simplified 54.5%

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

   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

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

      1. *-lft-identity N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-in N/A

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

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

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

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

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

      6. +-commutative N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 99.1

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

   5. Simplified 99.1%

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

   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 67.3%

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

4. Final simplification 81.3%

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

Alternative 5: 83.4% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sinh y) y)) (t_1 (* (sin x) t_0)))
   (if (<= t_1 (- INFINITY))
     (*
      (fma (* y y) (fma y (* y 0.008333333333333333) 0.16666666666666666) 1.0)
      (fma x (* x (* x -0.16666666666666666)) x))
     (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 = fma((y * y), fma(y, (y * 0.008333333333333333), 0.16666666666666666), 1.0) * fma(x, (x * (x * -0.16666666666666666)), x);
	} 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(fma(Float64(y * y), fma(y, Float64(y * 0.008333333333333333), 0.16666666666666666), 1.0) * fma(x, Float64(x * Float64(x * -0.16666666666666666)), x));
	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 * N[(y * 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x * N[(x * N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + x), $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

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

      1. distribute-rgt-in N/A

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

      2. *-rgt-identity N/A

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

      3. distribute-rgt-in N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out N/A

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

      7. +-commutative N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. distribute-lft-in N/A

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

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

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

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

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

      13. +-commutative N/A

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

   5. Simplified 69.3%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

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

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

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

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

      9. *-lowering-*.f64 54.5

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

   8. Simplified 54.5%

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

   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 98.3

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

   5. Simplified 98.3%

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

   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 67.3%

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

4. Final simplification 80.9%

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

Alternative 6: 81.1% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sin x) (/ (sinh y) y))))
   (if (<= t_0 (- INFINITY))
     (*
      (fma (* y y) (fma y (* y 0.008333333333333333) 0.16666666666666666) 1.0)
      (fma x (* x (* x -0.16666666666666666)) x))
     (if (<= t_0 1.0)
       (sin x)
       (*
        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 t_0 = sin(x) * (sinh(y) / y);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma((y * y), fma(y, (y * 0.008333333333333333), 0.16666666666666666), 1.0) * fma(x, (x * (x * -0.16666666666666666)), x);
	} else if (t_0 <= 1.0) {
		tmp = sin(x);
	} 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)
	t_0 = Float64(sin(x) * Float64(sinh(y) / y))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(fma(Float64(y * y), fma(y, Float64(y * 0.008333333333333333), 0.16666666666666666), 1.0) * fma(x, Float64(x * Float64(x * -0.16666666666666666)), x));
	elseif (t_0 <= 1.0)
		tmp = sin(x);
	else
		tmp = Float64(x * fma(Float64(y * y), fma(y, Float64(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 * N[(y * 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x * N[(x * N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1.0], N[Sin[x], $MachinePrecision], N[(x * N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * N[(y * N[(y * 0.0001984126984126984), $MachinePrecision] + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + 1.0), $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

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

      1. distribute-rgt-in N/A

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

      2. *-rgt-identity N/A

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

      3. distribute-rgt-in N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out N/A

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

      7. +-commutative N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. distribute-lft-in N/A

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

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

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

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

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

      13. +-commutative N/A

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

   5. Simplified 69.3%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

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

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

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

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

      9. *-lowering-*.f64 54.5

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

   8. Simplified 54.5%

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

   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 98.3

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

   5. Simplified 98.3%

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

   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 67.3%

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

   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. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. pow-plus N/A

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

      6. metadata-eval N/A

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

      7. cube-unmult N/A

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

      8. unpow2 N/A

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

      9. *-lft-identity N/A

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

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

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

   8. Simplified 61.7%

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

      1. clear-num N/A

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

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

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

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

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

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

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

   10. Applied egg-rr 61.7%

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

   11. Taylor expanded in y around 0

       \[\leadsto x \cdot \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)} \]
   12. 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. unpow2 N/A

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

       12. associate-*r* N/A

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

       13. *-commutative N/A

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

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

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

       15. *-commutative N/A

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

       16. *-lowering-*.f64 61.7

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

   13. Simplified 61.7%

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

4. Final simplification 79.7%

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

Alternative 7: 45.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin x \cdot \frac{\sinh y}{y}\\ t_1 := y \cdot \left(x \cdot y\right)\\ \mathbf{if}\;t\_0 \leq -0.1:\\ \;\;\;\;x \cdot \left(\left(x \cdot x\right) \cdot \mathsf{fma}\left(y \cdot y, -0.027777777777777776, -0.16666666666666666\right)\right)\\ \mathbf{elif}\;t\_0 \leq 0.98:\\ \;\;\;\;\mathsf{fma}\left(0.16666666666666666, t\_1, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y \cdot y\right) \cdot 0.008333333333333333\right) \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sin x) (/ (sinh y) y))) (t_1 (* y (* x y))))
   (if (<= t_0 -0.1)
     (* x (* (* x x) (fma (* y y) -0.027777777777777776 -0.16666666666666666)))
     (if (<= t_0 0.98)
       (fma 0.16666666666666666 t_1 x)
       (* (* (* y y) 0.008333333333333333) t_1)))))

C

double code(double x, double y) {
	double t_0 = sin(x) * (sinh(y) / y);
	double t_1 = y * (x * y);
	double tmp;
	if (t_0 <= -0.1) {
		tmp = x * ((x * x) * fma((y * y), -0.027777777777777776, -0.16666666666666666));
	} else if (t_0 <= 0.98) {
		tmp = fma(0.16666666666666666, t_1, x);
	} else {
		tmp = ((y * y) * 0.008333333333333333) * t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(x) * Float64(sinh(y) / y))
	t_1 = Float64(y * Float64(x * y))
	tmp = 0.0
	if (t_0 <= -0.1)
		tmp = Float64(x * Float64(Float64(x * x) * fma(Float64(y * y), -0.027777777777777776, -0.16666666666666666)));
	elseif (t_0 <= 0.98)
		tmp = fma(0.16666666666666666, t_1, x);
	else
		tmp = Float64(Float64(Float64(y * y) * 0.008333333333333333) * t_1);
	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]}, Block[{t$95$1 = N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.1], N[(x * N[(N[(x * x), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * -0.027777777777777776 + -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.98], N[(0.16666666666666666 * t$95$1 + x), $MachinePrecision], N[(N[(N[(y * y), $MachinePrecision] * 0.008333333333333333), $MachinePrecision] * t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 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

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

      1. *-lft-identity N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-in N/A

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

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

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

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

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

      6. +-commutative N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 70.2

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

   5. Simplified 70.2%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

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

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt1-in N/A

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

      10. +-commutative N/A

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

   8. Simplified 33.4%

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

   9. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

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

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

       5. unpow2 N/A

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

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

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

       7. +-commutative N/A

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

       8. distribute-rgt-in N/A

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

       9. metadata-eval N/A

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

       10. *-commutative N/A

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

       11. associate-*r* N/A

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

       12. *-commutative N/A

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

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

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

       14. unpow2 N/A

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

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

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

       16. metadata-eval 18.2

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

   11. Simplified 18.2%

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

   if -0.10000000000000001 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < 0.97999999999999998

   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 67.9%

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

   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. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. pow-plus N/A

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

      6. metadata-eval N/A

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

      7. cube-unmult N/A

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

      8. unpow2 N/A

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

      9. *-lft-identity N/A

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

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

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

   8. Simplified 67.1%

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

      1. clear-num N/A

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

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

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

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

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

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

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

   10. Applied egg-rr 67.1%

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

   11. Taylor expanded in y around 0

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

       1. associate-*r* N/A

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

       2. +-commutative N/A

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

       3. associate-*r* N/A

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

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

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

       5. unpow2 N/A

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

       6. associate-*r* N/A

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

       7. *-commutative N/A

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

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

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

       9. *-commutative N/A

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

       10. *-lowering-*.f64 66.9

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

   13. Simplified 66.9%

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

   if 0.97999999999999998 < (*.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 66.1%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

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

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

      3. unpow2 N/A

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

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

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt-in N/A

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

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

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 46.3

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

   8. Simplified 46.3%

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

      1. distribute-lft-in N/A

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

      2. associate-*l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

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

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

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

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

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

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

      8. *-lowering-*.f64 46.3

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

   10. Applied egg-rr 46.3%

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

   11. Taylor expanded in y around inf

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

       1. associate-*r* N/A

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

       2. metadata-eval N/A

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

       3. pow-sqr N/A

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

       4. associate-*l* N/A

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

       5. associate-*r* N/A

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

       6. *-commutative N/A

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

       7. associate-*r* N/A

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

       8. *-commutative N/A

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

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

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

       10. *-commutative N/A

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

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

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

       12. unpow2 N/A

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

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

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

       14. unpow2 N/A

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

       15. associate-*r* N/A

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

       16. *-commutative N/A

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

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

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

       18. *-commutative N/A

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

       19. *-lowering-*.f64 46.3

           \[\leadsto \left(\left(y \cdot y\right) \cdot 0.008333333333333333\right) \cdot \left(y \cdot \color{blue}{\left(y \cdot x\right)}\right) \]

   13. Simplified 46.3%

       \[\leadsto \color{blue}{\left(\left(y \cdot y\right) \cdot 0.008333333333333333\right) \cdot \left(y \cdot \left(y \cdot x\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 43.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin x \cdot \frac{\sinh y}{y} \leq -0.1:\\ \;\;\;\;x \cdot \left(\left(x \cdot x\right) \cdot \mathsf{fma}\left(y \cdot y, -0.027777777777777776, -0.16666666666666666\right)\right)\\ \mathbf{elif}\;\sin x \cdot \frac{\sinh y}{y} \leq 0.98:\\ \;\;\;\;\mathsf{fma}\left(0.16666666666666666, y \cdot \left(x \cdot y\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y \cdot y\right) \cdot 0.008333333333333333\right) \cdot \left(y \cdot \left(x \cdot y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 44.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin x \cdot \frac{\sinh y}{y}\\ t_1 := y \cdot \left(x \cdot y\right)\\ \mathbf{if}\;t\_0 \leq -0.1:\\ \;\;\;\;\left(y \cdot y\right) \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot -0.027777777777777776\right)\\ \mathbf{elif}\;t\_0 \leq 0.98:\\ \;\;\;\;\mathsf{fma}\left(0.16666666666666666, t\_1, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y \cdot y\right) \cdot 0.008333333333333333\right) \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sin x) (/ (sinh y) y))) (t_1 (* y (* x y))))
   (if (<= t_0 -0.1)
     (* (* y y) (* (* x (* x x)) -0.027777777777777776))
     (if (<= t_0 0.98)
       (fma 0.16666666666666666 t_1 x)
       (* (* (* y y) 0.008333333333333333) t_1)))))

C

double code(double x, double y) {
	double t_0 = sin(x) * (sinh(y) / y);
	double t_1 = y * (x * y);
	double tmp;
	if (t_0 <= -0.1) {
		tmp = (y * y) * ((x * (x * x)) * -0.027777777777777776);
	} else if (t_0 <= 0.98) {
		tmp = fma(0.16666666666666666, t_1, x);
	} else {
		tmp = ((y * y) * 0.008333333333333333) * t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(x) * Float64(sinh(y) / y))
	t_1 = Float64(y * Float64(x * y))
	tmp = 0.0
	if (t_0 <= -0.1)
		tmp = Float64(Float64(y * y) * Float64(Float64(x * Float64(x * x)) * -0.027777777777777776));
	elseif (t_0 <= 0.98)
		tmp = fma(0.16666666666666666, t_1, x);
	else
		tmp = Float64(Float64(Float64(y * y) * 0.008333333333333333) * t_1);
	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]}, Block[{t$95$1 = N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.1], N[(N[(y * y), $MachinePrecision] * N[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] * -0.027777777777777776), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.98], N[(0.16666666666666666 * t$95$1 + x), $MachinePrecision], N[(N[(N[(y * y), $MachinePrecision] * 0.008333333333333333), $MachinePrecision] * t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 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

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

      1. *-lft-identity N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-in N/A

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

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

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

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

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

      6. +-commutative N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 70.2

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

   5. Simplified 70.2%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

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

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt1-in N/A

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

      10. +-commutative N/A

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

   8. Simplified 33.4%

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

   9. Taylor expanded in y around inf

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. associate-*r* N/A

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

       6. distribute-lft-in N/A

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

       7. *-rgt-identity N/A

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

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

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

       9. unpow2 N/A

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

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

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

       11. *-rgt-identity N/A

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

       12. distribute-lft-in N/A

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

       13. *-commutative N/A

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

       14. associate-*l* N/A

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

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

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

       16. +-commutative N/A

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

       17. distribute-rgt-in N/A

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

       18. metadata-eval N/A

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

   11. Simplified 33.3%

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

   12. Taylor expanded in x around inf

       \[\leadsto \color{blue}{\frac{-1}{36} \cdot \left({x}^{3} \cdot {y}^{2}\right)} \]
   13. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{-1}{36} \cdot {x}^{3}\right) \cdot {y}^{2}} \]

       2. *-commutative N/A

          \[\leadsto \color{blue}{{y}^{2} \cdot \left(\frac{-1}{36} \cdot {x}^{3}\right)} \]

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

          \[\leadsto \color{blue}{{y}^{2} \cdot \left(\frac{-1}{36} \cdot {x}^{3}\right)} \]

       4. unpow2 N/A

          \[\leadsto \color{blue}{\left(y \cdot y\right)} \cdot \left(\frac{-1}{36} \cdot {x}^{3}\right) \]

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

          \[\leadsto \color{blue}{\left(y \cdot y\right)} \cdot \left(\frac{-1}{36} \cdot {x}^{3}\right) \]

       6. *-commutative N/A

          \[\leadsto \left(y \cdot y\right) \cdot \color{blue}{\left({x}^{3} \cdot \frac{-1}{36}\right)} \]

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

          \[\leadsto \left(y \cdot y\right) \cdot \color{blue}{\left({x}^{3} \cdot \frac{-1}{36}\right)} \]

       8. cube-mult N/A

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

       9. unpow2 N/A

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

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

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

       11. unpow2 N/A

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

       12. *-lowering-*.f64 18.1

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

   14. Simplified 18.1%

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

   if -0.10000000000000001 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < 0.97999999999999998

   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 67.9%

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

   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. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. pow-plus N/A

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

      6. metadata-eval N/A

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

      7. cube-unmult N/A

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

      8. unpow2 N/A

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

      9. *-lft-identity N/A

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

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

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

   8. Simplified 67.1%

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

      1. clear-num N/A

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

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

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

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

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

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

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

   10. Applied egg-rr 67.1%

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

   11. Taylor expanded in y around 0

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

       1. associate-*r* N/A

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

       2. +-commutative N/A

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

       3. associate-*r* N/A

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

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

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

       5. unpow2 N/A

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

       6. associate-*r* N/A

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

       7. *-commutative N/A

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

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

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

       9. *-commutative N/A

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

       10. *-lowering-*.f64 66.9

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

   13. Simplified 66.9%

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

   if 0.97999999999999998 < (*.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 66.1%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

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

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

      3. unpow2 N/A

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

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

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt-in N/A

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

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

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 46.3

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

   8. Simplified 46.3%

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

      1. distribute-lft-in N/A

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

      2. associate-*l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

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

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

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

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

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

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

      8. *-lowering-*.f64 46.3

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

   10. Applied egg-rr 46.3%

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

   11. Taylor expanded in y around inf

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

       1. associate-*r* N/A

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

       2. metadata-eval N/A

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

       3. pow-sqr N/A

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

       4. associate-*l* N/A

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

       5. associate-*r* N/A

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

       6. *-commutative N/A

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

       7. associate-*r* N/A

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

       8. *-commutative N/A

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

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

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

       10. *-commutative N/A

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

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

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

       12. unpow2 N/A

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

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

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

       14. unpow2 N/A

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

       15. associate-*r* N/A

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

       16. *-commutative N/A

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

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

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

       18. *-commutative N/A

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

       19. *-lowering-*.f64 46.3

           \[\leadsto \left(\left(y \cdot y\right) \cdot 0.008333333333333333\right) \cdot \left(y \cdot \color{blue}{\left(y \cdot x\right)}\right) \]

   13. Simplified 46.3%

       \[\leadsto \color{blue}{\left(\left(y \cdot y\right) \cdot 0.008333333333333333\right) \cdot \left(y \cdot \left(y \cdot x\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 43.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin x \cdot \frac{\sinh y}{y} \leq -0.1:\\ \;\;\;\;\left(y \cdot y\right) \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot -0.027777777777777776\right)\\ \mathbf{elif}\;\sin x \cdot \frac{\sinh y}{y} \leq 0.98:\\ \;\;\;\;\mathsf{fma}\left(0.16666666666666666, y \cdot \left(x \cdot y\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y \cdot y\right) \cdot 0.008333333333333333\right) \cdot \left(y \cdot \left(x \cdot y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 41.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin x \cdot \frac{\sinh y}{y}\\ \mathbf{if}\;t\_0 \leq -0.1:\\ \;\;\;\;-0.16666666666666666 \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \mathbf{elif}\;t\_0 \leq 0.98:\\ \;\;\;\;\mathsf{fma}\left(0.16666666666666666, y \cdot \left(x \cdot y\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot y\right) \cdot \left(x \cdot 0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sin x) (/ (sinh y) y))))
   (if (<= t_0 -0.1)
     (* -0.16666666666666666 (* x (* x x)))
     (if (<= t_0 0.98)
       (fma 0.16666666666666666 (* y (* x y)) x)
       (* (* y y) (* x 0.16666666666666666))))))

C

double code(double x, double y) {
	double t_0 = sin(x) * (sinh(y) / y);
	double tmp;
	if (t_0 <= -0.1) {
		tmp = -0.16666666666666666 * (x * (x * x));
	} else if (t_0 <= 0.98) {
		tmp = fma(0.16666666666666666, (y * (x * y)), x);
	} else {
		tmp = (y * y) * (x * 0.16666666666666666);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(x) * Float64(sinh(y) / y))
	tmp = 0.0
	if (t_0 <= -0.1)
		tmp = Float64(-0.16666666666666666 * Float64(x * Float64(x * x)));
	elseif (t_0 <= 0.98)
		tmp = fma(0.16666666666666666, Float64(y * Float64(x * y)), x);
	else
		tmp = Float64(Float64(y * y) * Float64(x * 0.16666666666666666));
	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, -0.1], N[(-0.16666666666666666 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.98], N[(0.16666666666666666 * N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(N[(y * y), $MachinePrecision] * N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 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

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

      1. sin-lowering-sin.f64 36.3

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

   5. Simplified 36.3%

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

   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. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

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

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

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

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

      9. *-lowering-*.f64 17.7

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

   8. Simplified 17.7%

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

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{-1}{6} \cdot {x}^{3}} \]
   10. Step-by-step derivation

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

          \[\leadsto \color{blue}{\frac{-1}{6} \cdot {x}^{3}} \]

       2. cube-mult N/A

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

       3. unpow2 N/A

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

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

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

       5. unpow2 N/A

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

       6. *-lowering-*.f64 17.4

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

   11. Simplified 17.4%

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

   if -0.10000000000000001 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < 0.97999999999999998

   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 67.9%

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

   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. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. pow-plus N/A

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

      6. metadata-eval N/A

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

      7. cube-unmult N/A

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

      8. unpow2 N/A

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

      9. *-lft-identity N/A

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

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

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

   8. Simplified 67.1%

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

      1. clear-num N/A

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

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

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

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

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

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

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

   10. Applied egg-rr 67.1%

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

   11. Taylor expanded in y around 0

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

       1. associate-*r* N/A

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

       2. +-commutative N/A

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

       3. associate-*r* N/A

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

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

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

       5. unpow2 N/A

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

       6. associate-*r* N/A

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

       7. *-commutative N/A

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

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

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

       9. *-commutative N/A

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

       10. *-lowering-*.f64 66.9

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

   13. Simplified 66.9%

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

   if 0.97999999999999998 < (*.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 y around 0

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

      1. *-lft-identity N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-in N/A

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

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

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

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

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

      6. +-commutative N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 58.2

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

   5. Simplified 58.2%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

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

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt1-in N/A

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

      10. +-commutative N/A

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

   8. Simplified 53.7%

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

   9. Taylor expanded in y around inf

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. associate-*r* N/A

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

       6. distribute-lft-in N/A

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

       7. *-rgt-identity N/A

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

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

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

       9. unpow2 N/A

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

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

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

       11. *-rgt-identity N/A

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

       12. distribute-lft-in N/A

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

       13. *-commutative N/A

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

       14. associate-*l* N/A

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

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

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

       16. +-commutative N/A

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

       17. distribute-rgt-in N/A

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

       18. metadata-eval N/A

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

   11. Simplified 53.7%

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

   12. Taylor expanded in x around 0

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

   14. Simplified 38.9%

       \[\leadsto \left(y \cdot y\right) \cdot \left(x \cdot \color{blue}{0.16666666666666666}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 41.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin x \cdot \frac{\sinh y}{y} \leq -0.1:\\ \;\;\;\;-0.16666666666666666 \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \mathbf{elif}\;\sin x \cdot \frac{\sinh y}{y} \leq 0.98:\\ \;\;\;\;\mathsf{fma}\left(0.16666666666666666, y \cdot \left(x \cdot y\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot y\right) \cdot \left(x \cdot 0.16666666666666666\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 41.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin x \cdot \frac{\sinh y}{y}\\ \mathbf{if}\;t\_0 \leq -0.1:\\ \;\;\;\;-0.16666666666666666 \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \mathbf{elif}\;t\_0 \leq 0.98:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot y\right) \cdot \left(x \cdot 0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sin x) (/ (sinh y) y))))
   (if (<= t_0 -0.1)
     (* -0.16666666666666666 (* x (* x x)))
     (if (<= t_0 0.98) x (* (* y y) (* x 0.16666666666666666))))))

C

double code(double x, double y) {
	double t_0 = sin(x) * (sinh(y) / y);
	double tmp;
	if (t_0 <= -0.1) {
		tmp = -0.16666666666666666 * (x * (x * x));
	} else if (t_0 <= 0.98) {
		tmp = x;
	} else {
		tmp = (y * y) * (x * 0.16666666666666666);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin(x) * (sinh(y) / y)
    if (t_0 <= (-0.1d0)) then
        tmp = (-0.16666666666666666d0) * (x * (x * x))
    else if (t_0 <= 0.98d0) then
        tmp = x
    else
        tmp = (y * y) * (x * 0.16666666666666666d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sin(x) * (Math.sinh(y) / y);
	double tmp;
	if (t_0 <= -0.1) {
		tmp = -0.16666666666666666 * (x * (x * x));
	} else if (t_0 <= 0.98) {
		tmp = x;
	} else {
		tmp = (y * y) * (x * 0.16666666666666666);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sin(x) * (math.sinh(y) / y)
	tmp = 0
	if t_0 <= -0.1:
		tmp = -0.16666666666666666 * (x * (x * x))
	elif t_0 <= 0.98:
		tmp = x
	else:
		tmp = (y * y) * (x * 0.16666666666666666)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sin(x) * Float64(sinh(y) / y))
	tmp = 0.0
	if (t_0 <= -0.1)
		tmp = Float64(-0.16666666666666666 * Float64(x * Float64(x * x)));
	elseif (t_0 <= 0.98)
		tmp = x;
	else
		tmp = Float64(Float64(y * y) * Float64(x * 0.16666666666666666));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sin(x) * (sinh(y) / y);
	tmp = 0.0;
	if (t_0 <= -0.1)
		tmp = -0.16666666666666666 * (x * (x * x));
	elseif (t_0 <= 0.98)
		tmp = x;
	else
		tmp = (y * y) * (x * 0.16666666666666666);
	end
	tmp_2 = 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, -0.1], N[(-0.16666666666666666 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.98], x, N[(N[(y * y), $MachinePrecision] * N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 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

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

      1. sin-lowering-sin.f64 36.3

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

   5. Simplified 36.3%

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

   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. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

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

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

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

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

      9. *-lowering-*.f64 17.7

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

   8. Simplified 17.7%

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

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{-1}{6} \cdot {x}^{3}} \]
   10. Step-by-step derivation

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

          \[\leadsto \color{blue}{\frac{-1}{6} \cdot {x}^{3}} \]

       2. cube-mult N/A

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

       3. unpow2 N/A

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

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

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

       5. unpow2 N/A

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

       6. *-lowering-*.f64 17.4

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

   11. Simplified 17.4%

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

   if -0.10000000000000001 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < 0.97999999999999998

   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 98.2

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

   5. Simplified 98.2%

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

   6. Taylor expanded in x around 0

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

   8. Simplified 66.1%

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

   if 0.97999999999999998 < (*.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 y around 0

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

      1. *-lft-identity N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-in N/A

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

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

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

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

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

      6. +-commutative N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 58.2

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

   5. Simplified 58.2%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

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

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt1-in N/A

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

      10. +-commutative N/A

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

   8. Simplified 53.7%

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

   9. Taylor expanded in y around inf

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. associate-*r* N/A

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

       6. distribute-lft-in N/A

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

       7. *-rgt-identity N/A

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

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

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

       9. unpow2 N/A

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

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

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

       11. *-rgt-identity N/A

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

       12. distribute-lft-in N/A

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

       13. *-commutative N/A

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

       14. associate-*l* N/A

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

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

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

       16. +-commutative N/A

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

       17. distribute-rgt-in N/A

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

       18. metadata-eval N/A

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

   11. Simplified 53.7%

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

   12. Taylor expanded in x around 0

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

   14. Simplified 38.9%

       \[\leadsto \left(y \cdot y\right) \cdot \left(x \cdot \color{blue}{0.16666666666666666}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 38.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin x \cdot \frac{\sinh y}{y}\\ \mathbf{if}\;t\_0 \leq -0.1:\\ \;\;\;\;-0.16666666666666666 \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \mathbf{elif}\;t\_0 \leq 0.98:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot \left(y \cdot \left(x \cdot y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sin x) (/ (sinh y) y))))
   (if (<= t_0 -0.1)
     (* -0.16666666666666666 (* x (* x x)))
     (if (<= t_0 0.98) x (* 0.16666666666666666 (* y (* x y)))))))

C

double code(double x, double y) {
	double t_0 = sin(x) * (sinh(y) / y);
	double tmp;
	if (t_0 <= -0.1) {
		tmp = -0.16666666666666666 * (x * (x * x));
	} else if (t_0 <= 0.98) {
		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) :: t_0
    real(8) :: tmp
    t_0 = sin(x) * (sinh(y) / y)
    if (t_0 <= (-0.1d0)) then
        tmp = (-0.16666666666666666d0) * (x * (x * x))
    else if (t_0 <= 0.98d0) 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 t_0 = Math.sin(x) * (Math.sinh(y) / y);
	double tmp;
	if (t_0 <= -0.1) {
		tmp = -0.16666666666666666 * (x * (x * x));
	} else if (t_0 <= 0.98) {
		tmp = x;
	} else {
		tmp = 0.16666666666666666 * (y * (x * y));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sin(x) * (math.sinh(y) / y)
	tmp = 0
	if t_0 <= -0.1:
		tmp = -0.16666666666666666 * (x * (x * x))
	elif t_0 <= 0.98:
		tmp = x
	else:
		tmp = 0.16666666666666666 * (y * (x * y))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sin(x) * Float64(sinh(y) / y))
	tmp = 0.0
	if (t_0 <= -0.1)
		tmp = Float64(-0.16666666666666666 * Float64(x * Float64(x * x)));
	elseif (t_0 <= 0.98)
		tmp = x;
	else
		tmp = Float64(0.16666666666666666 * Float64(y * Float64(x * y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sin(x) * (sinh(y) / y);
	tmp = 0.0;
	if (t_0 <= -0.1)
		tmp = -0.16666666666666666 * (x * (x * x));
	elseif (t_0 <= 0.98)
		tmp = x;
	else
		tmp = 0.16666666666666666 * (y * (x * y));
	end
	tmp_2 = 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, -0.1], N[(-0.16666666666666666 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.98], x, N[(0.16666666666666666 * N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 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

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

      1. sin-lowering-sin.f64 36.3

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

   5. Simplified 36.3%

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

   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. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

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

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

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

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

      9. *-lowering-*.f64 17.7

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

   8. Simplified 17.7%

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

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{-1}{6} \cdot {x}^{3}} \]
   10. Step-by-step derivation

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

          \[\leadsto \color{blue}{\frac{-1}{6} \cdot {x}^{3}} \]

       2. cube-mult N/A

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

       3. unpow2 N/A

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

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

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

       5. unpow2 N/A

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

       6. *-lowering-*.f64 17.4

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

   11. Simplified 17.4%

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

   if -0.10000000000000001 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < 0.97999999999999998

   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 98.2

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

   5. Simplified 98.2%

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

   6. Taylor expanded in x around 0

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

   8. Simplified 66.1%

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

   if 0.97999999999999998 < (*.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 y around 0

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

      1. *-lft-identity N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-in N/A

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

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

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

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

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

      6. +-commutative N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 58.2

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

   5. Simplified 58.2%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

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

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt1-in N/A

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

      10. +-commutative N/A

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

   8. Simplified 53.7%

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

   9. Taylor expanded in y around inf

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. associate-*r* N/A

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

       6. distribute-lft-in N/A

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

       7. *-rgt-identity N/A

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

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

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

       9. unpow2 N/A

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

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

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

       11. *-rgt-identity N/A

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

       12. distribute-lft-in N/A

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

       13. *-commutative N/A

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

       14. associate-*l* N/A

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

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

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

       16. +-commutative N/A

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

       17. distribute-rgt-in N/A

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

       18. metadata-eval N/A

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

   11. Simplified 53.7%

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

   12. Taylor expanded in x around 0

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

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

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

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

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

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

       6. *-commutative N/A

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

       7. *-lowering-*.f64 30.1

          \[\leadsto 0.16666666666666666 \cdot \left(y \cdot \color{blue}{\left(y \cdot x\right)}\right) \]

   14. Simplified 30.1%

       \[\leadsto \color{blue}{0.16666666666666666 \cdot \left(y \cdot \left(y \cdot x\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 39.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin x \cdot \frac{\sinh y}{y} \leq -0.1:\\ \;\;\;\;-0.16666666666666666 \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \mathbf{elif}\;\sin x \cdot \frac{\sinh y}{y} \leq 0.98:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot \left(y \cdot \left(x \cdot y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 57.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin x \cdot \frac{\sinh y}{y} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right) \cdot \mathsf{fma}\left(x, x \cdot \left(x \cdot -0.16666666666666666\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (sin x) (/ (sinh y) y)) 5e-6)
   (*
    (fma (* y y) (fma y (* y 0.008333333333333333) 0.16666666666666666) 1.0)
    (fma x (* x (* x -0.16666666666666666)) x))
   (*
    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)) <= 5e-6) {
		tmp = fma((y * y), fma(y, (y * 0.008333333333333333), 0.16666666666666666), 1.0) * fma(x, (x * (x * -0.16666666666666666)), x);
	} 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)) <= 5e-6)
		tmp = Float64(fma(Float64(y * y), fma(y, Float64(y * 0.008333333333333333), 0.16666666666666666), 1.0) * fma(x, Float64(x * Float64(x * -0.16666666666666666)), x));
	else
		tmp = Float64(x * fma(Float64(y * y), fma(y, Float64(y * fma(y, Float64(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], 5e-6], N[(N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x * N[(x * N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * N[(y * N[(y * 0.0001984126984126984), $MachinePrecision] + 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)) < 5.00000000000000041e-6

   1. Initial program 100.0%

      \[\sin x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\sin x + {y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{6} \cdot \sin x\right)} \]
   4. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \sin x + \color{blue}{\left(\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right)\right) \cdot {y}^{2} + \left(\frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}\right)} \]

      2. *-rgt-identity N/A

         \[\leadsto \color{blue}{\sin x \cdot 1} + \left(\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right)\right) \cdot {y}^{2} + \left(\frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \sin x \cdot 1 + \color{blue}{{y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{6} \cdot \sin x\right)} \]

      4. *-commutative N/A

         \[\leadsto \sin x \cdot 1 + \color{blue}{\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}} \]

      5. associate-*r* N/A

         \[\leadsto \sin x \cdot 1 + \left(\color{blue}{\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot \sin x} + \frac{1}{6} \cdot \sin x\right) \cdot {y}^{2} \]

      6. distribute-rgt-out N/A

         \[\leadsto \sin x \cdot 1 + \color{blue}{\left(\sin x \cdot \left(\frac{1}{120} \cdot {y}^{2} + \frac{1}{6}\right)\right)} \cdot {y}^{2} \]

      7. +-commutative N/A

         \[\leadsto \sin x \cdot 1 + \left(\sin x \cdot \color{blue}{\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)}\right) \cdot {y}^{2} \]

      8. associate-*l* N/A

         \[\leadsto \sin x \cdot 1 + \color{blue}{\sin x \cdot \left(\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) \cdot {y}^{2}\right)} \]

      9. *-commutative N/A

         \[\leadsto \sin x \cdot 1 + \sin x \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} \]

      10. distribute-lft-in N/A

          \[\leadsto \color{blue}{\sin x \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\sin x \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} \]

      12. sin-lowering-sin.f64 N/A

          \[\leadsto \color{blue}{\sin x} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \sin x \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) + 1\right)} \]

   5. Simplified 87.6%

      \[\leadsto \color{blue}{\sin x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)} \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(x \cdot \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2} + 1\right)}\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{\left(x \cdot \left(\frac{-1}{6} \cdot {x}^{2}\right) + x \cdot 1\right)} \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      3. *-rgt-identity N/A

         \[\leadsto \left(x \cdot \left(\frac{-1}{6} \cdot {x}^{2}\right) + \color{blue}{x}\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{-1}{6} \cdot {x}^{2}, x\right)} \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{{x}^{2} \cdot \frac{-1}{6}}, x\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\left(x \cdot x\right)} \cdot \frac{-1}{6}, x\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot \left(x \cdot \frac{-1}{6}\right)}, x\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot \left(x \cdot \frac{-1}{6}\right)}, x\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      9. *-lowering-*.f64 61.8

         \[\leadsto \mathsf{fma}\left(x, x \cdot \color{blue}{\left(x \cdot -0.16666666666666666\right)}, x\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right) \]

   8. Simplified 61.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot \left(x \cdot -0.16666666666666666\right), x\right)} \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right) \]

   if 5.00000000000000041e-6 < (*.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 41.7%

      \[\leadsto \color{blue}{x} \cdot \frac{\sinh y}{y} \]

   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. +-commutative N/A

         \[\leadsto x \cdot \frac{y \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) + 1\right)}}{y} \]

      2. distribute-rgt-in N/A

         \[\leadsto x \cdot \frac{\color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right) \cdot y + 1 \cdot y}}{y} \]

      3. *-commutative N/A

         \[\leadsto x \cdot \frac{\color{blue}{\left(\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot {y}^{2}\right)} \cdot y + 1 \cdot y}{y} \]

      4. associate-*l* N/A

         \[\leadsto x \cdot \frac{\color{blue}{\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot \left({y}^{2} \cdot y\right)} + 1 \cdot y}{y} \]

      5. pow-plus N/A

         \[\leadsto x \cdot \frac{\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot \color{blue}{{y}^{\left(2 + 1\right)}} + 1 \cdot y}{y} \]

      6. metadata-eval N/A

         \[\leadsto x \cdot \frac{\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot {y}^{\color{blue}{3}} + 1 \cdot y}{y} \]

      7. cube-unmult N/A

         \[\leadsto x \cdot \frac{\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot \color{blue}{\left(y \cdot \left(y \cdot y\right)\right)} + 1 \cdot y}{y} \]

      8. unpow2 N/A

         \[\leadsto x \cdot \frac{\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot \left(y \cdot \color{blue}{{y}^{2}}\right) + 1 \cdot y}{y} \]

      9. *-lft-identity N/A

         \[\leadsto x \cdot \frac{\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot \left(y \cdot {y}^{2}\right) + \color{blue}{y}}{y} \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto x \cdot \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right), y \cdot {y}^{2}, y\right)}}{y} \]

   8. Simplified 38.4%

      \[\leadsto x \cdot \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)}}{y} \]
   9. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto x \cdot \color{blue}{\frac{1}{\frac{y}{\left(\left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{5040} + \frac{1}{120}\right) + \frac{1}{6}\right) \cdot \left(y \cdot \left(y \cdot y\right)\right) + y}}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto x \cdot \color{blue}{\frac{1}{\frac{y}{\left(\left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{5040} + \frac{1}{120}\right) + \frac{1}{6}\right) \cdot \left(y \cdot \left(y \cdot y\right)\right) + y}}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto x \cdot \frac{1}{\color{blue}{\frac{y}{\left(\left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{5040} + \frac{1}{120}\right) + \frac{1}{6}\right) \cdot \left(y \cdot \left(y \cdot y\right)\right) + y}}} \]

      4. associate-*r* N/A

         \[\leadsto x \cdot \frac{1}{\frac{y}{\color{blue}{\left(\left(\left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{5040} + \frac{1}{120}\right) + \frac{1}{6}\right) \cdot y\right) \cdot \left(y \cdot y\right)} + y}} \]

      5. *-commutative N/A

         \[\leadsto x \cdot \frac{1}{\frac{y}{\color{blue}{\left(y \cdot y\right) \cdot \left(\left(\left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{5040} + \frac{1}{120}\right) + \frac{1}{6}\right) \cdot y\right)} + y}} \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \frac{1}{\frac{y}{\color{blue}{\mathsf{fma}\left(y \cdot y, \left(\left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{5040} + \frac{1}{120}\right) + \frac{1}{6}\right) \cdot y, y\right)}}} \]

   10. Applied egg-rr 38.4%

       \[\leadsto x \cdot \color{blue}{\frac{1}{\frac{y}{\mathsf{fma}\left(y \cdot y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y\right)}}} \]

   11. Taylor expanded in y around 0

       \[\leadsto x \cdot \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)} \]
   12. 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. unpow2 N/A

           \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \left(\frac{1}{5040} \cdot \color{blue}{\left(y \cdot y\right)} + \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

       12. associate-*r* N/A

           \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \left(\color{blue}{\left(\frac{1}{5040} \cdot y\right) \cdot y} + \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

       13. *-commutative N/A

           \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \left(\color{blue}{y \cdot \left(\frac{1}{5040} \cdot y\right)} + \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

       14. accelerator-lowering-fma.f64 N/A

           \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \color{blue}{\mathsf{fma}\left(y, \frac{1}{5040} \cdot y, \frac{1}{120}\right)}, \frac{1}{6}\right), 1\right) \]

       15. *-commutative N/A

           \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, \color{blue}{y \cdot \frac{1}{5040}}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

       16. *-lowering-*.f64 38.4

           \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, \color{blue}{y \cdot 0.0001984126984126984}, 0.008333333333333333\right), 0.16666666666666666\right), 1\right) \]

   13. Simplified 38.4%

       \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 53.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin x \cdot \frac{\sinh y}{y} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right) \cdot \mathsf{fma}\left(x, x \cdot \left(x \cdot -0.16666666666666666\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 53.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin x \cdot \frac{\sinh y}{y} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;x \cdot \left(\mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right) \cdot \mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (sin x) (/ (sinh y) y)) 5e-6)
   (*
    x
    (*
     (fma 0.16666666666666666 (* y y) 1.0)
     (fma x (* x -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)) <= 5e-6) {
		tmp = x * (fma(0.16666666666666666, (y * y), 1.0) * fma(x, (x * -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)) <= 5e-6)
		tmp = Float64(x * Float64(fma(0.16666666666666666, Float64(y * y), 1.0) * fma(x, Float64(x * -0.16666666666666666), 1.0)));
	else
		tmp = Float64(x * fma(Float64(y * y), fma(y, Float64(y * fma(y, Float64(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], 5e-6], N[(x * N[(N[(0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x * N[(x * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * N[(y * N[(y * 0.0001984126984126984), $MachinePrecision] + 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)) < 5.00000000000000041e-6

   1. Initial program 100.0%

      \[\sin x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\sin x + \frac{1}{6} \cdot \left({y}^{2} \cdot \sin x\right)} \]
   4. Step-by-step derivation

      1. *-lft-identity N/A

         \[\leadsto \color{blue}{1 \cdot \sin x} + \frac{1}{6} \cdot \left({y}^{2} \cdot \sin x\right) \]

      2. associate-*r* N/A

         \[\leadsto 1 \cdot \sin x + \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \sin x} \]

      3. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\sin x \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\sin x \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)} \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \color{blue}{\sin x} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right) \]

      6. +-commutative N/A

         \[\leadsto \sin x \cdot \color{blue}{\left(\frac{1}{6} \cdot {y}^{2} + 1\right)} \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \sin x \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{6}, {y}^{2}, 1\right)} \]

      8. unpow2 N/A

         \[\leadsto \sin x \cdot \mathsf{fma}\left(\frac{1}{6}, \color{blue}{y \cdot y}, 1\right) \]

      9. *-lowering-*.f64 81.7

         \[\leadsto \sin x \cdot \mathsf{fma}\left(0.16666666666666666, \color{blue}{y \cdot y}, 1\right) \]

   5. Simplified 81.7%

      \[\leadsto \color{blue}{\sin x \cdot \mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + \left(\frac{-1}{6} \cdot \left({x}^{2} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right) + \frac{1}{6} \cdot {y}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \left(1 + \color{blue}{\left(\frac{1}{6} \cdot {y}^{2} + \frac{-1}{6} \cdot \left({x}^{2} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)\right)}\right) \]

      2. associate-+r+ N/A

         \[\leadsto x \cdot \color{blue}{\left(\left(1 + \frac{1}{6} \cdot {y}^{2}\right) + \frac{-1}{6} \cdot \left({x}^{2} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)\right)} \]

      3. associate-*r* N/A

         \[\leadsto x \cdot \left(\left(1 + \frac{1}{6} \cdot {y}^{2}\right) + \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2}\right) \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto x \cdot \left(\left(1 + \frac{1}{6} \cdot {y}^{2}\right) + \color{blue}{\left({x}^{2} \cdot \frac{-1}{6}\right)} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto x \cdot \left(\left(1 + \frac{1}{6} \cdot {y}^{2}\right) + \color{blue}{{x}^{2} \cdot \left(\frac{-1}{6} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)}\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(\left(1 + \frac{1}{6} \cdot {y}^{2}\right) + {x}^{2} \cdot \left(\frac{-1}{6} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)\right)} \]

      7. associate-*r* N/A

         \[\leadsto x \cdot \left(\left(1 + \frac{1}{6} \cdot {y}^{2}\right) + \color{blue}{\left({x}^{2} \cdot \frac{-1}{6}\right) \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)}\right) \]

      8. *-commutative N/A

         \[\leadsto x \cdot \left(\left(1 + \frac{1}{6} \cdot {y}^{2}\right) + \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2}\right)} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right) \]

      9. distribute-rgt1-in N/A

         \[\leadsto x \cdot \color{blue}{\left(\left(\frac{-1}{6} \cdot {x}^{2} + 1\right) \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)} \]

      10. +-commutative N/A

          \[\leadsto x \cdot \left(\color{blue}{\left(1 + \frac{-1}{6} \cdot {x}^{2}\right)} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right) \]

   8. Simplified 60.0%

      \[\leadsto \color{blue}{x \cdot \left(\mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right)\right)} \]

   if 5.00000000000000041e-6 < (*.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 41.7%

      \[\leadsto \color{blue}{x} \cdot \frac{\sinh y}{y} \]

   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. +-commutative N/A

         \[\leadsto x \cdot \frac{y \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) + 1\right)}}{y} \]

      2. distribute-rgt-in N/A

         \[\leadsto x \cdot \frac{\color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right) \cdot y + 1 \cdot y}}{y} \]

      3. *-commutative N/A

         \[\leadsto x \cdot \frac{\color{blue}{\left(\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot {y}^{2}\right)} \cdot y + 1 \cdot y}{y} \]

      4. associate-*l* N/A

         \[\leadsto x \cdot \frac{\color{blue}{\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot \left({y}^{2} \cdot y\right)} + 1 \cdot y}{y} \]

      5. pow-plus N/A

         \[\leadsto x \cdot \frac{\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot \color{blue}{{y}^{\left(2 + 1\right)}} + 1 \cdot y}{y} \]

      6. metadata-eval N/A

         \[\leadsto x \cdot \frac{\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot {y}^{\color{blue}{3}} + 1 \cdot y}{y} \]

      7. cube-unmult N/A

         \[\leadsto x \cdot \frac{\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot \color{blue}{\left(y \cdot \left(y \cdot y\right)\right)} + 1 \cdot y}{y} \]

      8. unpow2 N/A

         \[\leadsto x \cdot \frac{\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot \left(y \cdot \color{blue}{{y}^{2}}\right) + 1 \cdot y}{y} \]

      9. *-lft-identity N/A

         \[\leadsto x \cdot \frac{\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot \left(y \cdot {y}^{2}\right) + \color{blue}{y}}{y} \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto x \cdot \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right), y \cdot {y}^{2}, y\right)}}{y} \]

   8. Simplified 38.4%

      \[\leadsto x \cdot \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)}}{y} \]
   9. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto x \cdot \color{blue}{\frac{1}{\frac{y}{\left(\left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{5040} + \frac{1}{120}\right) + \frac{1}{6}\right) \cdot \left(y \cdot \left(y \cdot y\right)\right) + y}}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto x \cdot \color{blue}{\frac{1}{\frac{y}{\left(\left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{5040} + \frac{1}{120}\right) + \frac{1}{6}\right) \cdot \left(y \cdot \left(y \cdot y\right)\right) + y}}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto x \cdot \frac{1}{\color{blue}{\frac{y}{\left(\left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{5040} + \frac{1}{120}\right) + \frac{1}{6}\right) \cdot \left(y \cdot \left(y \cdot y\right)\right) + y}}} \]

      4. associate-*r* N/A

         \[\leadsto x \cdot \frac{1}{\frac{y}{\color{blue}{\left(\left(\left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{5040} + \frac{1}{120}\right) + \frac{1}{6}\right) \cdot y\right) \cdot \left(y \cdot y\right)} + y}} \]

      5. *-commutative N/A

         \[\leadsto x \cdot \frac{1}{\frac{y}{\color{blue}{\left(y \cdot y\right) \cdot \left(\left(\left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{5040} + \frac{1}{120}\right) + \frac{1}{6}\right) \cdot y\right)} + y}} \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \frac{1}{\frac{y}{\color{blue}{\mathsf{fma}\left(y \cdot y, \left(\left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{5040} + \frac{1}{120}\right) + \frac{1}{6}\right) \cdot y, y\right)}}} \]

   10. Applied egg-rr 38.4%

       \[\leadsto x \cdot \color{blue}{\frac{1}{\frac{y}{\mathsf{fma}\left(y \cdot y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y\right)}}} \]

   11. Taylor expanded in y around 0

       \[\leadsto x \cdot \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)} \]
   12. 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. unpow2 N/A

           \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \left(\frac{1}{5040} \cdot \color{blue}{\left(y \cdot y\right)} + \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

       12. associate-*r* N/A

           \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \left(\color{blue}{\left(\frac{1}{5040} \cdot y\right) \cdot y} + \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

       13. *-commutative N/A

           \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \left(\color{blue}{y \cdot \left(\frac{1}{5040} \cdot y\right)} + \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

       14. accelerator-lowering-fma.f64 N/A

           \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \color{blue}{\mathsf{fma}\left(y, \frac{1}{5040} \cdot y, \frac{1}{120}\right)}, \frac{1}{6}\right), 1\right) \]

       15. *-commutative N/A

           \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, \color{blue}{y \cdot \frac{1}{5040}}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

       16. *-lowering-*.f64 38.4

           \[\leadsto x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, \color{blue}{y \cdot 0.0001984126984126984}, 0.008333333333333333\right), 0.16666666666666666\right), 1\right) \]

   13. Simplified 38.4%

       \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 52.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin x \cdot \frac{\sinh y}{y} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;x \cdot \left(\mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right) \cdot \mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 52.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin x \cdot \frac{\sinh y}{y} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;x \cdot \left(\mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right) \cdot \mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (sin x) (/ (sinh y) y)) 5e-6)
   (*
    x
    (*
     (fma 0.16666666666666666 (* y y) 1.0)
     (fma x (* x -0.16666666666666666) 1.0)))
   (*
    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) * (sinh(y) / y)) <= 5e-6) {
		tmp = x * (fma(0.16666666666666666, (y * y), 1.0) * fma(x, (x * -0.16666666666666666), 1.0));
	} 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 (Float64(sin(x) * Float64(sinh(y) / y)) <= 5e-6)
		tmp = Float64(x * Float64(fma(0.16666666666666666, Float64(y * y), 1.0) * fma(x, Float64(x * -0.16666666666666666), 1.0)));
	else
		tmp = Float64(x * fma(Float64(y * y), fma(y, Float64(y * 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], 5e-6], N[(x * N[(N[(0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x * N[(x * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * 0.008333333333333333), $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)) < 5.00000000000000041e-6

   1. Initial program 100.0%

      \[\sin x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\sin x + \frac{1}{6} \cdot \left({y}^{2} \cdot \sin x\right)} \]
   4. Step-by-step derivation

      1. *-lft-identity N/A

         \[\leadsto \color{blue}{1 \cdot \sin x} + \frac{1}{6} \cdot \left({y}^{2} \cdot \sin x\right) \]

      2. associate-*r* N/A

         \[\leadsto 1 \cdot \sin x + \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \sin x} \]

      3. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\sin x \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\sin x \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)} \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \color{blue}{\sin x} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right) \]

      6. +-commutative N/A

         \[\leadsto \sin x \cdot \color{blue}{\left(\frac{1}{6} \cdot {y}^{2} + 1\right)} \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \sin x \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{6}, {y}^{2}, 1\right)} \]

      8. unpow2 N/A

         \[\leadsto \sin x \cdot \mathsf{fma}\left(\frac{1}{6}, \color{blue}{y \cdot y}, 1\right) \]

      9. *-lowering-*.f64 81.7

         \[\leadsto \sin x \cdot \mathsf{fma}\left(0.16666666666666666, \color{blue}{y \cdot y}, 1\right) \]

   5. Simplified 81.7%

      \[\leadsto \color{blue}{\sin x \cdot \mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + \left(\frac{-1}{6} \cdot \left({x}^{2} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right) + \frac{1}{6} \cdot {y}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \left(1 + \color{blue}{\left(\frac{1}{6} \cdot {y}^{2} + \frac{-1}{6} \cdot \left({x}^{2} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)\right)}\right) \]

      2. associate-+r+ N/A

         \[\leadsto x \cdot \color{blue}{\left(\left(1 + \frac{1}{6} \cdot {y}^{2}\right) + \frac{-1}{6} \cdot \left({x}^{2} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)\right)} \]

      3. associate-*r* N/A

         \[\leadsto x \cdot \left(\left(1 + \frac{1}{6} \cdot {y}^{2}\right) + \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2}\right) \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto x \cdot \left(\left(1 + \frac{1}{6} \cdot {y}^{2}\right) + \color{blue}{\left({x}^{2} \cdot \frac{-1}{6}\right)} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto x \cdot \left(\left(1 + \frac{1}{6} \cdot {y}^{2}\right) + \color{blue}{{x}^{2} \cdot \left(\frac{-1}{6} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)}\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(\left(1 + \frac{1}{6} \cdot {y}^{2}\right) + {x}^{2} \cdot \left(\frac{-1}{6} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)\right)} \]

      7. associate-*r* N/A

         \[\leadsto x \cdot \left(\left(1 + \frac{1}{6} \cdot {y}^{2}\right) + \color{blue}{\left({x}^{2} \cdot \frac{-1}{6}\right) \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)}\right) \]

      8. *-commutative N/A

         \[\leadsto x \cdot \left(\left(1 + \frac{1}{6} \cdot {y}^{2}\right) + \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2}\right)} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right) \]

      9. distribute-rgt1-in N/A

         \[\leadsto x \cdot \color{blue}{\left(\left(\frac{-1}{6} \cdot {x}^{2} + 1\right) \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)} \]

      10. +-commutative N/A

          \[\leadsto x \cdot \left(\color{blue}{\left(1 + \frac{-1}{6} \cdot {x}^{2}\right)} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right) \]

   8. Simplified 60.0%

      \[\leadsto \color{blue}{x \cdot \left(\mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right)\right)} \]

   if 5.00000000000000041e-6 < (*.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 y around 0

      \[\leadsto \color{blue}{\sin x + {y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{6} \cdot \sin x\right)} \]
   4. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \sin x + \color{blue}{\left(\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right)\right) \cdot {y}^{2} + \left(\frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}\right)} \]

      2. *-rgt-identity N/A

         \[\leadsto \color{blue}{\sin x \cdot 1} + \left(\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right)\right) \cdot {y}^{2} + \left(\frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \sin x \cdot 1 + \color{blue}{{y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{6} \cdot \sin x\right)} \]

      4. *-commutative N/A

         \[\leadsto \sin x \cdot 1 + \color{blue}{\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}} \]

      5. associate-*r* N/A

         \[\leadsto \sin x \cdot 1 + \left(\color{blue}{\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot \sin x} + \frac{1}{6} \cdot \sin x\right) \cdot {y}^{2} \]

      6. distribute-rgt-out N/A

         \[\leadsto \sin x \cdot 1 + \color{blue}{\left(\sin x \cdot \left(\frac{1}{120} \cdot {y}^{2} + \frac{1}{6}\right)\right)} \cdot {y}^{2} \]

      7. +-commutative N/A

         \[\leadsto \sin x \cdot 1 + \left(\sin x \cdot \color{blue}{\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)}\right) \cdot {y}^{2} \]

      8. associate-*l* N/A

         \[\leadsto \sin x \cdot 1 + \color{blue}{\sin x \cdot \left(\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) \cdot {y}^{2}\right)} \]

      9. *-commutative N/A

         \[\leadsto \sin x \cdot 1 + \sin x \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} \]

      10. distribute-lft-in N/A

          \[\leadsto \color{blue}{\sin x \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\sin x \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} \]

      12. sin-lowering-sin.f64 N/A

          \[\leadsto \color{blue}{\sin x} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \sin x \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) + 1\right)} \]

   5. Simplified 83.6%

      \[\leadsto \color{blue}{\sin x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x} \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]
   7. Step-by-step derivation

   8. Simplified 35.1%

      \[\leadsto \color{blue}{x} \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin x \cdot \frac{\sinh y}{y} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;x \cdot \left(\mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right) \cdot \mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 46.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin x \cdot \frac{\sinh y}{y} \leq -0.1:\\ \;\;\;\;x \cdot \left(\left(x \cdot x\right) \cdot \mathsf{fma}\left(y \cdot y, -0.027777777777777776, -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (sin x) (/ (sinh y) y)) -0.1)
   (* 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) * (sinh(y) / y)) <= -0.1) {
		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 (Float64(sin(x) * Float64(sinh(y) / y)) <= -0.1)
		tmp = Float64(x * Float64(Float64(x * x) * fma(Float64(y * y), -0.027777777777777776, -0.16666666666666666)));
	else
		tmp = Float64(x * fma(Float64(y * y), fma(y, Float64(y * 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[(x * N[(N[(x * x), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * -0.027777777777777776 + -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * 0.008333333333333333), $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

      \[\leadsto \color{blue}{\sin x + \frac{1}{6} \cdot \left({y}^{2} \cdot \sin x\right)} \]
   4. Step-by-step derivation

      1. *-lft-identity N/A

         \[\leadsto \color{blue}{1 \cdot \sin x} + \frac{1}{6} \cdot \left({y}^{2} \cdot \sin x\right) \]

      2. associate-*r* N/A

         \[\leadsto 1 \cdot \sin x + \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \sin x} \]

      3. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\sin x \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\sin x \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)} \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \color{blue}{\sin x} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right) \]

      6. +-commutative N/A

         \[\leadsto \sin x \cdot \color{blue}{\left(\frac{1}{6} \cdot {y}^{2} + 1\right)} \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \sin x \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{6}, {y}^{2}, 1\right)} \]

      8. unpow2 N/A

         \[\leadsto \sin x \cdot \mathsf{fma}\left(\frac{1}{6}, \color{blue}{y \cdot y}, 1\right) \]

      9. *-lowering-*.f64 70.2

         \[\leadsto \sin x \cdot \mathsf{fma}\left(0.16666666666666666, \color{blue}{y \cdot y}, 1\right) \]

   5. Simplified 70.2%

      \[\leadsto \color{blue}{\sin x \cdot \mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + \left(\frac{-1}{6} \cdot \left({x}^{2} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right) + \frac{1}{6} \cdot {y}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \left(1 + \color{blue}{\left(\frac{1}{6} \cdot {y}^{2} + \frac{-1}{6} \cdot \left({x}^{2} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)\right)}\right) \]

      2. associate-+r+ N/A

         \[\leadsto x \cdot \color{blue}{\left(\left(1 + \frac{1}{6} \cdot {y}^{2}\right) + \frac{-1}{6} \cdot \left({x}^{2} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)\right)} \]

      3. associate-*r* N/A

         \[\leadsto x \cdot \left(\left(1 + \frac{1}{6} \cdot {y}^{2}\right) + \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2}\right) \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto x \cdot \left(\left(1 + \frac{1}{6} \cdot {y}^{2}\right) + \color{blue}{\left({x}^{2} \cdot \frac{-1}{6}\right)} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto x \cdot \left(\left(1 + \frac{1}{6} \cdot {y}^{2}\right) + \color{blue}{{x}^{2} \cdot \left(\frac{-1}{6} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)}\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(\left(1 + \frac{1}{6} \cdot {y}^{2}\right) + {x}^{2} \cdot \left(\frac{-1}{6} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)\right)} \]

      7. associate-*r* N/A

         \[\leadsto x \cdot \left(\left(1 + \frac{1}{6} \cdot {y}^{2}\right) + \color{blue}{\left({x}^{2} \cdot \frac{-1}{6}\right) \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)}\right) \]

      8. *-commutative N/A

         \[\leadsto x \cdot \left(\left(1 + \frac{1}{6} \cdot {y}^{2}\right) + \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2}\right)} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right) \]

      9. distribute-rgt1-in N/A

         \[\leadsto x \cdot \color{blue}{\left(\left(\frac{-1}{6} \cdot {x}^{2} + 1\right) \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)} \]

      10. +-commutative N/A

          \[\leadsto x \cdot \left(\color{blue}{\left(1 + \frac{-1}{6} \cdot {x}^{2}\right)} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right) \]

   8. Simplified 33.4%

      \[\leadsto \color{blue}{x \cdot \left(\mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right)\right)} \]

   9. Taylor expanded in x around inf

      \[\leadsto x \cdot \color{blue}{\left(\frac{-1}{6} \cdot \left({x}^{2} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)\right)} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto x \cdot \color{blue}{\left(\left({x}^{2} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right) \cdot \frac{-1}{6}\right)} \]

       2. associate-*r* N/A

          \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \left(\left(1 + \frac{1}{6} \cdot {y}^{2}\right) \cdot \frac{-1}{6}\right)\right)} \]

       3. *-commutative N/A

          \[\leadsto x \cdot \left({x}^{2} \cdot \color{blue}{\left(\frac{-1}{6} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)}\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{-1}{6} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)\right)} \]

       5. unpow2 N/A

          \[\leadsto x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{-1}{6} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{-1}{6} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)\right) \]

       7. +-commutative N/A

          \[\leadsto x \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{-1}{6} \cdot \color{blue}{\left(\frac{1}{6} \cdot {y}^{2} + 1\right)}\right)\right) \]

       8. distribute-rgt-in N/A

          \[\leadsto x \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{\left(\left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \frac{-1}{6} + 1 \cdot \frac{-1}{6}\right)}\right) \]

       9. metadata-eval N/A

          \[\leadsto x \cdot \left(\left(x \cdot x\right) \cdot \left(\left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \frac{-1}{6} + \color{blue}{\frac{-1}{6}}\right)\right) \]

       10. *-commutative N/A

           \[\leadsto x \cdot \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\frac{-1}{6} \cdot \left(\frac{1}{6} \cdot {y}^{2}\right)} + \frac{-1}{6}\right)\right) \]

       11. associate-*r* N/A

           \[\leadsto x \cdot \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\left(\frac{-1}{6} \cdot \frac{1}{6}\right) \cdot {y}^{2}} + \frac{-1}{6}\right)\right) \]

       12. *-commutative N/A

           \[\leadsto x \cdot \left(\left(x \cdot x\right) \cdot \left(\color{blue}{{y}^{2} \cdot \left(\frac{-1}{6} \cdot \frac{1}{6}\right)} + \frac{-1}{6}\right)\right) \]

       13. accelerator-lowering-fma.f64 N/A

           \[\leadsto x \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{-1}{6} \cdot \frac{1}{6}, \frac{-1}{6}\right)}\right) \]

       14. unpow2 N/A

           \[\leadsto x \cdot \left(\left(x \cdot x\right) \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{-1}{6} \cdot \frac{1}{6}, \frac{-1}{6}\right)\right) \]

       15. *-lowering-*.f64 N/A

           \[\leadsto x \cdot \left(\left(x \cdot x\right) \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{-1}{6} \cdot \frac{1}{6}, \frac{-1}{6}\right)\right) \]

       16. metadata-eval 18.2

           \[\leadsto x \cdot \left(\left(x \cdot x\right) \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{-0.027777777777777776}, -0.16666666666666666\right)\right) \]

   11. Simplified 18.2%

       \[\leadsto x \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot \mathsf{fma}\left(y \cdot y, -0.027777777777777776, -0.16666666666666666\right)\right)} \]

   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 y around 0

      \[\leadsto \color{blue}{\sin x + {y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{6} \cdot \sin x\right)} \]
   4. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \sin x + \color{blue}{\left(\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right)\right) \cdot {y}^{2} + \left(\frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}\right)} \]

      2. *-rgt-identity N/A

         \[\leadsto \color{blue}{\sin x \cdot 1} + \left(\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right)\right) \cdot {y}^{2} + \left(\frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \sin x \cdot 1 + \color{blue}{{y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{6} \cdot \sin x\right)} \]

      4. *-commutative N/A

         \[\leadsto \sin x \cdot 1 + \color{blue}{\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}} \]

      5. associate-*r* N/A

         \[\leadsto \sin x \cdot 1 + \left(\color{blue}{\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot \sin x} + \frac{1}{6} \cdot \sin x\right) \cdot {y}^{2} \]

      6. distribute-rgt-out N/A

         \[\leadsto \sin x \cdot 1 + \color{blue}{\left(\sin x \cdot \left(\frac{1}{120} \cdot {y}^{2} + \frac{1}{6}\right)\right)} \cdot {y}^{2} \]

      7. +-commutative N/A

         \[\leadsto \sin x \cdot 1 + \left(\sin x \cdot \color{blue}{\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)}\right) \cdot {y}^{2} \]

      8. associate-*l* N/A

         \[\leadsto \sin x \cdot 1 + \color{blue}{\sin x \cdot \left(\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) \cdot {y}^{2}\right)} \]

      9. *-commutative N/A

         \[\leadsto \sin x \cdot 1 + \sin x \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} \]

      10. distribute-lft-in N/A

          \[\leadsto \color{blue}{\sin x \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\sin x \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} \]

      12. sin-lowering-sin.f64 N/A

          \[\leadsto \color{blue}{\sin x} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \sin x \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) + 1\right)} \]

   5. Simplified 90.3%

      \[\leadsto \color{blue}{\sin x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x} \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]
   7. Step-by-step derivation

   8. Simplified 63.0%

      \[\leadsto \color{blue}{x} \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 44.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin x \cdot \frac{\sinh y}{y} \leq -0.1:\\ \;\;\;\;x \cdot \left(\left(x \cdot x\right) \cdot \mathsf{fma}\left(y \cdot y, -0.027777777777777776, -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, x \cdot \left(\left(y \cdot y\right) \cdot 0.008333333333333333\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (sin x) (/ (sinh y) y)) -0.1)
   (* x (* (* x x) (fma (* y y) -0.027777777777777776 -0.16666666666666666)))
   (fma (* y y) (* x (* (* y y) 0.008333333333333333)) x)))

C

double code(double x, double y) {
	double tmp;
	if ((sin(x) * (sinh(y) / y)) <= -0.1) {
		tmp = x * ((x * x) * fma((y * y), -0.027777777777777776, -0.16666666666666666));
	} else {
		tmp = fma((y * y), (x * ((y * y) * 0.008333333333333333)), x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(sin(x) * Float64(sinh(y) / y)) <= -0.1)
		tmp = Float64(x * Float64(Float64(x * x) * fma(Float64(y * y), -0.027777777777777776, -0.16666666666666666)));
	else
		tmp = fma(Float64(y * y), Float64(x * Float64(Float64(y * y) * 0.008333333333333333)), x);
	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[(x * N[(N[(x * x), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * -0.027777777777777776 + -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * y), $MachinePrecision] * N[(x * N[(N[(y * y), $MachinePrecision] * 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + x), $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

      \[\leadsto \color{blue}{\sin x + \frac{1}{6} \cdot \left({y}^{2} \cdot \sin x\right)} \]
   4. Step-by-step derivation

      1. *-lft-identity N/A

         \[\leadsto \color{blue}{1 \cdot \sin x} + \frac{1}{6} \cdot \left({y}^{2} \cdot \sin x\right) \]

      2. associate-*r* N/A

         \[\leadsto 1 \cdot \sin x + \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \sin x} \]

      3. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\sin x \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\sin x \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)} \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \color{blue}{\sin x} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right) \]

      6. +-commutative N/A

         \[\leadsto \sin x \cdot \color{blue}{\left(\frac{1}{6} \cdot {y}^{2} + 1\right)} \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \sin x \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{6}, {y}^{2}, 1\right)} \]

      8. unpow2 N/A

         \[\leadsto \sin x \cdot \mathsf{fma}\left(\frac{1}{6}, \color{blue}{y \cdot y}, 1\right) \]

      9. *-lowering-*.f64 70.2

         \[\leadsto \sin x \cdot \mathsf{fma}\left(0.16666666666666666, \color{blue}{y \cdot y}, 1\right) \]

   5. Simplified 70.2%

      \[\leadsto \color{blue}{\sin x \cdot \mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + \left(\frac{-1}{6} \cdot \left({x}^{2} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right) + \frac{1}{6} \cdot {y}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \left(1 + \color{blue}{\left(\frac{1}{6} \cdot {y}^{2} + \frac{-1}{6} \cdot \left({x}^{2} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)\right)}\right) \]

      2. associate-+r+ N/A

         \[\leadsto x \cdot \color{blue}{\left(\left(1 + \frac{1}{6} \cdot {y}^{2}\right) + \frac{-1}{6} \cdot \left({x}^{2} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)\right)} \]

      3. associate-*r* N/A

         \[\leadsto x \cdot \left(\left(1 + \frac{1}{6} \cdot {y}^{2}\right) + \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2}\right) \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto x \cdot \left(\left(1 + \frac{1}{6} \cdot {y}^{2}\right) + \color{blue}{\left({x}^{2} \cdot \frac{-1}{6}\right)} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto x \cdot \left(\left(1 + \frac{1}{6} \cdot {y}^{2}\right) + \color{blue}{{x}^{2} \cdot \left(\frac{-1}{6} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)}\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(\left(1 + \frac{1}{6} \cdot {y}^{2}\right) + {x}^{2} \cdot \left(\frac{-1}{6} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)\right)} \]

      7. associate-*r* N/A

         \[\leadsto x \cdot \left(\left(1 + \frac{1}{6} \cdot {y}^{2}\right) + \color{blue}{\left({x}^{2} \cdot \frac{-1}{6}\right) \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)}\right) \]

      8. *-commutative N/A

         \[\leadsto x \cdot \left(\left(1 + \frac{1}{6} \cdot {y}^{2}\right) + \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2}\right)} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right) \]

      9. distribute-rgt1-in N/A

         \[\leadsto x \cdot \color{blue}{\left(\left(\frac{-1}{6} \cdot {x}^{2} + 1\right) \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)} \]

      10. +-commutative N/A

          \[\leadsto x \cdot \left(\color{blue}{\left(1 + \frac{-1}{6} \cdot {x}^{2}\right)} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right) \]

   8. Simplified 33.4%

      \[\leadsto \color{blue}{x \cdot \left(\mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right)\right)} \]

   9. Taylor expanded in x around inf

      \[\leadsto x \cdot \color{blue}{\left(\frac{-1}{6} \cdot \left({x}^{2} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)\right)} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto x \cdot \color{blue}{\left(\left({x}^{2} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right) \cdot \frac{-1}{6}\right)} \]

       2. associate-*r* N/A

          \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \left(\left(1 + \frac{1}{6} \cdot {y}^{2}\right) \cdot \frac{-1}{6}\right)\right)} \]

       3. *-commutative N/A

          \[\leadsto x \cdot \left({x}^{2} \cdot \color{blue}{\left(\frac{-1}{6} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)}\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{-1}{6} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)\right)} \]

       5. unpow2 N/A

          \[\leadsto x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{-1}{6} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{-1}{6} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)\right) \]

       7. +-commutative N/A

          \[\leadsto x \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{-1}{6} \cdot \color{blue}{\left(\frac{1}{6} \cdot {y}^{2} + 1\right)}\right)\right) \]

       8. distribute-rgt-in N/A

          \[\leadsto x \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{\left(\left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \frac{-1}{6} + 1 \cdot \frac{-1}{6}\right)}\right) \]

       9. metadata-eval N/A

          \[\leadsto x \cdot \left(\left(x \cdot x\right) \cdot \left(\left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \frac{-1}{6} + \color{blue}{\frac{-1}{6}}\right)\right) \]

       10. *-commutative N/A

           \[\leadsto x \cdot \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\frac{-1}{6} \cdot \left(\frac{1}{6} \cdot {y}^{2}\right)} + \frac{-1}{6}\right)\right) \]

       11. associate-*r* N/A

           \[\leadsto x \cdot \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\left(\frac{-1}{6} \cdot \frac{1}{6}\right) \cdot {y}^{2}} + \frac{-1}{6}\right)\right) \]

       12. *-commutative N/A

           \[\leadsto x \cdot \left(\left(x \cdot x\right) \cdot \left(\color{blue}{{y}^{2} \cdot \left(\frac{-1}{6} \cdot \frac{1}{6}\right)} + \frac{-1}{6}\right)\right) \]

       13. accelerator-lowering-fma.f64 N/A

           \[\leadsto x \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{-1}{6} \cdot \frac{1}{6}, \frac{-1}{6}\right)}\right) \]

       14. unpow2 N/A

           \[\leadsto x \cdot \left(\left(x \cdot x\right) \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{-1}{6} \cdot \frac{1}{6}, \frac{-1}{6}\right)\right) \]

       15. *-lowering-*.f64 N/A

           \[\leadsto x \cdot \left(\left(x \cdot x\right) \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{-1}{6} \cdot \frac{1}{6}, \frac{-1}{6}\right)\right) \]

       16. metadata-eval 18.2

           \[\leadsto x \cdot \left(\left(x \cdot x\right) \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{-0.027777777777777776}, -0.16666666666666666\right)\right) \]

   11. Simplified 18.2%

       \[\leadsto x \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot \mathsf{fma}\left(y \cdot y, -0.027777777777777776, -0.16666666666666666\right)\right)} \]

   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 67.3%

      \[\leadsto \color{blue}{x} \cdot \frac{\sinh y}{y} \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + {y}^{2} \cdot \left(\frac{1}{120} \cdot \left(x \cdot {y}^{2}\right) + \frac{1}{6} \cdot x\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{y}^{2} \cdot \left(\frac{1}{120} \cdot \left(x \cdot {y}^{2}\right) + \frac{1}{6} \cdot x\right) + x} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{1}{120} \cdot \left(x \cdot {y}^{2}\right) + \frac{1}{6} \cdot x, x\right)} \]

      3. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{120} \cdot \left(x \cdot {y}^{2}\right) + \frac{1}{6} \cdot x, x\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{120} \cdot \left(x \cdot {y}^{2}\right) + \frac{1}{6} \cdot x, x\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(y \cdot y, \color{blue}{\frac{1}{6} \cdot x + \frac{1}{120} \cdot \left(x \cdot {y}^{2}\right)}, x\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y \cdot y, \frac{1}{6} \cdot x + \frac{1}{120} \cdot \color{blue}{\left({y}^{2} \cdot x\right)}, x\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(y \cdot y, \frac{1}{6} \cdot x + \color{blue}{\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot x}, x\right) \]

      8. distribute-rgt-in N/A

         \[\leadsto \mathsf{fma}\left(y \cdot y, \color{blue}{x \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)}, x\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y \cdot y, \color{blue}{x \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)}, x\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(y \cdot y, x \cdot \color{blue}{\left(\frac{1}{120} \cdot {y}^{2} + \frac{1}{6}\right)}, x\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(y \cdot y, x \cdot \left(\color{blue}{{y}^{2} \cdot \frac{1}{120}} + \frac{1}{6}\right), x\right) \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(y \cdot y, x \cdot \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{1}{120}, \frac{1}{6}\right)}, x\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(y \cdot y, x \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{120}, \frac{1}{6}\right), x\right) \]

      14. *-lowering-*.f64 60.0

          \[\leadsto \mathsf{fma}\left(y \cdot y, x \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, 0.008333333333333333, 0.16666666666666666\right), x\right) \]

   8. Simplified 60.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y \cdot y, x \cdot \mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), x\right)} \]

   9. Taylor expanded in y around inf

      \[\leadsto \mathsf{fma}\left(y \cdot y, x \cdot \color{blue}{\left(\frac{1}{120} \cdot {y}^{2}\right)}, x\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(y \cdot y, x \cdot \color{blue}{\left({y}^{2} \cdot \frac{1}{120}\right)}, x\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(y \cdot y, x \cdot \color{blue}{\left({y}^{2} \cdot \frac{1}{120}\right)}, x\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(y \cdot y, x \cdot \left(\color{blue}{\left(y \cdot y\right)} \cdot \frac{1}{120}\right), x\right) \]

       4. *-lowering-*.f64 59.4

          \[\leadsto \mathsf{fma}\left(y \cdot y, x \cdot \left(\color{blue}{\left(y \cdot y\right)} \cdot 0.008333333333333333\right), x\right) \]

   11. Simplified 59.4%

       \[\leadsto \mathsf{fma}\left(y \cdot y, x \cdot \color{blue}{\left(\left(y \cdot y\right) \cdot 0.008333333333333333\right)}, x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 42.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin x \cdot \frac{\sinh y}{y} \leq -0.1:\\ \;\;\;\;\left(y \cdot y\right) \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot -0.027777777777777776\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.16666666666666666, x \cdot \left(y \cdot y\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (sin x) (/ (sinh y) y)) -0.1)
   (* (* y y) (* (* x (* x x)) -0.027777777777777776))
   (fma 0.16666666666666666 (* x (* y y)) x)))

C

double code(double x, double y) {
	double tmp;
	if ((sin(x) * (sinh(y) / y)) <= -0.1) {
		tmp = (y * y) * ((x * (x * x)) * -0.027777777777777776);
	} else {
		tmp = fma(0.16666666666666666, (x * (y * y)), x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(sin(x) * Float64(sinh(y) / y)) <= -0.1)
		tmp = Float64(Float64(y * y) * Float64(Float64(x * Float64(x * x)) * -0.027777777777777776));
	else
		tmp = fma(0.16666666666666666, Float64(x * Float64(y * y)), x);
	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[(y * y), $MachinePrecision] * N[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] * -0.027777777777777776), $MachinePrecision]), $MachinePrecision], N[(0.16666666666666666 * N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision] + x), $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

      \[\leadsto \color{blue}{\sin x + \frac{1}{6} \cdot \left({y}^{2} \cdot \sin x\right)} \]
   4. Step-by-step derivation

      1. *-lft-identity N/A

         \[\leadsto \color{blue}{1 \cdot \sin x} + \frac{1}{6} \cdot \left({y}^{2} \cdot \sin x\right) \]

      2. associate-*r* N/A

         \[\leadsto 1 \cdot \sin x + \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \sin x} \]

      3. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\sin x \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\sin x \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)} \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \color{blue}{\sin x} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right) \]

      6. +-commutative N/A

         \[\leadsto \sin x \cdot \color{blue}{\left(\frac{1}{6} \cdot {y}^{2} + 1\right)} \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \sin x \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{6}, {y}^{2}, 1\right)} \]

      8. unpow2 N/A

         \[\leadsto \sin x \cdot \mathsf{fma}\left(\frac{1}{6}, \color{blue}{y \cdot y}, 1\right) \]

      9. *-lowering-*.f64 70.2

         \[\leadsto \sin x \cdot \mathsf{fma}\left(0.16666666666666666, \color{blue}{y \cdot y}, 1\right) \]

   5. Simplified 70.2%

      \[\leadsto \color{blue}{\sin x \cdot \mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + \left(\frac{-1}{6} \cdot \left({x}^{2} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right) + \frac{1}{6} \cdot {y}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \left(1 + \color{blue}{\left(\frac{1}{6} \cdot {y}^{2} + \frac{-1}{6} \cdot \left({x}^{2} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)\right)}\right) \]

      2. associate-+r+ N/A

         \[\leadsto x \cdot \color{blue}{\left(\left(1 + \frac{1}{6} \cdot {y}^{2}\right) + \frac{-1}{6} \cdot \left({x}^{2} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)\right)} \]

      3. associate-*r* N/A

         \[\leadsto x \cdot \left(\left(1 + \frac{1}{6} \cdot {y}^{2}\right) + \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2}\right) \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto x \cdot \left(\left(1 + \frac{1}{6} \cdot {y}^{2}\right) + \color{blue}{\left({x}^{2} \cdot \frac{-1}{6}\right)} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto x \cdot \left(\left(1 + \frac{1}{6} \cdot {y}^{2}\right) + \color{blue}{{x}^{2} \cdot \left(\frac{-1}{6} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)}\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(\left(1 + \frac{1}{6} \cdot {y}^{2}\right) + {x}^{2} \cdot \left(\frac{-1}{6} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)\right)} \]

      7. associate-*r* N/A

         \[\leadsto x \cdot \left(\left(1 + \frac{1}{6} \cdot {y}^{2}\right) + \color{blue}{\left({x}^{2} \cdot \frac{-1}{6}\right) \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)}\right) \]

      8. *-commutative N/A

         \[\leadsto x \cdot \left(\left(1 + \frac{1}{6} \cdot {y}^{2}\right) + \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2}\right)} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right) \]

      9. distribute-rgt1-in N/A

         \[\leadsto x \cdot \color{blue}{\left(\left(\frac{-1}{6} \cdot {x}^{2} + 1\right) \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)} \]

      10. +-commutative N/A

          \[\leadsto x \cdot \left(\color{blue}{\left(1 + \frac{-1}{6} \cdot {x}^{2}\right)} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right) \]

   8. Simplified 33.4%

      \[\leadsto \color{blue}{x \cdot \left(\mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right)\right)} \]

   9. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{1}{6} \cdot \left(x \cdot \left({y}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right)} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{1}{6} \cdot \left(x \cdot \color{blue}{\left(\left(1 + \frac{-1}{6} \cdot {x}^{2}\right) \cdot {y}^{2}\right)}\right) \]

       2. associate-*r* N/A

          \[\leadsto \frac{1}{6} \cdot \color{blue}{\left(\left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right) \cdot {y}^{2}\right)} \]

       3. associate-*l* N/A

          \[\leadsto \color{blue}{\left(\frac{1}{6} \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right) \cdot {y}^{2}} \]

       4. *-commutative N/A

          \[\leadsto \color{blue}{{y}^{2} \cdot \left(\frac{1}{6} \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right)} \]

       5. associate-*r* N/A

          \[\leadsto {y}^{2} \cdot \color{blue}{\left(\left(\frac{1}{6} \cdot x\right) \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)} \]

       6. distribute-lft-in N/A

          \[\leadsto {y}^{2} \cdot \color{blue}{\left(\left(\frac{1}{6} \cdot x\right) \cdot 1 + \left(\frac{1}{6} \cdot x\right) \cdot \left(\frac{-1}{6} \cdot {x}^{2}\right)\right)} \]

       7. *-rgt-identity N/A

          \[\leadsto {y}^{2} \cdot \left(\color{blue}{\frac{1}{6} \cdot x} + \left(\frac{1}{6} \cdot x\right) \cdot \left(\frac{-1}{6} \cdot {x}^{2}\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{{y}^{2} \cdot \left(\frac{1}{6} \cdot x + \left(\frac{1}{6} \cdot x\right) \cdot \left(\frac{-1}{6} \cdot {x}^{2}\right)\right)} \]

       9. unpow2 N/A

          \[\leadsto \color{blue}{\left(y \cdot y\right)} \cdot \left(\frac{1}{6} \cdot x + \left(\frac{1}{6} \cdot x\right) \cdot \left(\frac{-1}{6} \cdot {x}^{2}\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \color{blue}{\left(y \cdot y\right)} \cdot \left(\frac{1}{6} \cdot x + \left(\frac{1}{6} \cdot x\right) \cdot \left(\frac{-1}{6} \cdot {x}^{2}\right)\right) \]

       11. *-rgt-identity N/A

           \[\leadsto \left(y \cdot y\right) \cdot \left(\color{blue}{\left(\frac{1}{6} \cdot x\right) \cdot 1} + \left(\frac{1}{6} \cdot x\right) \cdot \left(\frac{-1}{6} \cdot {x}^{2}\right)\right) \]

       12. distribute-lft-in N/A

           \[\leadsto \left(y \cdot y\right) \cdot \color{blue}{\left(\left(\frac{1}{6} \cdot x\right) \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)} \]

       13. *-commutative N/A

           \[\leadsto \left(y \cdot y\right) \cdot \left(\color{blue}{\left(x \cdot \frac{1}{6}\right)} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right) \]

       14. associate-*l* N/A

           \[\leadsto \left(y \cdot y\right) \cdot \color{blue}{\left(x \cdot \left(\frac{1}{6} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right)} \]

       15. *-lowering-*.f64 N/A

           \[\leadsto \left(y \cdot y\right) \cdot \color{blue}{\left(x \cdot \left(\frac{1}{6} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right)} \]

       16. +-commutative N/A

           \[\leadsto \left(y \cdot y\right) \cdot \left(x \cdot \left(\frac{1}{6} \cdot \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2} + 1\right)}\right)\right) \]

       17. distribute-rgt-in N/A

           \[\leadsto \left(y \cdot y\right) \cdot \left(x \cdot \color{blue}{\left(\left(\frac{-1}{6} \cdot {x}^{2}\right) \cdot \frac{1}{6} + 1 \cdot \frac{1}{6}\right)}\right) \]

       18. metadata-eval N/A

           \[\leadsto \left(y \cdot y\right) \cdot \left(x \cdot \left(\left(\frac{-1}{6} \cdot {x}^{2}\right) \cdot \frac{1}{6} + \color{blue}{\frac{1}{6}}\right)\right) \]

   11. Simplified 33.3%

       \[\leadsto \color{blue}{\left(y \cdot y\right) \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, -0.027777777777777776, 0.16666666666666666\right)\right)} \]

   12. Taylor expanded in x around inf

       \[\leadsto \color{blue}{\frac{-1}{36} \cdot \left({x}^{3} \cdot {y}^{2}\right)} \]
   13. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{-1}{36} \cdot {x}^{3}\right) \cdot {y}^{2}} \]

       2. *-commutative N/A

          \[\leadsto \color{blue}{{y}^{2} \cdot \left(\frac{-1}{36} \cdot {x}^{3}\right)} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{{y}^{2} \cdot \left(\frac{-1}{36} \cdot {x}^{3}\right)} \]

       4. unpow2 N/A

          \[\leadsto \color{blue}{\left(y \cdot y\right)} \cdot \left(\frac{-1}{36} \cdot {x}^{3}\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\left(y \cdot y\right)} \cdot \left(\frac{-1}{36} \cdot {x}^{3}\right) \]

       6. *-commutative N/A

          \[\leadsto \left(y \cdot y\right) \cdot \color{blue}{\left({x}^{3} \cdot \frac{-1}{36}\right)} \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \left(y \cdot y\right) \cdot \color{blue}{\left({x}^{3} \cdot \frac{-1}{36}\right)} \]

       8. cube-mult N/A

          \[\leadsto \left(y \cdot y\right) \cdot \left(\color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot \frac{-1}{36}\right) \]

       9. unpow2 N/A

          \[\leadsto \left(y \cdot y\right) \cdot \left(\left(x \cdot \color{blue}{{x}^{2}}\right) \cdot \frac{-1}{36}\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \left(y \cdot y\right) \cdot \left(\color{blue}{\left(x \cdot {x}^{2}\right)} \cdot \frac{-1}{36}\right) \]

       11. unpow2 N/A

           \[\leadsto \left(y \cdot y\right) \cdot \left(\left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \frac{-1}{36}\right) \]

       12. *-lowering-*.f64 18.1

           \[\leadsto \left(y \cdot y\right) \cdot \left(\left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot -0.027777777777777776\right) \]

   14. Simplified 18.1%

       \[\leadsto \color{blue}{\left(y \cdot y\right) \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot -0.027777777777777776\right)} \]

   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 67.3%

      \[\leadsto \color{blue}{x} \cdot \frac{\sinh y}{y} \]

   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. +-commutative N/A

         \[\leadsto x \cdot \frac{y \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) + 1\right)}}{y} \]

      2. distribute-rgt-in N/A

         \[\leadsto x \cdot \frac{\color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right) \cdot y + 1 \cdot y}}{y} \]

      3. *-commutative N/A

         \[\leadsto x \cdot \frac{\color{blue}{\left(\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot {y}^{2}\right)} \cdot y + 1 \cdot y}{y} \]

      4. associate-*l* N/A

         \[\leadsto x \cdot \frac{\color{blue}{\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot \left({y}^{2} \cdot y\right)} + 1 \cdot y}{y} \]

      5. pow-plus N/A

         \[\leadsto x \cdot \frac{\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot \color{blue}{{y}^{\left(2 + 1\right)}} + 1 \cdot y}{y} \]

      6. metadata-eval N/A

         \[\leadsto x \cdot \frac{\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot {y}^{\color{blue}{3}} + 1 \cdot y}{y} \]

      7. cube-unmult N/A

         \[\leadsto x \cdot \frac{\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot \color{blue}{\left(y \cdot \left(y \cdot y\right)\right)} + 1 \cdot y}{y} \]

      8. unpow2 N/A

         \[\leadsto x \cdot \frac{\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot \left(y \cdot \color{blue}{{y}^{2}}\right) + 1 \cdot y}{y} \]

      9. *-lft-identity N/A

         \[\leadsto x \cdot \frac{\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot \left(y \cdot {y}^{2}\right) + \color{blue}{y}}{y} \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto x \cdot \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right), y \cdot {y}^{2}, y\right)}}{y} \]

   8. Simplified 64.9%

      \[\leadsto x \cdot \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)}}{y} \]

   9. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + \frac{1}{6} \cdot \left(x \cdot {y}^{2}\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto x + \color{blue}{\left(\frac{1}{6} \cdot x\right) \cdot {y}^{2}} \]

       2. +-commutative N/A

          \[\leadsto \color{blue}{\left(\frac{1}{6} \cdot x\right) \cdot {y}^{2} + x} \]

       3. associate-*r* N/A

          \[\leadsto \color{blue}{\frac{1}{6} \cdot \left(x \cdot {y}^{2}\right)} + x \]

       4. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{6}, x \cdot {y}^{2}, x\right)} \]

       5. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{6}, \color{blue}{{y}^{2} \cdot x}, x\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{6}, \color{blue}{{y}^{2} \cdot x}, x\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{6}, \color{blue}{\left(y \cdot y\right)} \cdot x, x\right) \]

       8. *-lowering-*.f64 57.4

          \[\leadsto \mathsf{fma}\left(0.16666666666666666, \color{blue}{\left(y \cdot y\right)} \cdot x, x\right) \]

   11. Simplified 57.4%

       \[\leadsto \color{blue}{\mathsf{fma}\left(0.16666666666666666, \left(y \cdot y\right) \cdot x, x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin x \cdot \frac{\sinh y}{y} \leq -0.1:\\ \;\;\;\;\left(y \cdot y\right) \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot -0.027777777777777776\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.16666666666666666, x \cdot \left(y \cdot y\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 41.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin x \cdot \frac{\sinh y}{y} \leq -0.1:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot \left(x \cdot -0.16666666666666666\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.16666666666666666, x \cdot \left(y \cdot y\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (sin x) (/ (sinh y) y)) -0.1)
   (fma x (* x (* x -0.16666666666666666)) x)
   (fma 0.16666666666666666 (* x (* y y)) x)))

C

double code(double x, double y) {
	double tmp;
	if ((sin(x) * (sinh(y) / y)) <= -0.1) {
		tmp = fma(x, (x * (x * -0.16666666666666666)), x);
	} else {
		tmp = fma(0.16666666666666666, (x * (y * y)), x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(sin(x) * Float64(sinh(y) / y)) <= -0.1)
		tmp = fma(x, Float64(x * Float64(x * -0.16666666666666666)), x);
	else
		tmp = fma(0.16666666666666666, Float64(x * Float64(y * y)), x);
	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[(x * N[(x * N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(0.16666666666666666 * N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision] + x), $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

      \[\leadsto \color{blue}{\sin x} \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 36.3

         \[\leadsto \color{blue}{\sin x} \]

   5. Simplified 36.3%

      \[\leadsto \color{blue}{\sin x} \]

   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. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2} + 1\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{x \cdot \left(\frac{-1}{6} \cdot {x}^{2}\right) + x \cdot 1} \]

      3. *-rgt-identity N/A

         \[\leadsto x \cdot \left(\frac{-1}{6} \cdot {x}^{2}\right) + \color{blue}{x} \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{-1}{6} \cdot {x}^{2}, x\right)} \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{{x}^{2} \cdot \frac{-1}{6}}, x\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\left(x \cdot x\right)} \cdot \frac{-1}{6}, x\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot \left(x \cdot \frac{-1}{6}\right)}, x\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot \left(x \cdot \frac{-1}{6}\right)}, x\right) \]

      9. *-lowering-*.f64 17.7

         \[\leadsto \mathsf{fma}\left(x, x \cdot \color{blue}{\left(x \cdot -0.16666666666666666\right)}, x\right) \]

   8. Simplified 17.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot \left(x \cdot -0.16666666666666666\right), x\right)} \]

   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 67.3%

      \[\leadsto \color{blue}{x} \cdot \frac{\sinh y}{y} \]

   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. +-commutative N/A

         \[\leadsto x \cdot \frac{y \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) + 1\right)}}{y} \]

      2. distribute-rgt-in N/A

         \[\leadsto x \cdot \frac{\color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right) \cdot y + 1 \cdot y}}{y} \]

      3. *-commutative N/A

         \[\leadsto x \cdot \frac{\color{blue}{\left(\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot {y}^{2}\right)} \cdot y + 1 \cdot y}{y} \]

      4. associate-*l* N/A

         \[\leadsto x \cdot \frac{\color{blue}{\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot \left({y}^{2} \cdot y\right)} + 1 \cdot y}{y} \]

      5. pow-plus N/A

         \[\leadsto x \cdot \frac{\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot \color{blue}{{y}^{\left(2 + 1\right)}} + 1 \cdot y}{y} \]

      6. metadata-eval N/A

         \[\leadsto x \cdot \frac{\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot {y}^{\color{blue}{3}} + 1 \cdot y}{y} \]

      7. cube-unmult N/A

         \[\leadsto x \cdot \frac{\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot \color{blue}{\left(y \cdot \left(y \cdot y\right)\right)} + 1 \cdot y}{y} \]

      8. unpow2 N/A

         \[\leadsto x \cdot \frac{\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot \left(y \cdot \color{blue}{{y}^{2}}\right) + 1 \cdot y}{y} \]

      9. *-lft-identity N/A

         \[\leadsto x \cdot \frac{\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot \left(y \cdot {y}^{2}\right) + \color{blue}{y}}{y} \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto x \cdot \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right), y \cdot {y}^{2}, y\right)}}{y} \]

   8. Simplified 64.9%

      \[\leadsto x \cdot \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)}}{y} \]

   9. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + \frac{1}{6} \cdot \left(x \cdot {y}^{2}\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto x + \color{blue}{\left(\frac{1}{6} \cdot x\right) \cdot {y}^{2}} \]

       2. +-commutative N/A

          \[\leadsto \color{blue}{\left(\frac{1}{6} \cdot x\right) \cdot {y}^{2} + x} \]

       3. associate-*r* N/A

          \[\leadsto \color{blue}{\frac{1}{6} \cdot \left(x \cdot {y}^{2}\right)} + x \]

       4. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{6}, x \cdot {y}^{2}, x\right)} \]

       5. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{6}, \color{blue}{{y}^{2} \cdot x}, x\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{6}, \color{blue}{{y}^{2} \cdot x}, x\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{6}, \color{blue}{\left(y \cdot y\right)} \cdot x, x\right) \]

       8. *-lowering-*.f64 57.4

          \[\leadsto \mathsf{fma}\left(0.16666666666666666, \color{blue}{\left(y \cdot y\right)} \cdot x, x\right) \]

   11. Simplified 57.4%

       \[\leadsto \color{blue}{\mathsf{fma}\left(0.16666666666666666, \left(y \cdot y\right) \cdot x, x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 41.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin x \cdot \frac{\sinh y}{y} \leq -0.1:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot \left(x \cdot -0.16666666666666666\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.16666666666666666, x \cdot \left(y \cdot y\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 41.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin x \cdot \frac{\sinh y}{y} \leq -0.1:\\ \;\;\;\;-0.16666666666666666 \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.16666666666666666, x \cdot \left(y \cdot y\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (sin x) (/ (sinh y) y)) -0.1)
   (* -0.16666666666666666 (* x (* x x)))
   (fma 0.16666666666666666 (* x (* y y)) x)))

C

double code(double x, double y) {
	double tmp;
	if ((sin(x) * (sinh(y) / y)) <= -0.1) {
		tmp = -0.16666666666666666 * (x * (x * x));
	} else {
		tmp = fma(0.16666666666666666, (x * (y * y)), x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(sin(x) * Float64(sinh(y) / y)) <= -0.1)
		tmp = Float64(-0.16666666666666666 * Float64(x * Float64(x * x)));
	else
		tmp = fma(0.16666666666666666, Float64(x * Float64(y * y)), x);
	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[(-0.16666666666666666 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.16666666666666666 * N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision] + x), $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

      \[\leadsto \color{blue}{\sin x} \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 36.3

         \[\leadsto \color{blue}{\sin x} \]

   5. Simplified 36.3%

      \[\leadsto \color{blue}{\sin x} \]

   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. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2} + 1\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{x \cdot \left(\frac{-1}{6} \cdot {x}^{2}\right) + x \cdot 1} \]

      3. *-rgt-identity N/A

         \[\leadsto x \cdot \left(\frac{-1}{6} \cdot {x}^{2}\right) + \color{blue}{x} \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{-1}{6} \cdot {x}^{2}, x\right)} \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{{x}^{2} \cdot \frac{-1}{6}}, x\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\left(x \cdot x\right)} \cdot \frac{-1}{6}, x\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot \left(x \cdot \frac{-1}{6}\right)}, x\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot \left(x \cdot \frac{-1}{6}\right)}, x\right) \]

      9. *-lowering-*.f64 17.7

         \[\leadsto \mathsf{fma}\left(x, x \cdot \color{blue}{\left(x \cdot -0.16666666666666666\right)}, x\right) \]

   8. Simplified 17.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot \left(x \cdot -0.16666666666666666\right), x\right)} \]

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{-1}{6} \cdot {x}^{3}} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\frac{-1}{6} \cdot {x}^{3}} \]

       2. cube-mult N/A

          \[\leadsto \frac{-1}{6} \cdot \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \]

       3. unpow2 N/A

          \[\leadsto \frac{-1}{6} \cdot \left(x \cdot \color{blue}{{x}^{2}}\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \frac{-1}{6} \cdot \color{blue}{\left(x \cdot {x}^{2}\right)} \]

       5. unpow2 N/A

          \[\leadsto \frac{-1}{6} \cdot \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \]

       6. *-lowering-*.f64 17.4

          \[\leadsto -0.16666666666666666 \cdot \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \]

   11. Simplified 17.4%

       \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(x \cdot \left(x \cdot x\right)\right)} \]

   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 67.3%

      \[\leadsto \color{blue}{x} \cdot \frac{\sinh y}{y} \]

   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. +-commutative N/A

         \[\leadsto x \cdot \frac{y \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) + 1\right)}}{y} \]

      2. distribute-rgt-in N/A

         \[\leadsto x \cdot \frac{\color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right) \cdot y + 1 \cdot y}}{y} \]

      3. *-commutative N/A

         \[\leadsto x \cdot \frac{\color{blue}{\left(\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot {y}^{2}\right)} \cdot y + 1 \cdot y}{y} \]

      4. associate-*l* N/A

         \[\leadsto x \cdot \frac{\color{blue}{\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot \left({y}^{2} \cdot y\right)} + 1 \cdot y}{y} \]

      5. pow-plus N/A

         \[\leadsto x \cdot \frac{\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot \color{blue}{{y}^{\left(2 + 1\right)}} + 1 \cdot y}{y} \]

      6. metadata-eval N/A

         \[\leadsto x \cdot \frac{\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot {y}^{\color{blue}{3}} + 1 \cdot y}{y} \]

      7. cube-unmult N/A

         \[\leadsto x \cdot \frac{\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot \color{blue}{\left(y \cdot \left(y \cdot y\right)\right)} + 1 \cdot y}{y} \]

      8. unpow2 N/A

         \[\leadsto x \cdot \frac{\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot \left(y \cdot \color{blue}{{y}^{2}}\right) + 1 \cdot y}{y} \]

      9. *-lft-identity N/A

         \[\leadsto x \cdot \frac{\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot \left(y \cdot {y}^{2}\right) + \color{blue}{y}}{y} \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto x \cdot \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right), y \cdot {y}^{2}, y\right)}}{y} \]

   8. Simplified 64.9%

      \[\leadsto x \cdot \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)}}{y} \]

   9. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + \frac{1}{6} \cdot \left(x \cdot {y}^{2}\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto x + \color{blue}{\left(\frac{1}{6} \cdot x\right) \cdot {y}^{2}} \]

       2. +-commutative N/A

          \[\leadsto \color{blue}{\left(\frac{1}{6} \cdot x\right) \cdot {y}^{2} + x} \]

       3. associate-*r* N/A

          \[\leadsto \color{blue}{\frac{1}{6} \cdot \left(x \cdot {y}^{2}\right)} + x \]

       4. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{6}, x \cdot {y}^{2}, x\right)} \]

       5. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{6}, \color{blue}{{y}^{2} \cdot x}, x\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{6}, \color{blue}{{y}^{2} \cdot x}, x\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{6}, \color{blue}{\left(y \cdot y\right)} \cdot x, x\right) \]

       8. *-lowering-*.f64 57.4

          \[\leadsto \mathsf{fma}\left(0.16666666666666666, \color{blue}{\left(y \cdot y\right)} \cdot x, x\right) \]

   11. Simplified 57.4%

       \[\leadsto \color{blue}{\mathsf{fma}\left(0.16666666666666666, \left(y \cdot y\right) \cdot x, x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 41.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin x \cdot \frac{\sinh y}{y} \leq -0.1:\\ \;\;\;\;-0.16666666666666666 \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.16666666666666666, x \cdot \left(y \cdot y\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 30.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin x \cdot \frac{\sinh y}{y} \leq -0.1:\\ \;\;\;\;-0.16666666666666666 \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (sin x) (/ (sinh y) y)) -0.1)
   (* -0.16666666666666666 (* x (* x x)))
   x))

C

double code(double x, double y) {
	double tmp;
	if ((sin(x) * (sinh(y) / y)) <= -0.1) {
		tmp = -0.16666666666666666 * (x * (x * x));
	} else {
		tmp = x;
	}
	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)) <= (-0.1d0)) then
        tmp = (-0.16666666666666666d0) * (x * (x * x))
    else
        tmp = x
    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)) <= -0.1) {
		tmp = -0.16666666666666666 * (x * (x * x));
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (math.sin(x) * (math.sinh(y) / y)) <= -0.1:
		tmp = -0.16666666666666666 * (x * (x * x))
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(sin(x) * Float64(sinh(y) / y)) <= -0.1)
		tmp = Float64(-0.16666666666666666 * Float64(x * Float64(x * x)));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((sin(x) * (sinh(y) / y)) <= -0.1)
		tmp = -0.16666666666666666 * (x * (x * x));
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Sin[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], -0.1], N[(-0.16666666666666666 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]

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

      \[\leadsto \color{blue}{\sin x} \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 36.3

         \[\leadsto \color{blue}{\sin x} \]

   5. Simplified 36.3%

      \[\leadsto \color{blue}{\sin x} \]

   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. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2} + 1\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{x \cdot \left(\frac{-1}{6} \cdot {x}^{2}\right) + x \cdot 1} \]

      3. *-rgt-identity N/A

         \[\leadsto x \cdot \left(\frac{-1}{6} \cdot {x}^{2}\right) + \color{blue}{x} \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{-1}{6} \cdot {x}^{2}, x\right)} \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{{x}^{2} \cdot \frac{-1}{6}}, x\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\left(x \cdot x\right)} \cdot \frac{-1}{6}, x\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot \left(x \cdot \frac{-1}{6}\right)}, x\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot \left(x \cdot \frac{-1}{6}\right)}, x\right) \]

      9. *-lowering-*.f64 17.7

         \[\leadsto \mathsf{fma}\left(x, x \cdot \color{blue}{\left(x \cdot -0.16666666666666666\right)}, x\right) \]

   8. Simplified 17.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot \left(x \cdot -0.16666666666666666\right), x\right)} \]

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{-1}{6} \cdot {x}^{3}} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\frac{-1}{6} \cdot {x}^{3}} \]

       2. cube-mult N/A

          \[\leadsto \frac{-1}{6} \cdot \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \]

       3. unpow2 N/A

          \[\leadsto \frac{-1}{6} \cdot \left(x \cdot \color{blue}{{x}^{2}}\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \frac{-1}{6} \cdot \color{blue}{\left(x \cdot {x}^{2}\right)} \]

       5. unpow2 N/A

          \[\leadsto \frac{-1}{6} \cdot \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \]

       6. *-lowering-*.f64 17.4

          \[\leadsto -0.16666666666666666 \cdot \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \]

   11. Simplified 17.4%

       \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(x \cdot \left(x \cdot x\right)\right)} \]

   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 y around 0

      \[\leadsto \color{blue}{\sin x} \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 66.4

         \[\leadsto \color{blue}{\sin x} \]

   5. Simplified 66.4%

      \[\leadsto \color{blue}{\sin x} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x} \]
   7. Step-by-step derivation

   8. Simplified 44.4%

      \[\leadsto \color{blue}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 26.9% 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 54.6

      \[\leadsto \color{blue}{\sin x} \]

5. Simplified 54.6%

   \[\leadsto \color{blue}{\sin x} \]

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x} \]
7. Step-by-step derivation

8. Simplified 28.1%

   \[\leadsto \color{blue}{x} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024204 
(FPCore (x y)
  :name "Linear.Quaternion:$ccos from linear-1.19.1.3"
  :precision binary64
  (* (sin x) (/ (sinh y) y)))