Linear.Quaternion:$ccos from linear-1.19.1.3

Percentage Accurate: 100.0% → 100.0%

Time: 15.0s

Alternatives: 24

Speedup: 1.0×

• 50.3% 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: 74.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y}{y}\\ t_1 := \sin x \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;t\_0 \cdot \left(x \cdot \left(-0.16666666666666666 \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{elif}\;t\_1 \leq 1:\\ \;\;\;\;\sin x \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, x \cdot 0.008333333333333333, -0.16666666666666666\right), x \cdot \left(x \cdot x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sinh y) y)) (t_1 (* (sin x) t_0)))
   (if (<= t_1 (- INFINITY))
     (* t_0 (* x (* -0.16666666666666666 (* x x))))
     (if (<= t_1 1.0)
       (*
        (sin x)
        (fma
         y
         (*
          y
          (fma
           (* y y)
           (fma (* y y) 0.0001984126984126984 0.008333333333333333)
           0.16666666666666666))
         1.0))
       (*
        t_0
        (fma
         (fma x (* x 0.008333333333333333) -0.16666666666666666)
         (* x (* x x))
         x))))))

C

double code(double x, double y) {
	double t_0 = sinh(y) / y;
	double t_1 = sin(x) * t_0;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_0 * (x * (-0.16666666666666666 * (x * x)));
	} else if (t_1 <= 1.0) {
		tmp = sin(x) * fma(y, (y * fma((y * y), fma((y * y), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666)), 1.0);
	} else {
		tmp = t_0 * fma(fma(x, (x * 0.008333333333333333), -0.16666666666666666), (x * (x * x)), x);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sinh(y) / y)
	t_1 = Float64(sin(x) * t_0)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(t_0 * Float64(x * Float64(-0.16666666666666666 * Float64(x * x))));
	elseif (t_1 <= 1.0)
		tmp = Float64(sin(x) * fma(y, Float64(y * fma(Float64(y * y), fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666)), 1.0));
	else
		tmp = Float64(t_0 * fma(fma(x, Float64(x * 0.008333333333333333), -0.16666666666666666), Float64(x * Float64(x * x)), x));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[x], $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(t$95$0 * N[(x * N[(-0.16666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1.0], N[(N[Sin[x], $MachinePrecision] * N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(N[(x * N[(x * 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $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 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. lower-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. lower-*.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. lower-*.f64 71.4

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

   5. Applied rewrites 71.4%

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

   6. Taylor expanded in x around inf

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

      1. unpow3 N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 14.3

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

   8. Applied rewrites 14.3%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. distribute-lft-in N/A

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

      2. associate-+r+ N/A

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

      3. *-lft-identity N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. distribute-rgt-in N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. distribute-rgt-out N/A

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

      11. +-commutative N/A

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

   5. Applied rewrites 100.0%

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

   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

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. sub-neg N/A

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

      9. *-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 71.8

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

   5. Applied rewrites 71.8%

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

4. Final simplification 72.6%

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

Alternative 3: 83.7% accurate, 0.4× speedup

Math

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

FPCore

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

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

      1. distribute-lft-in N/A

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

      2. associate-+r+ N/A

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

      3. *-lft-identity N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. distribute-rgt-in N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. distribute-rgt-out N/A

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

      11. +-commutative N/A

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

   5. Applied rewrites 82.8%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. unpow2 N/A

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

      6. unpow3 N/A

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

      7. *-lft-identity N/A

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

      8. lower-fma.f64 N/A

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

   8. Applied rewrites 62.8%

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

   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. lower-*.f64 N/A

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

      5. lower-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. lower-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. lower-*.f64 99.7

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

   5. Applied rewrites 99.7%

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

   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 y around 0

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

      1. distribute-lft-in N/A

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

      2. associate-+r+ N/A

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

      3. *-lft-identity N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. distribute-rgt-in N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. distribute-rgt-out N/A

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

      11. +-commutative N/A

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

   5. Applied rewrites 82.8%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-*l* N/A

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

      4. *-lft-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 60.7

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

   8. Applied rewrites 60.7%

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

   9. Taylor expanded in y around inf

      \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{1}{120}, x \cdot x, \frac{-1}{6}\right) \cdot x, x\right) \cdot \mathsf{fma}\left(y, y \cdot \color{blue}{\left(\frac{1}{5040} \cdot {y}^{4}\right)}, 1\right) \]
   10. Step-by-step derivation

       1. metadata-eval N/A

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

       2. pow-sqr N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f64 60.7

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

   11. Applied rewrites 60.7%

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

4. Final simplification 79.7%

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

Alternative 4: 83.4% accurate, 0.4× speedup

Math

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

FPCore

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

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

      1. distribute-lft-in N/A

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

      2. associate-+r+ N/A

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

      3. *-lft-identity N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. distribute-rgt-in N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. distribute-rgt-out N/A

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

      11. +-commutative N/A

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

   5. Applied rewrites 82.8%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. unpow2 N/A

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

      6. unpow3 N/A

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

      7. *-lft-identity N/A

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

      8. lower-fma.f64 N/A

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

   8. Applied rewrites 62.8%

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

   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. lower-sin.f64 98.6

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

   5. Applied rewrites 98.6%

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

   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 y around 0

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

      1. distribute-lft-in N/A

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

      2. associate-+r+ N/A

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

      3. *-lft-identity N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. distribute-rgt-in N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. distribute-rgt-out N/A

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

      11. +-commutative N/A

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

   5. Applied rewrites 82.8%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-*l* N/A

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

      4. *-lft-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 60.7

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

   8. Applied rewrites 60.7%

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

   9. Taylor expanded in y around inf

      \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{1}{120}, x \cdot x, \frac{-1}{6}\right) \cdot x, x\right) \cdot \mathsf{fma}\left(y, y \cdot \color{blue}{\left(\frac{1}{5040} \cdot {y}^{4}\right)}, 1\right) \]
   10. Step-by-step derivation

       1. metadata-eval N/A

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

       2. pow-sqr N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f64 60.7

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

   11. Applied rewrites 60.7%

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

4. Final simplification 79.2%

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

Alternative 5: 58.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin x \cdot \frac{\sinh y}{y} \leq 0.02:\\ \;\;\;\;x \cdot \left(\mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \left(y \cdot y\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (sin x) (/ (sinh y) y)) 0.02)
   (*
    x
    (*
     (fma x (* x -0.16666666666666666) 1.0)
     (fma (* y y) (fma (* y y) 0.008333333333333333 0.16666666666666666) 1.0)))
   (fma
    x
    (*
     (* y y)
     (fma
      y
      (* y (fma (* y y) 0.0001984126984126984 0.008333333333333333))
      0.16666666666666666))
    x)))

C

double code(double x, double y) {
	double tmp;
	if ((sin(x) * (sinh(y) / y)) <= 0.02) {
		tmp = x * (fma(x, (x * -0.16666666666666666), 1.0) * fma((y * y), fma((y * y), 0.008333333333333333, 0.16666666666666666), 1.0));
	} else {
		tmp = fma(x, ((y * y) * fma(y, (y * fma((y * y), 0.0001984126984126984, 0.008333333333333333)), 0.16666666666666666)), x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(sin(x) * Float64(sinh(y) / y)) <= 0.02)
		tmp = Float64(x * Float64(fma(x, Float64(x * -0.16666666666666666), 1.0) * fma(Float64(y * y), fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666), 1.0)));
	else
		tmp = fma(x, Float64(Float64(y * y) * fma(y, Float64(y * fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333)), 0.16666666666666666)), x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Sin[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], 0.02], N[(x * N[(N[(x * N[(x * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < 0.0200000000000000004

   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. lower-*.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. lower-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. Applied rewrites 91.0%

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

   6. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt1-in N/A

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

      6. +-commutative N/A

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

      7. lower-*.f64 N/A

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

   8. Applied rewrites 67.8%

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

   if 0.0200000000000000004 < (*.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

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

      1. distribute-lft-in N/A

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

      2. associate-+r+ N/A

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

      3. *-lft-identity N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. distribute-rgt-in N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. distribute-rgt-out N/A

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

      11. +-commutative N/A

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

   5. Applied rewrites 87.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 15.2%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 45.9%

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

Alternative 6: 51.6% accurate, 0.8× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	double tmp;
	if ((sin(x) * (sinh(y) / y)) <= -0.1) {
		tmp = 0.16666666666666666 * (y * (y * fma(x, (-0.16666666666666666 * (x * x)), x)));
	} else {
		tmp = fma(x, ((y * y) * fma(y, (y * fma((y * y), 0.0001984126984126984, 0.008333333333333333)), 0.16666666666666666)), 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(y * Float64(y * fma(x, Float64(-0.16666666666666666 * Float64(x * x)), x))));
	else
		tmp = fma(x, Float64(Float64(y * y) * fma(y, Float64(y * fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333)), 0.16666666666666666)), 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[(y * N[(y * N[(x * N[(-0.16666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $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 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. lower-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. lower-*.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. lower-*.f64 50.6

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

   5. Applied rewrites 50.6%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 30.1

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

   8. Applied rewrites 30.1%

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

   9. Taylor expanded in y around inf

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

       1. lower-*.f64 N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. +-commutative N/A

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

       7. unpow3 N/A

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

       8. unpow2 N/A

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

       9. associate-*r* N/A

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

       10. *-commutative N/A

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

       11. lower-fma.f64 N/A

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

       12. lower-*.f64 N/A

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

       13. unpow2 N/A

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

       14. lower-*.f64 19.6

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

   11. Applied rewrites 19.6%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. distribute-lft-in N/A

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

      2. associate-+r+ N/A

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

      3. *-lft-identity N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. distribute-rgt-in N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. distribute-rgt-out N/A

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

      11. +-commutative N/A

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

   5. Applied rewrites 92.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 48.8%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 66.8%

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

Alternative 7: 54.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin x \cdot \frac{\sinh y}{y} \leq 0.02:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, x \cdot \left(x \cdot x\right), x\right) \cdot \mathsf{fma}\left(y, y \cdot 0.16666666666666666, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, x \cdot \left(\left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (sin x) (/ (sinh y) y)) 0.02)
   (*
    (fma -0.16666666666666666 (* x (* x x)) x)
    (fma y (* y 0.16666666666666666) 1.0))
   (fma (* y y) (* x (* (* y y) (* (* y y) 0.0001984126984126984))) x)))

C

double code(double x, double y) {
	double tmp;
	if ((sin(x) * (sinh(y) / y)) <= 0.02) {
		tmp = fma(-0.16666666666666666, (x * (x * x)), x) * fma(y, (y * 0.16666666666666666), 1.0);
	} else {
		tmp = fma((y * y), (x * ((y * y) * ((y * y) * 0.0001984126984126984))), x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(sin(x) * Float64(sinh(y) / y)) <= 0.02)
		tmp = Float64(fma(-0.16666666666666666, Float64(x * Float64(x * x)), x) * fma(y, Float64(y * 0.16666666666666666), 1.0));
	else
		tmp = fma(Float64(y * y), Float64(x * Float64(Float64(y * y) * Float64(Float64(y * y) * 0.0001984126984126984))), x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Sin[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], 0.02], N[(N[(-0.16666666666666666 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] * N[(y * N[(y * 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(y * y), $MachinePrecision] * N[(x * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < 0.0200000000000000004

   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. lower-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. lower-*.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. lower-*.f64 72.9

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

   5. Applied rewrites 72.9%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 61.8

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

   8. Applied rewrites 61.8%

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

   if 0.0200000000000000004 < (*.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

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

      1. distribute-lft-in N/A

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

      2. associate-+r+ N/A

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

      3. *-lft-identity N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. distribute-rgt-in N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. distribute-rgt-out N/A

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

      11. +-commutative N/A

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

   5. Applied rewrites 87.2%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 45.0

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

   8. Applied rewrites 45.0%

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

   9. Taylor expanded in y around inf

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

       1. metadata-eval N/A

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

       2. pow-sqr N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f64 45.0

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

   11. Applied rewrites 45.0%

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

4. Final simplification 54.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin x \cdot \frac{\sinh y}{y} \leq 0.02:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, x \cdot \left(x \cdot x\right), x\right) \cdot \mathsf{fma}\left(y, y \cdot 0.16666666666666666, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, x \cdot \left(\left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 54.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin x \cdot \frac{\sinh y}{y} \leq 0.02:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, x \cdot \left(x \cdot x\right), x\right) \cdot \mathsf{fma}\left(y, y \cdot 0.16666666666666666, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, y \cdot \left(y \cdot \left(x \cdot \left(\left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (sin x) (/ (sinh y) y)) 0.02)
   (*
    (fma -0.16666666666666666 (* x (* x x)) x)
    (fma y (* y 0.16666666666666666) 1.0))
   (fma (* y y) (* y (* y (* x (* (* y y) 0.0001984126984126984)))) x)))

C

double code(double x, double y) {
	double tmp;
	if ((sin(x) * (sinh(y) / y)) <= 0.02) {
		tmp = fma(-0.16666666666666666, (x * (x * x)), x) * fma(y, (y * 0.16666666666666666), 1.0);
	} else {
		tmp = fma((y * y), (y * (y * (x * ((y * y) * 0.0001984126984126984)))), x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(sin(x) * Float64(sinh(y) / y)) <= 0.02)
		tmp = Float64(fma(-0.16666666666666666, Float64(x * Float64(x * x)), x) * fma(y, Float64(y * 0.16666666666666666), 1.0));
	else
		tmp = fma(Float64(y * y), Float64(y * Float64(y * Float64(x * Float64(Float64(y * y) * 0.0001984126984126984)))), x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Sin[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], 0.02], N[(N[(-0.16666666666666666 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] * N[(y * N[(y * 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * N[(x * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < 0.0200000000000000004

   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. lower-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. lower-*.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. lower-*.f64 72.9

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

   5. Applied rewrites 72.9%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 61.8

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

   8. Applied rewrites 61.8%

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

   if 0.0200000000000000004 < (*.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

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

      1. distribute-lft-in N/A

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

      2. associate-+r+ N/A

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

      3. *-lft-identity N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. distribute-rgt-in N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. distribute-rgt-out N/A

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

      11. +-commutative N/A

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

   5. Applied rewrites 87.2%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 45.0

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

   8. Applied rewrites 45.0%

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

   9. Taylor expanded in y around inf

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

       1. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(y \cdot y, \color{blue}{\left(\frac{1}{5040} \cdot x\right) \cdot {y}^{4}}, x\right) \]

       2. metadata-eval N/A

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

       3. pow-sqr N/A

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

       4. associate-*l* N/A

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

       5. associate-*r* N/A

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

       6. *-commutative N/A

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

       7. unpow2 N/A

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

       8. associate-*l* N/A

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

       9. lower-*.f64 N/A

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

       10. lower-*.f64 N/A

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

       11. *-commutative N/A

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

       12. associate-*l* N/A

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

       13. *-commutative N/A

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

       14. lower-*.f64 N/A

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

       15. *-commutative N/A

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

       16. lower-*.f64 N/A

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

       17. unpow2 N/A

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

       18. lower-*.f64 43.2

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

   11. Applied rewrites 43.2%

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

Alternative 9: 49.0% accurate, 0.9× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	double tmp;
	if ((sin(x) * (sinh(y) / y)) <= -0.1) {
		tmp = 0.16666666666666666 * (y * (y * fma(x, (-0.16666666666666666 * (x * x)), x)));
	} else {
		tmp = x * (fma(0.16666666666666666, (y * (y * y)), y) / y);
	}
	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(y * Float64(y * fma(x, Float64(-0.16666666666666666 * Float64(x * x)), x))));
	else
		tmp = Float64(x * Float64(fma(0.16666666666666666, Float64(y * Float64(y * y)), y) / y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Sin[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], -0.1], N[(0.16666666666666666 * N[(y * N[(y * N[(x * N[(-0.16666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(0.16666666666666666 * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] / y), $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 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. lower-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. lower-*.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. lower-*.f64 50.6

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

   5. Applied rewrites 50.6%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 30.1

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

   8. Applied rewrites 30.1%

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

   9. Taylor expanded in y around inf

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

       1. lower-*.f64 N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. +-commutative N/A

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

       7. unpow3 N/A

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

       8. unpow2 N/A

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

       9. associate-*r* N/A

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

       10. *-commutative N/A

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

       11. lower-fma.f64 N/A

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

       12. lower-*.f64 N/A

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

       13. unpow2 N/A

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

       14. lower-*.f64 19.6

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

   11. Applied rewrites 19.6%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-rgt-identity N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 87.7

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

   5. Applied rewrites 87.7%

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

   6. Taylor expanded in x around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 63.8

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

   8. Applied rewrites 63.8%

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

Alternative 10: 49.1% accurate, 0.9× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	double tmp;
	if ((sin(x) * (sinh(y) / y)) <= -0.1) {
		tmp = 0.16666666666666666 * (y * (y * fma(x, (-0.16666666666666666 * (x * x)), x)));
	} else {
		tmp = fma((y * y), (x * fma(y, (y * 0.008333333333333333), 0.16666666666666666)), 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(y * Float64(y * fma(x, Float64(-0.16666666666666666 * Float64(x * x)), x))));
	else
		tmp = fma(Float64(y * y), Float64(x * fma(y, Float64(y * 0.008333333333333333), 0.16666666666666666)), 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[(y * N[(y * N[(x * N[(-0.16666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * y), $MachinePrecision] * N[(x * N[(y * N[(y * 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $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 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. lower-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. lower-*.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. lower-*.f64 50.6

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

   5. Applied rewrites 50.6%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 30.1

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

   8. Applied rewrites 30.1%

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

   9. Taylor expanded in y around inf

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

       1. lower-*.f64 N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. +-commutative N/A

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

       7. unpow3 N/A

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

       8. unpow2 N/A

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

       9. associate-*r* N/A

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

       10. *-commutative N/A

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

       11. lower-fma.f64 N/A

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

       12. lower-*.f64 N/A

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

       13. unpow2 N/A

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

       14. lower-*.f64 19.6

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

   11. Applied rewrites 19.6%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. distribute-lft-in N/A

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

      2. associate-+r+ N/A

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

      3. *-lft-identity N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. distribute-rgt-in N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. distribute-rgt-out N/A

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

      11. +-commutative N/A

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

   5. Applied rewrites 92.4%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 66.3

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

   8. Applied rewrites 66.3%

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

   9. Taylor expanded in y around 0

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

       1. +-commutative N/A

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

   11. Applied rewrites 63.0%

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

4. Final simplification 49.4%

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

Alternative 11: 45.3% accurate, 0.9× speedup

Math

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

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 (fma y (* y 0.008333333333333333) 0.16666666666666666))
    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 * fma(y, (y * 0.008333333333333333), 0.16666666666666666)), x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(sin(x) * Float64(sinh(y) / y)) <= -0.1)
		tmp = Float64(Float64(x * Float64(x * x)) * fma(Float64(y * y), -0.027777777777777776, -0.16666666666666666));
	else
		tmp = fma(Float64(y * y), Float64(x * fma(y, Float64(y * 0.008333333333333333), 0.16666666666666666)), 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[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * -0.027777777777777776 + -0.16666666666666666), $MachinePrecision]), $MachinePrecision], N[(N[(y * y), $MachinePrecision] * N[(x * N[(y * N[(y * 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $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 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. lower-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. lower-*.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. lower-*.f64 50.6

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

   5. Applied rewrites 50.6%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 30.1

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

   8. Applied rewrites 30.1%

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

   9. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. lower-*.f64 N/A

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

       4. distribute-rgt-in N/A

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

       5. metadata-eval N/A

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

       6. +-commutative N/A

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

       7. *-commutative N/A

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

       8. associate-*l* N/A

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

       9. metadata-eval N/A

          \[\leadsto \left({y}^{2} \cdot \color{blue}{\frac{-1}{36}} + \frac{-1}{6}\right) \cdot {x}^{3} \]

       10. metadata-eval N/A

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

       11. lower-fma.f64 N/A

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

       12. unpow2 N/A

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

       13. lower-*.f64 N/A

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

       14. metadata-eval N/A

           \[\leadsto \mathsf{fma}\left(y \cdot y, \color{blue}{\frac{-1}{36}}, \frac{-1}{6}\right) \cdot {x}^{3} \]

       15. cube-mult N/A

           \[\leadsto \mathsf{fma}\left(y \cdot y, \frac{-1}{36}, \frac{-1}{6}\right) \cdot \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \]

       16. unpow2 N/A

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

       17. lower-*.f64 N/A

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

       18. unpow2 N/A

           \[\leadsto \mathsf{fma}\left(y \cdot y, \frac{-1}{36}, \frac{-1}{6}\right) \cdot \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \]

       19. lower-*.f64 10.9

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

   11. Applied rewrites 10.9%

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

   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

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

      1. distribute-lft-in N/A

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

      2. associate-+r+ N/A

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

      3. *-lft-identity N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. distribute-rgt-in N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. distribute-rgt-out N/A

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

      11. +-commutative N/A

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

   5. Applied rewrites 92.4%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 66.3

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

   8. Applied rewrites 66.3%

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

   9. Taylor expanded in y around 0

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

       1. +-commutative N/A

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

   11. Applied rewrites 63.0%

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

4. Final simplification 46.7%

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

Alternative 12: 43.0% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (sin x) (/ (sinh y) y)) -0.1)
   (* (* x (* x x)) (fma (* y y) -0.027777777777777776 -0.16666666666666666))
   (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 = (x * (x * x)) * fma((y * y), -0.027777777777777776, -0.16666666666666666);
	} 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(x * Float64(x * x)) * fma(Float64(y * y), -0.027777777777777776, -0.16666666666666666));
	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[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * -0.027777777777777776 + -0.16666666666666666), $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 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. lower-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. lower-*.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. lower-*.f64 50.6

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

   5. Applied rewrites 50.6%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 30.1

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

   8. Applied rewrites 30.1%

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

   9. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. lower-*.f64 N/A

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

       4. distribute-rgt-in N/A

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

       5. metadata-eval N/A

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

       6. +-commutative N/A

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

       7. *-commutative N/A

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

       8. associate-*l* N/A

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

       9. metadata-eval N/A

          \[\leadsto \left({y}^{2} \cdot \color{blue}{\frac{-1}{36}} + \frac{-1}{6}\right) \cdot {x}^{3} \]

       10. metadata-eval N/A

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

       11. lower-fma.f64 N/A

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

       12. unpow2 N/A

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

       13. lower-*.f64 N/A

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

       14. metadata-eval N/A

           \[\leadsto \mathsf{fma}\left(y \cdot y, \color{blue}{\frac{-1}{36}}, \frac{-1}{6}\right) \cdot {x}^{3} \]

       15. cube-mult N/A

           \[\leadsto \mathsf{fma}\left(y \cdot y, \frac{-1}{36}, \frac{-1}{6}\right) \cdot \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \]

       16. unpow2 N/A

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

       17. lower-*.f64 N/A

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

       18. unpow2 N/A

           \[\leadsto \mathsf{fma}\left(y \cdot y, \frac{-1}{36}, \frac{-1}{6}\right) \cdot \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \]

       19. lower-*.f64 10.9

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

   11. Applied rewrites 10.9%

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

   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

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

      1. distribute-lft-in N/A

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

      2. associate-+r+ N/A

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

      3. *-lft-identity N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. distribute-rgt-in N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. distribute-rgt-out N/A

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

      11. +-commutative N/A

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

   5. Applied rewrites 92.4%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 66.3

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

   8. Applied rewrites 66.3%

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

   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. lower-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. lower-*.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. lower-*.f64 61.1

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

   11. Applied rewrites 61.1%

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

4. Final simplification 45.4%

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

Alternative 13: 60.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin x \leq 5 \cdot 10^{-17}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, x \cdot \left(x \cdot x\right), x\right) \cdot \frac{\mathsf{fma}\left(y \cdot y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, x \cdot \mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right), x\right) \cdot \mathsf{fma}\left(y, y \cdot \left(\left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sin x) 5e-17)
   (*
    (fma -0.16666666666666666 (* x (* x x)) x)
    (/
     (fma
      (* y y)
      (*
       y
       (fma
        (* y y)
        (fma (* y y) 0.0001984126984126984 0.008333333333333333)
        0.16666666666666666))
      y)
     y))
   (*
    (fma
     (* x x)
     (* x (fma 0.008333333333333333 (* x x) -0.16666666666666666))
     x)
    (fma y (* y (* (* y y) (* (* y y) 0.0001984126984126984))) 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (sin(x) <= 5e-17) {
		tmp = fma(-0.16666666666666666, (x * (x * x)), x) * (fma((y * y), (y * fma((y * y), fma((y * y), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666)), y) / y);
	} else {
		tmp = fma((x * x), (x * fma(0.008333333333333333, (x * x), -0.16666666666666666)), x) * fma(y, (y * ((y * y) * ((y * y) * 0.0001984126984126984))), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (sin(x) <= 5e-17)
		tmp = Float64(fma(-0.16666666666666666, Float64(x * Float64(x * x)), x) * Float64(fma(Float64(y * y), Float64(y * fma(Float64(y * y), fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666)), y) / y));
	else
		tmp = Float64(fma(Float64(x * x), Float64(x * fma(0.008333333333333333, Float64(x * x), -0.16666666666666666)), x) * fma(y, Float64(y * Float64(Float64(y * y) * Float64(Float64(y * y) * 0.0001984126984126984))), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[Sin[x], $MachinePrecision], 5e-17], N[(N[(-0.16666666666666666 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] * N[(N[(N[(y * y), $MachinePrecision] * N[(y * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * x), $MachinePrecision] * N[(x * N[(0.008333333333333333 * N[(x * x), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] * N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 x) < 4.9999999999999999e-17

   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. lower-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. lower-*.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. lower-*.f64 78.6

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

   5. Applied rewrites 78.6%

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

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

   8. Applied rewrites 73.9%

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

   if 4.9999999999999999e-17 < (sin.f64 x)

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. distribute-lft-in N/A

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

      2. associate-+r+ N/A

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

      3. *-lft-identity N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. distribute-rgt-in N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. distribute-rgt-out N/A

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

      11. +-commutative N/A

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

   5. Applied rewrites 91.4%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-*l* N/A

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

      4. *-lft-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 26.3

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

   8. Applied rewrites 26.3%

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

   9. Taylor expanded in y around inf

      \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{1}{120}, x \cdot x, \frac{-1}{6}\right) \cdot x, x\right) \cdot \mathsf{fma}\left(y, y \cdot \color{blue}{\left(\frac{1}{5040} \cdot {y}^{4}\right)}, 1\right) \]
   10. Step-by-step derivation

       1. metadata-eval N/A

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

       2. pow-sqr N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f64 26.3

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

   11. Applied rewrites 26.3%

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

4. Final simplification 61.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin x \leq 5 \cdot 10^{-17}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, x \cdot \left(x \cdot x\right), x\right) \cdot \frac{\mathsf{fma}\left(y \cdot y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, x \cdot \mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right), x\right) \cdot \mathsf{fma}\left(y, y \cdot \left(\left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 59.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin x \leq -0.005:\\ \;\;\;\;0.16666666666666666 \cdot \left(y \cdot \left(y \cdot \mathsf{fma}\left(x, -0.16666666666666666 \cdot \left(x \cdot x\right), x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right) \cdot \mathsf{fma}\left(x \cdot x, x \cdot \mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sin x) -0.005)
   (*
    0.16666666666666666
    (* y (* y (fma x (* -0.16666666666666666 (* x x)) x))))
   (*
    (fma
     y
     (*
      y
      (fma
       (* y y)
       (fma (* y y) 0.0001984126984126984 0.008333333333333333)
       0.16666666666666666))
     1.0)
    (fma
     (* x x)
     (* x (fma 0.008333333333333333 (* x x) -0.16666666666666666))
     x))))

C

double code(double x, double y) {
	double tmp;
	if (sin(x) <= -0.005) {
		tmp = 0.16666666666666666 * (y * (y * fma(x, (-0.16666666666666666 * (x * x)), x)));
	} else {
		tmp = fma(y, (y * fma((y * y), fma((y * y), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666)), 1.0) * fma((x * x), (x * fma(0.008333333333333333, (x * x), -0.16666666666666666)), x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (sin(x) <= -0.005)
		tmp = Float64(0.16666666666666666 * Float64(y * Float64(y * fma(x, Float64(-0.16666666666666666 * Float64(x * x)), x))));
	else
		tmp = Float64(fma(y, Float64(y * fma(Float64(y * y), fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666)), 1.0) * fma(Float64(x * x), Float64(x * fma(0.008333333333333333, Float64(x * x), -0.16666666666666666)), x));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[Sin[x], $MachinePrecision], -0.005], N[(0.16666666666666666 * N[(y * N[(y * N[(x * N[(-0.16666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * N[(0.008333333333333333 * N[(x * x), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 x) < -0.0050000000000000001

   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. lower-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. lower-*.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. lower-*.f64 17.3

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

   5. Applied rewrites 17.3%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 17.4

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

   8. Applied rewrites 17.4%

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

   9. Taylor expanded in y around inf

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

       1. lower-*.f64 N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. +-commutative N/A

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

       7. unpow3 N/A

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

       8. unpow2 N/A

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

       9. associate-*r* N/A

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

       10. *-commutative N/A

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

       11. lower-fma.f64 N/A

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

       12. lower-*.f64 N/A

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

       13. unpow2 N/A

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

       14. lower-*.f64 17.3

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

   11. Applied rewrites 17.3%

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

   if -0.0050000000000000001 < (sin.f64 x)

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. distribute-lft-in N/A

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

      2. associate-+r+ N/A

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

      3. *-lft-identity N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. distribute-rgt-in N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. distribute-rgt-out N/A

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

      11. +-commutative N/A

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

   5. Applied rewrites 91.2%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-*l* N/A

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

      4. *-lft-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 70.1

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

   8. Applied rewrites 70.1%

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

4. Final simplification 60.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin x \leq -0.005:\\ \;\;\;\;0.16666666666666666 \cdot \left(y \cdot \left(y \cdot \mathsf{fma}\left(x, -0.16666666666666666 \cdot \left(x \cdot x\right), x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right) \cdot \mathsf{fma}\left(x \cdot x, x \cdot \mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 59.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin x \leq 5 \cdot 10^{-17}:\\ \;\;\;\;\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right) \cdot \mathsf{fma}\left(x, -0.16666666666666666 \cdot \left(x \cdot x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, x \cdot \mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right), x\right) \cdot \mathsf{fma}\left(y, y \cdot \left(\left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sin x) 5e-17)
   (*
    (fma
     y
     (*
      y
      (fma
       (* y y)
       (fma (* y y) 0.0001984126984126984 0.008333333333333333)
       0.16666666666666666))
     1.0)
    (fma x (* -0.16666666666666666 (* x x)) x))
   (*
    (fma
     (* x x)
     (* x (fma 0.008333333333333333 (* x x) -0.16666666666666666))
     x)
    (fma y (* y (* (* y y) (* (* y y) 0.0001984126984126984))) 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (sin(x) <= 5e-17) {
		tmp = fma(y, (y * fma((y * y), fma((y * y), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666)), 1.0) * fma(x, (-0.16666666666666666 * (x * x)), x);
	} else {
		tmp = fma((x * x), (x * fma(0.008333333333333333, (x * x), -0.16666666666666666)), x) * fma(y, (y * ((y * y) * ((y * y) * 0.0001984126984126984))), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (sin(x) <= 5e-17)
		tmp = Float64(fma(y, Float64(y * fma(Float64(y * y), fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666)), 1.0) * fma(x, Float64(-0.16666666666666666 * Float64(x * x)), x));
	else
		tmp = Float64(fma(Float64(x * x), Float64(x * fma(0.008333333333333333, Float64(x * x), -0.16666666666666666)), x) * fma(y, Float64(y * Float64(Float64(y * y) * Float64(Float64(y * y) * 0.0001984126984126984))), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[Sin[x], $MachinePrecision], 5e-17], N[(N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x * N[(-0.16666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * x), $MachinePrecision] * N[(x * N[(0.008333333333333333 * N[(x * x), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] * N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 x) < 4.9999999999999999e-17

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. distribute-lft-in N/A

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

      2. associate-+r+ N/A

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

      3. *-lft-identity N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. distribute-rgt-in N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. distribute-rgt-out N/A

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

      11. +-commutative N/A

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

   5. Applied rewrites 90.8%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 71.9

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

   8. Applied rewrites 71.9%

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

   if 4.9999999999999999e-17 < (sin.f64 x)

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. distribute-lft-in N/A

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

      2. associate-+r+ N/A

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

      3. *-lft-identity N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. distribute-rgt-in N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. distribute-rgt-out N/A

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

      11. +-commutative N/A

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

   5. Applied rewrites 91.4%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-*l* N/A

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

      4. *-lft-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 26.3

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

   8. Applied rewrites 26.3%

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

   9. Taylor expanded in y around inf

      \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{1}{120}, x \cdot x, \frac{-1}{6}\right) \cdot x, x\right) \cdot \mathsf{fma}\left(y, y \cdot \color{blue}{\left(\frac{1}{5040} \cdot {y}^{4}\right)}, 1\right) \]
   10. Step-by-step derivation

       1. metadata-eval N/A

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

       2. pow-sqr N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{1}{120}, x \cdot x, \frac{-1}{6}\right) \cdot x, x\right) \cdot \mathsf{fma}\left(y, y \cdot \left(\color{blue}{\left(y \cdot y\right)} \cdot \left(\frac{1}{5040} \cdot {y}^{2}\right)\right), 1\right) \]

       7. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{1}{120}, x \cdot x, \frac{-1}{6}\right) \cdot x, x\right) \cdot \mathsf{fma}\left(y, y \cdot \left(\color{blue}{\left(y \cdot y\right)} \cdot \left(\frac{1}{5040} \cdot {y}^{2}\right)\right), 1\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{1}{120}, x \cdot x, \frac{-1}{6}\right) \cdot x, x\right) \cdot \mathsf{fma}\left(y, y \cdot \left(\left(y \cdot y\right) \cdot \color{blue}{\left({y}^{2} \cdot \frac{1}{5040}\right)}\right), 1\right) \]

       9. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{1}{120}, x \cdot x, \frac{-1}{6}\right) \cdot x, x\right) \cdot \mathsf{fma}\left(y, y \cdot \left(\left(y \cdot y\right) \cdot \color{blue}{\left({y}^{2} \cdot \frac{1}{5040}\right)}\right), 1\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{1}{120}, x \cdot x, \frac{-1}{6}\right) \cdot x, x\right) \cdot \mathsf{fma}\left(y, y \cdot \left(\left(y \cdot y\right) \cdot \left(\color{blue}{\left(y \cdot y\right)} \cdot \frac{1}{5040}\right)\right), 1\right) \]

       11. lower-*.f64 26.3

           \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right) \cdot x, x\right) \cdot \mathsf{fma}\left(y, y \cdot \left(\left(y \cdot y\right) \cdot \left(\color{blue}{\left(y \cdot y\right)} \cdot 0.0001984126984126984\right)\right), 1\right) \]

   11. Applied rewrites 26.3%

       \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right) \cdot x, x\right) \cdot \mathsf{fma}\left(y, y \cdot \color{blue}{\left(\left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right)}, 1\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin x \leq 5 \cdot 10^{-17}:\\ \;\;\;\;\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right) \cdot \mathsf{fma}\left(x, -0.16666666666666666 \cdot \left(x \cdot x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, x \cdot \mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right), x\right) \cdot \mathsf{fma}\left(y, y \cdot \left(\left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 59.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin x \leq 0.005:\\ \;\;\;\;\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right) \cdot \mathsf{fma}\left(x, -0.16666666666666666 \cdot \left(x \cdot x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right) \cdot \mathsf{fma}\left(x \cdot x, x \cdot \mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sin x) 0.005)
   (*
    (fma
     y
     (*
      y
      (fma
       (* y y)
       (fma (* y y) 0.0001984126984126984 0.008333333333333333)
       0.16666666666666666))
     1.0)
    (fma x (* -0.16666666666666666 (* x x)) x))
   (*
    (fma (* y y) (fma y (* y 0.008333333333333333) 0.16666666666666666) 1.0)
    (fma
     (* x x)
     (* x (fma 0.008333333333333333 (* x x) -0.16666666666666666))
     x))))

C

double code(double x, double y) {
	double tmp;
	if (sin(x) <= 0.005) {
		tmp = fma(y, (y * fma((y * y), fma((y * y), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666)), 1.0) * fma(x, (-0.16666666666666666 * (x * x)), x);
	} else {
		tmp = fma((y * y), fma(y, (y * 0.008333333333333333), 0.16666666666666666), 1.0) * fma((x * x), (x * fma(0.008333333333333333, (x * x), -0.16666666666666666)), x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (sin(x) <= 0.005)
		tmp = Float64(fma(y, Float64(y * fma(Float64(y * y), fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666)), 1.0) * fma(x, Float64(-0.16666666666666666 * Float64(x * x)), x));
	else
		tmp = Float64(fma(Float64(y * y), fma(y, Float64(y * 0.008333333333333333), 0.16666666666666666), 1.0) * fma(Float64(x * x), Float64(x * fma(0.008333333333333333, Float64(x * x), -0.16666666666666666)), x));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[Sin[x], $MachinePrecision], 0.005], N[(N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x * N[(-0.16666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * N[(0.008333333333333333 * N[(x * x), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 x) < 0.0050000000000000001

   1. Initial program 100.0%

      \[\sin x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\sin x + {y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x + {y}^{2} \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{120} \cdot \sin x\right)\right)} \]
   4. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \sin x + \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x\right) + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{120} \cdot \sin x\right)\right)\right)} \]

      2. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\sin x + {y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x\right)\right) + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{120} \cdot \sin x\right)\right)} \]

      3. *-lft-identity N/A

         \[\leadsto \left(\color{blue}{1 \cdot \sin x} + {y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x\right)\right) + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{120} \cdot \sin x\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \left(1 \cdot \sin x + \color{blue}{\left({y}^{2} \cdot \frac{1}{6}\right) \cdot \sin x}\right) + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{120} \cdot \sin x\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \left(1 \cdot \sin x + \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right)} \cdot \sin x\right) + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{120} \cdot \sin x\right)\right) \]

      6. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\sin x \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)} + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{120} \cdot \sin x\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \sin x \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right) + \color{blue}{\left({y}^{2} \cdot {y}^{2}\right) \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{120} \cdot \sin x\right)} \]

      8. *-commutative N/A

         \[\leadsto \sin x \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right) + \color{blue}{\left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{120} \cdot \sin x\right) \cdot \left({y}^{2} \cdot {y}^{2}\right)} \]

      9. associate-*r* N/A

         \[\leadsto \sin x \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right) + \left(\color{blue}{\left(\frac{1}{5040} \cdot {y}^{2}\right) \cdot \sin x} + \frac{1}{120} \cdot \sin x\right) \cdot \left({y}^{2} \cdot {y}^{2}\right) \]

      10. distribute-rgt-out N/A

          \[\leadsto \sin x \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right) + \color{blue}{\left(\sin x \cdot \left(\frac{1}{5040} \cdot {y}^{2} + \frac{1}{120}\right)\right)} \cdot \left({y}^{2} \cdot {y}^{2}\right) \]

      11. +-commutative N/A

          \[\leadsto \sin x \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right) + \left(\sin x \cdot \color{blue}{\left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)}\right) \cdot \left({y}^{2} \cdot {y}^{2}\right) \]

   5. Applied rewrites 90.8%

      \[\leadsto \color{blue}{\sin x \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)} \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(x \cdot \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2} + 1\right)}\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{\left(x \cdot \left(\frac{-1}{6} \cdot {x}^{2}\right) + x \cdot 1\right)} \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      3. *-rgt-identity N/A

         \[\leadsto \left(x \cdot \left(\frac{-1}{6} \cdot {x}^{2}\right) + \color{blue}{x}\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{-1}{6} \cdot {x}^{2}, x\right)} \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{-1}{6} \cdot {x}^{2}}, x\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(x, \frac{-1}{6} \cdot \color{blue}{\left(x \cdot x\right)}, x\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      7. lower-*.f64 71.9

         \[\leadsto \mathsf{fma}\left(x, -0.16666666666666666 \cdot \color{blue}{\left(x \cdot x\right)}, x\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right) \]

   8. Applied rewrites 71.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, -0.16666666666666666 \cdot \left(x \cdot x\right), x\right)} \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right) \]

   if 0.0050000000000000001 < (sin.f64 x)

   1. Initial program 100.0%

      \[\sin x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\sin x + {y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{6} \cdot \sin x\right)} \]
   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. lower-*.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. lower-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. Applied rewrites 91.4%

      \[\leadsto \color{blue}{\sin x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right)\right)} \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) + 1\right)}\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(\left({x}^{2} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right) \cdot x + 1 \cdot x\right)} \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      3. associate-*l* N/A

         \[\leadsto \left(\color{blue}{{x}^{2} \cdot \left(\left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) \cdot x\right)} + 1 \cdot x\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      4. *-lft-identity N/A

         \[\leadsto \left({x}^{2} \cdot \left(\left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) \cdot x\right) + \color{blue}{x}\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{2}, \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) \cdot x, x\right)} \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) \cdot x, x\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) \cdot x, x\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{\left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) \cdot x}, x\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{\left(\frac{1}{120} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)} \cdot x, x\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \left(\frac{1}{120} \cdot {x}^{2} + \color{blue}{\frac{-1}{6}}\right) \cdot x, x\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(\frac{1}{120}, {x}^{2}, \frac{-1}{6}\right)} \cdot x, x\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{1}{120}, \color{blue}{x \cdot x}, \frac{-1}{6}\right) \cdot x, x\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      13. lower-*.f64 26.3

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.008333333333333333, \color{blue}{x \cdot x}, -0.16666666666666666\right) \cdot x, x\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right) \]

   8. Applied rewrites 26.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right) \cdot x, x\right)} \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin x \leq 0.005:\\ \;\;\;\;\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right) \cdot \mathsf{fma}\left(x, -0.16666666666666666 \cdot \left(x \cdot x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right) \cdot \mathsf{fma}\left(x \cdot x, x \cdot \mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 59.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin x \leq 0.005:\\ \;\;\;\;\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right) \cdot \mathsf{fma}\left(x, -0.16666666666666666 \cdot \left(x \cdot x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right) \cdot \mathsf{fma}\left(x \cdot x, x \cdot \left(\left(x \cdot x\right) \cdot 0.008333333333333333\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sin x) 0.005)
   (*
    (fma
     y
     (*
      y
      (fma
       (* y y)
       (fma (* y y) 0.0001984126984126984 0.008333333333333333)
       0.16666666666666666))
     1.0)
    (fma x (* -0.16666666666666666 (* x x)) x))
   (*
    (fma (* y y) (fma y (* y 0.008333333333333333) 0.16666666666666666) 1.0)
    (fma (* x x) (* x (* (* x x) 0.008333333333333333)) x))))

C

double code(double x, double y) {
	double tmp;
	if (sin(x) <= 0.005) {
		tmp = fma(y, (y * fma((y * y), fma((y * y), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666)), 1.0) * fma(x, (-0.16666666666666666 * (x * x)), x);
	} else {
		tmp = fma((y * y), fma(y, (y * 0.008333333333333333), 0.16666666666666666), 1.0) * fma((x * x), (x * ((x * x) * 0.008333333333333333)), x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (sin(x) <= 0.005)
		tmp = Float64(fma(y, Float64(y * fma(Float64(y * y), fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666)), 1.0) * fma(x, Float64(-0.16666666666666666 * Float64(x * x)), x));
	else
		tmp = Float64(fma(Float64(y * y), fma(y, Float64(y * 0.008333333333333333), 0.16666666666666666), 1.0) * fma(Float64(x * x), Float64(x * Float64(Float64(x * x) * 0.008333333333333333)), x));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[Sin[x], $MachinePrecision], 0.005], N[(N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x * N[(-0.16666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * N[(N[(x * x), $MachinePrecision] * 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 x) < 0.0050000000000000001

   1. Initial program 100.0%

      \[\sin x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\sin x + {y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x + {y}^{2} \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{120} \cdot \sin x\right)\right)} \]
   4. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \sin x + \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x\right) + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{120} \cdot \sin x\right)\right)\right)} \]

      2. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\sin x + {y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x\right)\right) + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{120} \cdot \sin x\right)\right)} \]

      3. *-lft-identity N/A

         \[\leadsto \left(\color{blue}{1 \cdot \sin x} + {y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x\right)\right) + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{120} \cdot \sin x\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \left(1 \cdot \sin x + \color{blue}{\left({y}^{2} \cdot \frac{1}{6}\right) \cdot \sin x}\right) + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{120} \cdot \sin x\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \left(1 \cdot \sin x + \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right)} \cdot \sin x\right) + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{120} \cdot \sin x\right)\right) \]

      6. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\sin x \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)} + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{120} \cdot \sin x\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \sin x \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right) + \color{blue}{\left({y}^{2} \cdot {y}^{2}\right) \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{120} \cdot \sin x\right)} \]

      8. *-commutative N/A

         \[\leadsto \sin x \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right) + \color{blue}{\left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{120} \cdot \sin x\right) \cdot \left({y}^{2} \cdot {y}^{2}\right)} \]

      9. associate-*r* N/A

         \[\leadsto \sin x \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right) + \left(\color{blue}{\left(\frac{1}{5040} \cdot {y}^{2}\right) \cdot \sin x} + \frac{1}{120} \cdot \sin x\right) \cdot \left({y}^{2} \cdot {y}^{2}\right) \]

      10. distribute-rgt-out N/A

          \[\leadsto \sin x \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right) + \color{blue}{\left(\sin x \cdot \left(\frac{1}{5040} \cdot {y}^{2} + \frac{1}{120}\right)\right)} \cdot \left({y}^{2} \cdot {y}^{2}\right) \]

      11. +-commutative N/A

          \[\leadsto \sin x \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right) + \left(\sin x \cdot \color{blue}{\left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)}\right) \cdot \left({y}^{2} \cdot {y}^{2}\right) \]

   5. Applied rewrites 90.8%

      \[\leadsto \color{blue}{\sin x \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)} \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(x \cdot \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2} + 1\right)}\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{\left(x \cdot \left(\frac{-1}{6} \cdot {x}^{2}\right) + x \cdot 1\right)} \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      3. *-rgt-identity N/A

         \[\leadsto \left(x \cdot \left(\frac{-1}{6} \cdot {x}^{2}\right) + \color{blue}{x}\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{-1}{6} \cdot {x}^{2}, x\right)} \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{-1}{6} \cdot {x}^{2}}, x\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(x, \frac{-1}{6} \cdot \color{blue}{\left(x \cdot x\right)}, x\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      7. lower-*.f64 71.9

         \[\leadsto \mathsf{fma}\left(x, -0.16666666666666666 \cdot \color{blue}{\left(x \cdot x\right)}, x\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right) \]

   8. Applied rewrites 71.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, -0.16666666666666666 \cdot \left(x \cdot x\right), x\right)} \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right) \]

   if 0.0050000000000000001 < (sin.f64 x)

   1. Initial program 100.0%

      \[\sin x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\sin x + {y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{6} \cdot \sin x\right)} \]
   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. lower-*.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. lower-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. Applied rewrites 91.4%

      \[\leadsto \color{blue}{\sin x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right)\right)} \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) + 1\right)}\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(\left({x}^{2} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right) \cdot x + 1 \cdot x\right)} \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      3. associate-*l* N/A

         \[\leadsto \left(\color{blue}{{x}^{2} \cdot \left(\left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) \cdot x\right)} + 1 \cdot x\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      4. *-lft-identity N/A

         \[\leadsto \left({x}^{2} \cdot \left(\left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) \cdot x\right) + \color{blue}{x}\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{2}, \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) \cdot x, x\right)} \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) \cdot x, x\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) \cdot x, x\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{\left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) \cdot x}, x\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{\left(\frac{1}{120} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)} \cdot x, x\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \left(\frac{1}{120} \cdot {x}^{2} + \color{blue}{\frac{-1}{6}}\right) \cdot x, x\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(\frac{1}{120}, {x}^{2}, \frac{-1}{6}\right)} \cdot x, x\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{1}{120}, \color{blue}{x \cdot x}, \frac{-1}{6}\right) \cdot x, x\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

      13. lower-*.f64 26.3

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.008333333333333333, \color{blue}{x \cdot x}, -0.16666666666666666\right) \cdot x, x\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right) \]

   8. Applied rewrites 26.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right) \cdot x, x\right)} \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right) \]

   9. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{\left(\frac{1}{120} \cdot {x}^{2}\right)} \cdot x, x\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{\left({x}^{2} \cdot \frac{1}{120}\right)} \cdot x, x\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{\left({x}^{2} \cdot \frac{1}{120}\right)} \cdot x, x\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{120}\right) \cdot x, x\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{1}{120}, \frac{1}{6}\right), 1\right) \]

       4. lower-*.f64 25.4

          \[\leadsto \mathsf{fma}\left(x \cdot x, \left(\color{blue}{\left(x \cdot x\right)} \cdot 0.008333333333333333\right) \cdot x, x\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right) \]

   11. Applied rewrites 25.4%

       \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{\left(\left(x \cdot x\right) \cdot 0.008333333333333333\right)} \cdot x, x\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin x \leq 0.005:\\ \;\;\;\;\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right) \cdot \mathsf{fma}\left(x, -0.16666666666666666 \cdot \left(x \cdot x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right) \cdot \mathsf{fma}\left(x \cdot x, x \cdot \left(\left(x \cdot x\right) \cdot 0.008333333333333333\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 94.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1200000:\\ \;\;\;\;\sin x \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+77}:\\ \;\;\;\;\frac{\sinh y}{y} \cdot \mathsf{fma}\left(-0.16666666666666666, x \cdot \left(x \cdot x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1200000.0)
   (*
    (sin x)
    (fma
     y
     (*
      y
      (fma
       (* y y)
       (fma (* y y) 0.0001984126984126984 0.008333333333333333)
       0.16666666666666666))
     1.0))
   (if (<= y 5e+77)
     (* (/ (sinh y) y) (fma -0.16666666666666666 (* x (* x x)) x))
     (*
      (sin x)
      (fma
       (* y y)
       (fma y (* y 0.008333333333333333) 0.16666666666666666)
       1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1200000.0) {
		tmp = sin(x) * fma(y, (y * fma((y * y), fma((y * y), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666)), 1.0);
	} else if (y <= 5e+77) {
		tmp = (sinh(y) / y) * fma(-0.16666666666666666, (x * (x * x)), x);
	} else {
		tmp = sin(x) * fma((y * y), fma(y, (y * 0.008333333333333333), 0.16666666666666666), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1200000.0)
		tmp = Float64(sin(x) * fma(y, Float64(y * fma(Float64(y * y), fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666)), 1.0));
	elseif (y <= 5e+77)
		tmp = Float64(Float64(sinh(y) / y) * fma(-0.16666666666666666, Float64(x * Float64(x * x)), x));
	else
		tmp = Float64(sin(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[y, 1200000.0], N[(N[Sin[x], $MachinePrecision] * N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5e+77], N[(N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision] * N[(-0.16666666666666666 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], N[(N[Sin[x], $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.2e6

   1. Initial program 100.0%

      \[\sin x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\sin x + {y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x + {y}^{2} \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{120} \cdot \sin x\right)\right)} \]
   4. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \sin x + \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x\right) + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{120} \cdot \sin x\right)\right)\right)} \]

      2. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\sin x + {y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x\right)\right) + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{120} \cdot \sin x\right)\right)} \]

      3. *-lft-identity N/A

         \[\leadsto \left(\color{blue}{1 \cdot \sin x} + {y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x\right)\right) + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{120} \cdot \sin x\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \left(1 \cdot \sin x + \color{blue}{\left({y}^{2} \cdot \frac{1}{6}\right) \cdot \sin x}\right) + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{120} \cdot \sin x\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \left(1 \cdot \sin x + \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right)} \cdot \sin x\right) + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{120} \cdot \sin x\right)\right) \]

      6. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\sin x \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)} + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{120} \cdot \sin x\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \sin x \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right) + \color{blue}{\left({y}^{2} \cdot {y}^{2}\right) \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{120} \cdot \sin x\right)} \]

      8. *-commutative N/A

         \[\leadsto \sin x \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right) + \color{blue}{\left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{120} \cdot \sin x\right) \cdot \left({y}^{2} \cdot {y}^{2}\right)} \]

      9. associate-*r* N/A

         \[\leadsto \sin x \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right) + \left(\color{blue}{\left(\frac{1}{5040} \cdot {y}^{2}\right) \cdot \sin x} + \frac{1}{120} \cdot \sin x\right) \cdot \left({y}^{2} \cdot {y}^{2}\right) \]

      10. distribute-rgt-out N/A

          \[\leadsto \sin x \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right) + \color{blue}{\left(\sin x \cdot \left(\frac{1}{5040} \cdot {y}^{2} + \frac{1}{120}\right)\right)} \cdot \left({y}^{2} \cdot {y}^{2}\right) \]

      11. +-commutative N/A

          \[\leadsto \sin x \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right) + \left(\sin x \cdot \color{blue}{\left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)}\right) \cdot \left({y}^{2} \cdot {y}^{2}\right) \]

   5. Applied rewrites 94.7%

      \[\leadsto \color{blue}{\sin x \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)} \]

   if 1.2e6 < y < 5.00000000000000004e77

   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. lower-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. lower-*.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. lower-*.f64 89.5

          \[\leadsto \mathsf{fma}\left(-0.16666666666666666, x \cdot \color{blue}{\left(x \cdot x\right)}, x\right) \cdot \frac{\sinh y}{y} \]

   5. Applied rewrites 89.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.16666666666666666, x \cdot \left(x \cdot x\right), x\right)} \cdot \frac{\sinh y}{y} \]

   if 5.00000000000000004e77 < 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. lower-*.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. lower-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. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\sin x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1200000:\\ \;\;\;\;\sin x \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+77}:\\ \;\;\;\;\frac{\sinh y}{y} \cdot \mathsf{fma}\left(-0.16666666666666666, x \cdot \left(x \cdot x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 90.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 7.6 \cdot 10^{-6}:\\ \;\;\;\;\sin x \cdot \frac{\mathsf{fma}\left(0.16666666666666666, y \cdot \left(y \cdot y\right), y\right)}{y}\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+77}:\\ \;\;\;\;\frac{\sinh y}{y} \cdot \mathsf{fma}\left(-0.16666666666666666, x \cdot \left(x \cdot x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 7.6e-6)
   (* (sin x) (/ (fma 0.16666666666666666 (* y (* y y)) y) y))
   (if (<= y 5e+77)
     (* (/ (sinh y) y) (fma -0.16666666666666666 (* x (* x x)) x))
     (*
      (sin x)
      (fma
       (* y y)
       (fma y (* y 0.008333333333333333) 0.16666666666666666)
       1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 7.6e-6) {
		tmp = sin(x) * (fma(0.16666666666666666, (y * (y * y)), y) / y);
	} else if (y <= 5e+77) {
		tmp = (sinh(y) / y) * fma(-0.16666666666666666, (x * (x * x)), x);
	} else {
		tmp = sin(x) * fma((y * y), fma(y, (y * 0.008333333333333333), 0.16666666666666666), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 7.6e-6)
		tmp = Float64(sin(x) * Float64(fma(0.16666666666666666, Float64(y * Float64(y * y)), y) / y));
	elseif (y <= 5e+77)
		tmp = Float64(Float64(sinh(y) / y) * fma(-0.16666666666666666, Float64(x * Float64(x * x)), x));
	else
		tmp = Float64(sin(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[y, 7.6e-6], N[(N[Sin[x], $MachinePrecision] * N[(N[(0.16666666666666666 * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5e+77], N[(N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision] * N[(-0.16666666666666666 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], N[(N[Sin[x], $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 7.6000000000000001e-6

   1. Initial program 100.0%

      \[\sin x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \sin x \cdot \frac{\color{blue}{y \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)}}{y} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \sin x \cdot \frac{y \cdot \color{blue}{\left(\frac{1}{6} \cdot {y}^{2} + 1\right)}}{y} \]

      2. distribute-lft-in N/A

         \[\leadsto \sin x \cdot \frac{\color{blue}{y \cdot \left(\frac{1}{6} \cdot {y}^{2}\right) + y \cdot 1}}{y} \]

      3. *-commutative N/A

         \[\leadsto \sin x \cdot \frac{y \cdot \color{blue}{\left({y}^{2} \cdot \frac{1}{6}\right)} + y \cdot 1}{y} \]

      4. associate-*r* N/A

         \[\leadsto \sin x \cdot \frac{\color{blue}{\left(y \cdot {y}^{2}\right) \cdot \frac{1}{6}} + y \cdot 1}{y} \]

      5. *-commutative N/A

         \[\leadsto \sin x \cdot \frac{\color{blue}{\frac{1}{6} \cdot \left(y \cdot {y}^{2}\right)} + y \cdot 1}{y} \]

      6. *-rgt-identity N/A

         \[\leadsto \sin x \cdot \frac{\frac{1}{6} \cdot \left(y \cdot {y}^{2}\right) + \color{blue}{y}}{y} \]

      7. lower-fma.f64 N/A

         \[\leadsto \sin x \cdot \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{6}, y \cdot {y}^{2}, y\right)}}{y} \]

      8. lower-*.f64 N/A

         \[\leadsto \sin x \cdot \frac{\mathsf{fma}\left(\frac{1}{6}, \color{blue}{y \cdot {y}^{2}}, y\right)}{y} \]

      9. unpow2 N/A

         \[\leadsto \sin x \cdot \frac{\mathsf{fma}\left(\frac{1}{6}, y \cdot \color{blue}{\left(y \cdot y\right)}, y\right)}{y} \]

      10. lower-*.f64 92.3

          \[\leadsto \sin x \cdot \frac{\mathsf{fma}\left(0.16666666666666666, y \cdot \color{blue}{\left(y \cdot y\right)}, y\right)}{y} \]

   5. Applied rewrites 92.3%

      \[\leadsto \sin x \cdot \frac{\color{blue}{\mathsf{fma}\left(0.16666666666666666, y \cdot \left(y \cdot y\right), y\right)}}{y} \]

   if 7.6000000000000001e-6 < y < 5.00000000000000004e77

   1. Initial program 99.9%

      \[\sin x \cdot \frac{\sinh y}{y} \]
   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. lower-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. lower-*.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. lower-*.f64 85.6

          \[\leadsto \mathsf{fma}\left(-0.16666666666666666, x \cdot \color{blue}{\left(x \cdot x\right)}, x\right) \cdot \frac{\sinh y}{y} \]

   5. Applied rewrites 85.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.16666666666666666, x \cdot \left(x \cdot x\right), x\right)} \cdot \frac{\sinh y}{y} \]

   if 5.00000000000000004e77 < 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. lower-*.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. lower-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. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\sin x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7.6 \cdot 10^{-6}:\\ \;\;\;\;\sin x \cdot \frac{\mathsf{fma}\left(0.16666666666666666, y \cdot \left(y \cdot y\right), y\right)}{y}\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+77}:\\ \;\;\;\;\frac{\sinh y}{y} \cdot \mathsf{fma}\left(-0.16666666666666666, x \cdot \left(x \cdot x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 89.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right)\\ \mathbf{if}\;y \leq 7.6 \cdot 10^{-6}:\\ \;\;\;\;\sin x \cdot \frac{\mathsf{fma}\left(0.16666666666666666, y \cdot \left(y \cdot y\right), y\right)}{y}\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{+77}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, x \cdot \mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right), x\right) \cdot \mathsf{fma}\left(y, \frac{y \cdot \mathsf{fma}\left(y \cdot y, t\_0 \cdot t\_0, -0.027777777777777776\right)}{\mathsf{fma}\left(y, t\_0, -0.16666666666666666\right)}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (fma y (* y 0.0001984126984126984) 0.008333333333333333))))
   (if (<= y 7.6e-6)
     (* (sin x) (/ (fma 0.16666666666666666 (* y (* y y)) y) y))
     (if (<= y 3.8e+77)
       (*
        (fma
         (* x x)
         (* x (fma 0.008333333333333333 (* x x) -0.16666666666666666))
         x)
        (fma
         y
         (/
          (* y (fma (* y y) (* t_0 t_0) -0.027777777777777776))
          (fma y t_0 -0.16666666666666666))
         1.0))
       (*
        (sin x)
        (fma
         (* y y)
         (fma y (* y 0.008333333333333333) 0.16666666666666666)
         1.0))))))

C

double code(double x, double y) {
	double t_0 = y * fma(y, (y * 0.0001984126984126984), 0.008333333333333333);
	double tmp;
	if (y <= 7.6e-6) {
		tmp = sin(x) * (fma(0.16666666666666666, (y * (y * y)), y) / y);
	} else if (y <= 3.8e+77) {
		tmp = fma((x * x), (x * fma(0.008333333333333333, (x * x), -0.16666666666666666)), x) * fma(y, ((y * fma((y * y), (t_0 * t_0), -0.027777777777777776)) / fma(y, t_0, -0.16666666666666666)), 1.0);
	} else {
		tmp = sin(x) * fma((y * y), fma(y, (y * 0.008333333333333333), 0.16666666666666666), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(y * fma(y, Float64(y * 0.0001984126984126984), 0.008333333333333333))
	tmp = 0.0
	if (y <= 7.6e-6)
		tmp = Float64(sin(x) * Float64(fma(0.16666666666666666, Float64(y * Float64(y * y)), y) / y));
	elseif (y <= 3.8e+77)
		tmp = Float64(fma(Float64(x * x), Float64(x * fma(0.008333333333333333, Float64(x * x), -0.16666666666666666)), x) * fma(y, Float64(Float64(y * fma(Float64(y * y), Float64(t_0 * t_0), -0.027777777777777776)) / fma(y, t_0, -0.16666666666666666)), 1.0));
	else
		tmp = Float64(sin(x) * fma(Float64(y * y), fma(y, Float64(y * 0.008333333333333333), 0.16666666666666666), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(y * N[(y * 0.0001984126984126984), $MachinePrecision] + 0.008333333333333333), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 7.6e-6], N[(N[Sin[x], $MachinePrecision] * N[(N[(0.16666666666666666 * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.8e+77], N[(N[(N[(x * x), $MachinePrecision] * N[(x * N[(0.008333333333333333 * N[(x * x), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] * N[(y * N[(N[(y * N[(N[(y * y), $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision] + -0.027777777777777776), $MachinePrecision]), $MachinePrecision] / N[(y * t$95$0 + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Sin[x], $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 7.6000000000000001e-6

   1. Initial program 100.0%

      \[\sin x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \sin x \cdot \frac{\color{blue}{y \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)}}{y} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \sin x \cdot \frac{y \cdot \color{blue}{\left(\frac{1}{6} \cdot {y}^{2} + 1\right)}}{y} \]

      2. distribute-lft-in N/A

         \[\leadsto \sin x \cdot \frac{\color{blue}{y \cdot \left(\frac{1}{6} \cdot {y}^{2}\right) + y \cdot 1}}{y} \]

      3. *-commutative N/A

         \[\leadsto \sin x \cdot \frac{y \cdot \color{blue}{\left({y}^{2} \cdot \frac{1}{6}\right)} + y \cdot 1}{y} \]

      4. associate-*r* N/A

         \[\leadsto \sin x \cdot \frac{\color{blue}{\left(y \cdot {y}^{2}\right) \cdot \frac{1}{6}} + y \cdot 1}{y} \]

      5. *-commutative N/A

         \[\leadsto \sin x \cdot \frac{\color{blue}{\frac{1}{6} \cdot \left(y \cdot {y}^{2}\right)} + y \cdot 1}{y} \]

      6. *-rgt-identity N/A

         \[\leadsto \sin x \cdot \frac{\frac{1}{6} \cdot \left(y \cdot {y}^{2}\right) + \color{blue}{y}}{y} \]

      7. lower-fma.f64 N/A

         \[\leadsto \sin x \cdot \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{6}, y \cdot {y}^{2}, y\right)}}{y} \]

      8. lower-*.f64 N/A

         \[\leadsto \sin x \cdot \frac{\mathsf{fma}\left(\frac{1}{6}, \color{blue}{y \cdot {y}^{2}}, y\right)}{y} \]

      9. unpow2 N/A

         \[\leadsto \sin x \cdot \frac{\mathsf{fma}\left(\frac{1}{6}, y \cdot \color{blue}{\left(y \cdot y\right)}, y\right)}{y} \]

      10. lower-*.f64 92.3

          \[\leadsto \sin x \cdot \frac{\mathsf{fma}\left(0.16666666666666666, y \cdot \color{blue}{\left(y \cdot y\right)}, y\right)}{y} \]

   5. Applied rewrites 92.3%

      \[\leadsto \sin x \cdot \frac{\color{blue}{\mathsf{fma}\left(0.16666666666666666, y \cdot \left(y \cdot y\right), y\right)}}{y} \]

   if 7.6000000000000001e-6 < y < 3.8000000000000001e77

   1. Initial program 99.9%

      \[\sin x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\sin x + {y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x + {y}^{2} \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{120} \cdot \sin x\right)\right)} \]
   4. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \sin x + \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x\right) + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{120} \cdot \sin x\right)\right)\right)} \]

      2. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\sin x + {y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x\right)\right) + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{120} \cdot \sin x\right)\right)} \]

      3. *-lft-identity N/A

         \[\leadsto \left(\color{blue}{1 \cdot \sin x} + {y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x\right)\right) + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{120} \cdot \sin x\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \left(1 \cdot \sin x + \color{blue}{\left({y}^{2} \cdot \frac{1}{6}\right) \cdot \sin x}\right) + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{120} \cdot \sin x\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \left(1 \cdot \sin x + \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right)} \cdot \sin x\right) + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{120} \cdot \sin x\right)\right) \]

      6. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\sin x \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)} + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{120} \cdot \sin x\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \sin x \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right) + \color{blue}{\left({y}^{2} \cdot {y}^{2}\right) \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{120} \cdot \sin x\right)} \]

      8. *-commutative N/A

         \[\leadsto \sin x \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right) + \color{blue}{\left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{120} \cdot \sin x\right) \cdot \left({y}^{2} \cdot {y}^{2}\right)} \]

      9. associate-*r* N/A

         \[\leadsto \sin x \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right) + \left(\color{blue}{\left(\frac{1}{5040} \cdot {y}^{2}\right) \cdot \sin x} + \frac{1}{120} \cdot \sin x\right) \cdot \left({y}^{2} \cdot {y}^{2}\right) \]

      10. distribute-rgt-out N/A

          \[\leadsto \sin x \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right) + \color{blue}{\left(\sin x \cdot \left(\frac{1}{5040} \cdot {y}^{2} + \frac{1}{120}\right)\right)} \cdot \left({y}^{2} \cdot {y}^{2}\right) \]

      11. +-commutative N/A

          \[\leadsto \sin x \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right) + \left(\sin x \cdot \color{blue}{\left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)}\right) \cdot \left({y}^{2} \cdot {y}^{2}\right) \]

   5. Applied rewrites 31.7%

      \[\leadsto \color{blue}{\sin x \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right)\right)} \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) + 1\right)}\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(\left({x}^{2} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right) \cdot x + 1 \cdot x\right)} \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      3. associate-*l* N/A

         \[\leadsto \left(\color{blue}{{x}^{2} \cdot \left(\left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) \cdot x\right)} + 1 \cdot x\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      4. *-lft-identity N/A

         \[\leadsto \left({x}^{2} \cdot \left(\left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) \cdot x\right) + \color{blue}{x}\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{2}, \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) \cdot x, x\right)} \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) \cdot x, x\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) \cdot x, x\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{\left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) \cdot x}, x\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{\left(\frac{1}{120} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)} \cdot x, x\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \left(\frac{1}{120} \cdot {x}^{2} + \color{blue}{\frac{-1}{6}}\right) \cdot x, x\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(\frac{1}{120}, {x}^{2}, \frac{-1}{6}\right)} \cdot x, x\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{1}{120}, \color{blue}{x \cdot x}, \frac{-1}{6}\right) \cdot x, x\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      13. lower-*.f64 34.9

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.008333333333333333, \color{blue}{x \cdot x}, -0.16666666666666666\right) \cdot x, x\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right) \]

   8. Applied rewrites 34.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right) \cdot x, x\right)} \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right) \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{1}{120}, x \cdot x, \frac{-1}{6}\right) \cdot x, x\right) \cdot \mathsf{fma}\left(y, y \cdot \left(\color{blue}{\left(y \cdot y\right)} \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{5040} + \frac{1}{120}\right) + \frac{1}{6}\right), 1\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{1}{120}, x \cdot x, \frac{-1}{6}\right) \cdot x, x\right) \cdot \mathsf{fma}\left(y, y \cdot \left(\left(y \cdot y\right) \cdot \left(\color{blue}{\left(y \cdot y\right)} \cdot \frac{1}{5040} + \frac{1}{120}\right) + \frac{1}{6}\right), 1\right) \]

      3. lift-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{1}{120}, x \cdot x, \frac{-1}{6}\right) \cdot x, x\right) \cdot \mathsf{fma}\left(y, y \cdot \left(\left(y \cdot y\right) \cdot \color{blue}{\mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right)} + \frac{1}{6}\right), 1\right) \]

      4. lift-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{1}{120}, x \cdot x, \frac{-1}{6}\right) \cdot x, x\right) \cdot \mathsf{fma}\left(y, y \cdot \color{blue}{\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right)}, 1\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{1}{120}, x \cdot x, \frac{-1}{6}\right) \cdot x, x\right) \cdot \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right) \cdot y}, 1\right) \]

      6. lift-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{1}{120}, x \cdot x, \frac{-1}{6}\right) \cdot x, x\right) \cdot \mathsf{fma}\left(y, \color{blue}{\left(\left(y \cdot y\right) \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right) + \frac{1}{6}\right)} \cdot y, 1\right) \]

      7. flip-+ N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{1}{120}, x \cdot x, \frac{-1}{6}\right) \cdot x, x\right) \cdot \mathsf{fma}\left(y, \color{blue}{\frac{\left(\left(y \cdot y\right) \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right)\right) \cdot \left(\left(y \cdot y\right) \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right)\right) - \frac{1}{6} \cdot \frac{1}{6}}{\left(y \cdot y\right) \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right) - \frac{1}{6}}} \cdot y, 1\right) \]

      8. associate-*l/ N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{1}{120}, x \cdot x, \frac{-1}{6}\right) \cdot x, x\right) \cdot \mathsf{fma}\left(y, \color{blue}{\frac{\left(\left(\left(y \cdot y\right) \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right)\right) \cdot \left(\left(y \cdot y\right) \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right)\right) - \frac{1}{6} \cdot \frac{1}{6}\right) \cdot y}{\left(y \cdot y\right) \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right) - \frac{1}{6}}}, 1\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{1}{120}, x \cdot x, \frac{-1}{6}\right) \cdot x, x\right) \cdot \mathsf{fma}\left(y, \color{blue}{\frac{\left(\left(\left(y \cdot y\right) \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right)\right) \cdot \left(\left(y \cdot y\right) \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right)\right) - \frac{1}{6} \cdot \frac{1}{6}\right) \cdot y}{\left(y \cdot y\right) \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right) - \frac{1}{6}}}, 1\right) \]

   10. Applied rewrites 62.4%

       \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right) \cdot x, x\right) \cdot \mathsf{fma}\left(y, \color{blue}{\frac{\mathsf{fma}\left(y \cdot y, \left(y \cdot \mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right)\right) \cdot \left(y \cdot \mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right)\right), -0.027777777777777776\right) \cdot y}{\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right)}}, 1\right) \]

   if 3.8000000000000001e77 < 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. lower-*.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. lower-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. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\sin x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7.6 \cdot 10^{-6}:\\ \;\;\;\;\sin x \cdot \frac{\mathsf{fma}\left(0.16666666666666666, y \cdot \left(y \cdot y\right), y\right)}{y}\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{+77}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, x \cdot \mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right), x\right) \cdot \mathsf{fma}\left(y, \frac{y \cdot \mathsf{fma}\left(y \cdot y, \left(y \cdot \mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right)\right) \cdot \left(y \cdot \mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right)\right), -0.027777777777777776\right)}{\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right)}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 85.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right)\\ \mathbf{if}\;y \leq 7.6 \cdot 10^{-6}:\\ \;\;\;\;\sin x \cdot \mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{+77}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, x \cdot \mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right), x\right) \cdot \mathsf{fma}\left(y, \frac{y \cdot \mathsf{fma}\left(y \cdot y, t\_0 \cdot t\_0, -0.027777777777777776\right)}{\mathsf{fma}\left(y, t\_0, -0.16666666666666666\right)}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (fma y (* y 0.0001984126984126984) 0.008333333333333333))))
   (if (<= y 7.6e-6)
     (* (sin x) (fma 0.16666666666666666 (* y y) 1.0))
     (if (<= y 3.8e+77)
       (*
        (fma
         (* x x)
         (* x (fma 0.008333333333333333 (* x x) -0.16666666666666666))
         x)
        (fma
         y
         (/
          (* y (fma (* y y) (* t_0 t_0) -0.027777777777777776))
          (fma y t_0 -0.16666666666666666))
         1.0))
       (*
        (sin x)
        (fma
         (* y y)
         (fma y (* y 0.008333333333333333) 0.16666666666666666)
         1.0))))))

C

double code(double x, double y) {
	double t_0 = y * fma(y, (y * 0.0001984126984126984), 0.008333333333333333);
	double tmp;
	if (y <= 7.6e-6) {
		tmp = sin(x) * fma(0.16666666666666666, (y * y), 1.0);
	} else if (y <= 3.8e+77) {
		tmp = fma((x * x), (x * fma(0.008333333333333333, (x * x), -0.16666666666666666)), x) * fma(y, ((y * fma((y * y), (t_0 * t_0), -0.027777777777777776)) / fma(y, t_0, -0.16666666666666666)), 1.0);
	} else {
		tmp = sin(x) * fma((y * y), fma(y, (y * 0.008333333333333333), 0.16666666666666666), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(y * fma(y, Float64(y * 0.0001984126984126984), 0.008333333333333333))
	tmp = 0.0
	if (y <= 7.6e-6)
		tmp = Float64(sin(x) * fma(0.16666666666666666, Float64(y * y), 1.0));
	elseif (y <= 3.8e+77)
		tmp = Float64(fma(Float64(x * x), Float64(x * fma(0.008333333333333333, Float64(x * x), -0.16666666666666666)), x) * fma(y, Float64(Float64(y * fma(Float64(y * y), Float64(t_0 * t_0), -0.027777777777777776)) / fma(y, t_0, -0.16666666666666666)), 1.0));
	else
		tmp = Float64(sin(x) * fma(Float64(y * y), fma(y, Float64(y * 0.008333333333333333), 0.16666666666666666), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(y * N[(y * 0.0001984126984126984), $MachinePrecision] + 0.008333333333333333), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 7.6e-6], N[(N[Sin[x], $MachinePrecision] * N[(0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.8e+77], N[(N[(N[(x * x), $MachinePrecision] * N[(x * N[(0.008333333333333333 * N[(x * x), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] * N[(y * N[(N[(y * N[(N[(y * y), $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision] + -0.027777777777777776), $MachinePrecision]), $MachinePrecision] / N[(y * t$95$0 + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Sin[x], $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 7.6000000000000001e-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. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sin x \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)} \]

      5. lower-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. lower-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. lower-*.f64 87.6

         \[\leadsto \sin x \cdot \mathsf{fma}\left(0.16666666666666666, \color{blue}{y \cdot y}, 1\right) \]

   5. Applied rewrites 87.6%

      \[\leadsto \color{blue}{\sin x \cdot \mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right)} \]

   if 7.6000000000000001e-6 < y < 3.8000000000000001e77

   1. Initial program 99.9%

      \[\sin x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\sin x + {y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x + {y}^{2} \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{120} \cdot \sin x\right)\right)} \]
   4. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \sin x + \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x\right) + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{120} \cdot \sin x\right)\right)\right)} \]

      2. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\sin x + {y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x\right)\right) + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{120} \cdot \sin x\right)\right)} \]

      3. *-lft-identity N/A

         \[\leadsto \left(\color{blue}{1 \cdot \sin x} + {y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x\right)\right) + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{120} \cdot \sin x\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \left(1 \cdot \sin x + \color{blue}{\left({y}^{2} \cdot \frac{1}{6}\right) \cdot \sin x}\right) + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{120} \cdot \sin x\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \left(1 \cdot \sin x + \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right)} \cdot \sin x\right) + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{120} \cdot \sin x\right)\right) \]

      6. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\sin x \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)} + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{120} \cdot \sin x\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \sin x \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right) + \color{blue}{\left({y}^{2} \cdot {y}^{2}\right) \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{120} \cdot \sin x\right)} \]

      8. *-commutative N/A

         \[\leadsto \sin x \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right) + \color{blue}{\left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{120} \cdot \sin x\right) \cdot \left({y}^{2} \cdot {y}^{2}\right)} \]

      9. associate-*r* N/A

         \[\leadsto \sin x \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right) + \left(\color{blue}{\left(\frac{1}{5040} \cdot {y}^{2}\right) \cdot \sin x} + \frac{1}{120} \cdot \sin x\right) \cdot \left({y}^{2} \cdot {y}^{2}\right) \]

      10. distribute-rgt-out N/A

          \[\leadsto \sin x \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right) + \color{blue}{\left(\sin x \cdot \left(\frac{1}{5040} \cdot {y}^{2} + \frac{1}{120}\right)\right)} \cdot \left({y}^{2} \cdot {y}^{2}\right) \]

      11. +-commutative N/A

          \[\leadsto \sin x \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right) + \left(\sin x \cdot \color{blue}{\left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)}\right) \cdot \left({y}^{2} \cdot {y}^{2}\right) \]

   5. Applied rewrites 31.7%

      \[\leadsto \color{blue}{\sin x \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right)\right)} \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) + 1\right)}\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(\left({x}^{2} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right) \cdot x + 1 \cdot x\right)} \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      3. associate-*l* N/A

         \[\leadsto \left(\color{blue}{{x}^{2} \cdot \left(\left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) \cdot x\right)} + 1 \cdot x\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      4. *-lft-identity N/A

         \[\leadsto \left({x}^{2} \cdot \left(\left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) \cdot x\right) + \color{blue}{x}\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{2}, \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) \cdot x, x\right)} \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) \cdot x, x\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) \cdot x, x\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{\left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) \cdot x}, x\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{\left(\frac{1}{120} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)} \cdot x, x\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \left(\frac{1}{120} \cdot {x}^{2} + \color{blue}{\frac{-1}{6}}\right) \cdot x, x\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(\frac{1}{120}, {x}^{2}, \frac{-1}{6}\right)} \cdot x, x\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{1}{120}, \color{blue}{x \cdot x}, \frac{-1}{6}\right) \cdot x, x\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right) \]

      13. lower-*.f64 34.9

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.008333333333333333, \color{blue}{x \cdot x}, -0.16666666666666666\right) \cdot x, x\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right) \]

   8. Applied rewrites 34.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right) \cdot x, x\right)} \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right) \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{1}{120}, x \cdot x, \frac{-1}{6}\right) \cdot x, x\right) \cdot \mathsf{fma}\left(y, y \cdot \left(\color{blue}{\left(y \cdot y\right)} \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{5040} + \frac{1}{120}\right) + \frac{1}{6}\right), 1\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{1}{120}, x \cdot x, \frac{-1}{6}\right) \cdot x, x\right) \cdot \mathsf{fma}\left(y, y \cdot \left(\left(y \cdot y\right) \cdot \left(\color{blue}{\left(y \cdot y\right)} \cdot \frac{1}{5040} + \frac{1}{120}\right) + \frac{1}{6}\right), 1\right) \]

      3. lift-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{1}{120}, x \cdot x, \frac{-1}{6}\right) \cdot x, x\right) \cdot \mathsf{fma}\left(y, y \cdot \left(\left(y \cdot y\right) \cdot \color{blue}{\mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right)} + \frac{1}{6}\right), 1\right) \]

      4. lift-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{1}{120}, x \cdot x, \frac{-1}{6}\right) \cdot x, x\right) \cdot \mathsf{fma}\left(y, y \cdot \color{blue}{\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right)}, 1\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{1}{120}, x \cdot x, \frac{-1}{6}\right) \cdot x, x\right) \cdot \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right) \cdot y}, 1\right) \]

      6. lift-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{1}{120}, x \cdot x, \frac{-1}{6}\right) \cdot x, x\right) \cdot \mathsf{fma}\left(y, \color{blue}{\left(\left(y \cdot y\right) \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right) + \frac{1}{6}\right)} \cdot y, 1\right) \]

      7. flip-+ N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{1}{120}, x \cdot x, \frac{-1}{6}\right) \cdot x, x\right) \cdot \mathsf{fma}\left(y, \color{blue}{\frac{\left(\left(y \cdot y\right) \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right)\right) \cdot \left(\left(y \cdot y\right) \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right)\right) - \frac{1}{6} \cdot \frac{1}{6}}{\left(y \cdot y\right) \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right) - \frac{1}{6}}} \cdot y, 1\right) \]

      8. associate-*l/ N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{1}{120}, x \cdot x, \frac{-1}{6}\right) \cdot x, x\right) \cdot \mathsf{fma}\left(y, \color{blue}{\frac{\left(\left(\left(y \cdot y\right) \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right)\right) \cdot \left(\left(y \cdot y\right) \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right)\right) - \frac{1}{6} \cdot \frac{1}{6}\right) \cdot y}{\left(y \cdot y\right) \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right) - \frac{1}{6}}}, 1\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{1}{120}, x \cdot x, \frac{-1}{6}\right) \cdot x, x\right) \cdot \mathsf{fma}\left(y, \color{blue}{\frac{\left(\left(\left(y \cdot y\right) \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right)\right) \cdot \left(\left(y \cdot y\right) \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right)\right) - \frac{1}{6} \cdot \frac{1}{6}\right) \cdot y}{\left(y \cdot y\right) \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right) - \frac{1}{6}}}, 1\right) \]

   10. Applied rewrites 62.4%

       \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right) \cdot x, x\right) \cdot \mathsf{fma}\left(y, \color{blue}{\frac{\mathsf{fma}\left(y \cdot y, \left(y \cdot \mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right)\right) \cdot \left(y \cdot \mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right)\right), -0.027777777777777776\right) \cdot y}{\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right)}}, 1\right) \]

   if 3.8000000000000001e77 < 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. lower-*.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. lower-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. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\sin x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7.6 \cdot 10^{-6}:\\ \;\;\;\;\sin x \cdot \mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{+77}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, x \cdot \mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right), x\right) \cdot \mathsf{fma}\left(y, \frac{y \cdot \mathsf{fma}\left(y \cdot y, \left(y \cdot \mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right)\right) \cdot \left(y \cdot \mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right)\right), -0.027777777777777776\right)}{\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right)}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 48.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin x \leq -0.005:\\ \;\;\;\;x \cdot \left(-0.16666666666666666 \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.16666666666666666, x \cdot \left(y \cdot y\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sin x) -0.005)
   (* x (* -0.16666666666666666 (* x x)))
   (fma 0.16666666666666666 (* x (* y y)) x)))

C

double code(double x, double y) {
	double tmp;
	if (sin(x) <= -0.005) {
		tmp = x * (-0.16666666666666666 * (x * x));
	} else {
		tmp = fma(0.16666666666666666, (x * (y * y)), x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (sin(x) <= -0.005)
		tmp = Float64(x * Float64(-0.16666666666666666 * 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[Sin[x], $MachinePrecision], -0.005], N[(x * N[(-0.16666666666666666 * 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 (sin.f64 x) < -0.0050000000000000001

   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. lower-sin.f64 52.5

         \[\leadsto \color{blue}{\sin x} \]

   5. Applied rewrites 52.5%

      \[\leadsto \color{blue}{\sin x} \]

   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. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{-1}{6} \cdot {x}^{2}, x\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{-1}{6} \cdot {x}^{2}}, x\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(x, \frac{-1}{6} \cdot \color{blue}{\left(x \cdot x\right)}, x\right) \]

      7. lower-*.f64 11.6

         \[\leadsto \mathsf{fma}\left(x, -0.16666666666666666 \cdot \color{blue}{\left(x \cdot x\right)}, x\right) \]

   8. Applied rewrites 11.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, -0.16666666666666666 \cdot \left(x \cdot x\right), x\right)} \]

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{-1}{6} \cdot {x}^{3}} \]
   10. Step-by-step derivation

       1. unpow3 N/A

          \[\leadsto \frac{-1}{6} \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot x\right)} \]

       2. unpow2 N/A

          \[\leadsto \frac{-1}{6} \cdot \left(\color{blue}{{x}^{2}} \cdot x\right) \]

       3. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2}\right) \cdot x} \]

       4. *-commutative N/A

          \[\leadsto \color{blue}{x \cdot \left(\frac{-1}{6} \cdot {x}^{2}\right)} \]

       5. lower-*.f64 N/A

          \[\leadsto \color{blue}{x \cdot \left(\frac{-1}{6} \cdot {x}^{2}\right)} \]

       6. lower-*.f64 N/A

          \[\leadsto x \cdot \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2}\right)} \]

       7. unpow2 N/A

          \[\leadsto x \cdot \left(\frac{-1}{6} \cdot \color{blue}{\left(x \cdot x\right)}\right) \]

       8. lower-*.f64 11.6

          \[\leadsto x \cdot \left(-0.16666666666666666 \cdot \color{blue}{\left(x \cdot x\right)}\right) \]

   11. Applied rewrites 11.6%

       \[\leadsto \color{blue}{x \cdot \left(-0.16666666666666666 \cdot \left(x \cdot x\right)\right)} \]

   if -0.0050000000000000001 < (sin.f64 x)

   1. Initial program 100.0%

      \[\sin x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\sin x + {y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x + {y}^{2} \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{120} \cdot \sin x\right)\right)} \]
   4. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \sin x + \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x\right) + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{120} \cdot \sin x\right)\right)\right)} \]

      2. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\sin x + {y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x\right)\right) + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{120} \cdot \sin x\right)\right)} \]

      3. *-lft-identity N/A

         \[\leadsto \left(\color{blue}{1 \cdot \sin x} + {y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x\right)\right) + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{120} \cdot \sin x\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \left(1 \cdot \sin x + \color{blue}{\left({y}^{2} \cdot \frac{1}{6}\right) \cdot \sin x}\right) + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{120} \cdot \sin x\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \left(1 \cdot \sin x + \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right)} \cdot \sin x\right) + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{120} \cdot \sin x\right)\right) \]

      6. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\sin x \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)} + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{120} \cdot \sin x\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \sin x \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right) + \color{blue}{\left({y}^{2} \cdot {y}^{2}\right) \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{120} \cdot \sin x\right)} \]

      8. *-commutative N/A

         \[\leadsto \sin x \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right) + \color{blue}{\left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{120} \cdot \sin x\right) \cdot \left({y}^{2} \cdot {y}^{2}\right)} \]

      9. associate-*r* N/A

         \[\leadsto \sin x \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right) + \left(\color{blue}{\left(\frac{1}{5040} \cdot {y}^{2}\right) \cdot \sin x} + \frac{1}{120} \cdot \sin x\right) \cdot \left({y}^{2} \cdot {y}^{2}\right) \]

      10. distribute-rgt-out N/A

          \[\leadsto \sin x \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right) + \color{blue}{\left(\sin x \cdot \left(\frac{1}{5040} \cdot {y}^{2} + \frac{1}{120}\right)\right)} \cdot \left({y}^{2} \cdot {y}^{2}\right) \]

      11. +-commutative N/A

          \[\leadsto \sin x \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right) + \left(\sin x \cdot \color{blue}{\left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)}\right) \cdot \left({y}^{2} \cdot {y}^{2}\right) \]

   5. Applied rewrites 91.2%

      \[\leadsto \color{blue}{\sin x \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right) \cdot x} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) + 1\right)} \cdot x \]

      3. distribute-lft1-in N/A

         \[\leadsto \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right) \cdot x + x} \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{{y}^{2} \cdot \left(\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot x\right)} + x \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({y}^{2}, \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot x, x\right)} \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{y \cdot y}, \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot x, x\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{y \cdot y}, \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot x, x\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y \cdot y, \color{blue}{\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot x}, x\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(y \cdot y, \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right) + \frac{1}{6}\right)} \cdot x, x\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(y \cdot y, \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}, \frac{1}{6}\right)} \cdot x, x\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}, \frac{1}{6}\right) \cdot x, x\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}, \frac{1}{6}\right) \cdot x, x\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \color{blue}{\frac{1}{5040} \cdot {y}^{2} + \frac{1}{120}}, \frac{1}{6}\right) \cdot x, x\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \color{blue}{{y}^{2} \cdot \frac{1}{5040}} + \frac{1}{120}, \frac{1}{6}\right) \cdot x, x\right) \]

      15. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{1}{5040}, \frac{1}{120}\right)}, \frac{1}{6}\right) \cdot x, x\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right) \cdot x, x\right) \]

      17. lower-*.f64 68.6

          \[\leadsto \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right) \cdot x, x\right) \]

   8. Applied rewrites 68.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right) \cdot x, x\right)} \]

   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. lower-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. lower-*.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. lower-*.f64 59.4

          \[\leadsto \mathsf{fma}\left(0.16666666666666666, \color{blue}{\left(y \cdot y\right)} \cdot x, x\right) \]

   11. Applied rewrites 59.4%

       \[\leadsto \color{blue}{\mathsf{fma}\left(0.16666666666666666, \left(y \cdot y\right) \cdot x, x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin x \leq -0.005:\\ \;\;\;\;x \cdot \left(-0.16666666666666666 \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.16666666666666666, x \cdot \left(y \cdot y\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 34.3% accurate, 12.8× speedup

Math

\[\begin{array}{l} \\ x \cdot \mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* x (fma x (* x -0.16666666666666666) 1.0)))

C

double code(double x, double y) {
	return x * fma(x, (x * -0.16666666666666666), 1.0);
}

Julia

function code(x, y)
	return Float64(x * fma(x, Float64(x * -0.16666666666666666), 1.0))
end

Wolfram

code[x_, y_] := N[(x * N[(x * N[(x * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\sin x \cdot \frac{\sinh y}{y} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\sin x} \]
4. Step-by-step derivation

   1. lower-sin.f64 48.4

      \[\leadsto \color{blue}{\sin x} \]

5. Applied rewrites 48.4%

   \[\leadsto \color{blue}{\sin x} \]

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. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{-1}{6} \cdot {x}^{2}, x\right)} \]

   5. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{-1}{6} \cdot {x}^{2}}, x\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(x, \frac{-1}{6} \cdot \color{blue}{\left(x \cdot x\right)}, x\right) \]

   7. lower-*.f64 34.7

      \[\leadsto \mathsf{fma}\left(x, -0.16666666666666666 \cdot \color{blue}{\left(x \cdot x\right)}, x\right) \]

8. Applied rewrites 34.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, -0.16666666666666666 \cdot \left(x \cdot x\right), x\right)} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto x \cdot \left(\frac{-1}{6} \cdot \color{blue}{\left(x \cdot x\right)}\right) + x \]

   2. lift-*.f64 N/A

      \[\leadsto x \cdot \color{blue}{\left(\frac{-1}{6} \cdot \left(x \cdot x\right)\right)} + x \]

   3. *-commutative N/A

      \[\leadsto \color{blue}{\left(\frac{-1}{6} \cdot \left(x \cdot x\right)\right) \cdot x} + x \]

   4. distribute-lft1-in N/A

      \[\leadsto \color{blue}{\left(\frac{-1}{6} \cdot \left(x \cdot x\right) + 1\right) \cdot x} \]

   5. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{-1}{6} \cdot \left(x \cdot x\right) + 1\right) \cdot x} \]

   6. lift-*.f64 N/A

      \[\leadsto \left(\color{blue}{\frac{-1}{6} \cdot \left(x \cdot x\right)} + 1\right) \cdot x \]

   7. *-commutative N/A

      \[\leadsto \left(\color{blue}{\left(x \cdot x\right) \cdot \frac{-1}{6}} + 1\right) \cdot x \]

   8. lift-*.f64 N/A

      \[\leadsto \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{-1}{6} + 1\right) \cdot x \]

   9. associate-*l* N/A

      \[\leadsto \left(\color{blue}{x \cdot \left(x \cdot \frac{-1}{6}\right)} + 1\right) \cdot x \]

   10. lower-fma.f64 N/A

       \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)} \cdot x \]

   11. lower-*.f64 34.7

       \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot -0.16666666666666666}, 1\right) \cdot x \]

10. Applied rewrites 34.7%

    \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right) \cdot x} \]

11. Final simplification 34.7%

    \[\leadsto x \cdot \mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right) \]
12. Add Preprocessing

Alternative 24: 10.9% accurate, 13.6× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(-0.16666666666666666 \cdot \left(x \cdot x\right)\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* x (* -0.16666666666666666 (* x x))))

C

double code(double x, double y) {
	return x * (-0.16666666666666666 * (x * x));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x * ((-0.16666666666666666d0) * (x * x))
end function

Java

public static double code(double x, double y) {
	return x * (-0.16666666666666666 * (x * x));
}

Python

def code(x, y):
	return x * (-0.16666666666666666 * (x * x))

Julia

function code(x, y)
	return Float64(x * Float64(-0.16666666666666666 * Float64(x * x)))
end

MATLAB

function tmp = code(x, y)
	tmp = x * (-0.16666666666666666 * (x * x));
end

Wolfram

code[x_, y_] := N[(x * N[(-0.16666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\sin x \cdot \frac{\sinh y}{y} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\sin x} \]
4. Step-by-step derivation

   1. lower-sin.f64 48.4

      \[\leadsto \color{blue}{\sin x} \]

5. Applied rewrites 48.4%

   \[\leadsto \color{blue}{\sin x} \]

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. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{-1}{6} \cdot {x}^{2}, x\right)} \]

   5. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{-1}{6} \cdot {x}^{2}}, x\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(x, \frac{-1}{6} \cdot \color{blue}{\left(x \cdot x\right)}, x\right) \]

   7. lower-*.f64 34.7

      \[\leadsto \mathsf{fma}\left(x, -0.16666666666666666 \cdot \color{blue}{\left(x \cdot x\right)}, x\right) \]

8. Applied rewrites 34.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, -0.16666666666666666 \cdot \left(x \cdot x\right), x\right)} \]

9. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{-1}{6} \cdot {x}^{3}} \]
10. Step-by-step derivation

    1. unpow3 N/A

       \[\leadsto \frac{-1}{6} \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot x\right)} \]

    2. unpow2 N/A

       \[\leadsto \frac{-1}{6} \cdot \left(\color{blue}{{x}^{2}} \cdot x\right) \]

    3. associate-*r* N/A

       \[\leadsto \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2}\right) \cdot x} \]

    4. *-commutative N/A

       \[\leadsto \color{blue}{x \cdot \left(\frac{-1}{6} \cdot {x}^{2}\right)} \]

    5. lower-*.f64 N/A

       \[\leadsto \color{blue}{x \cdot \left(\frac{-1}{6} \cdot {x}^{2}\right)} \]

    6. lower-*.f64 N/A

       \[\leadsto x \cdot \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2}\right)} \]

    7. unpow2 N/A

       \[\leadsto x \cdot \left(\frac{-1}{6} \cdot \color{blue}{\left(x \cdot x\right)}\right) \]

    8. lower-*.f64 9.1

       \[\leadsto x \cdot \left(-0.16666666666666666 \cdot \color{blue}{\left(x \cdot x\right)}\right) \]

11. Applied rewrites 9.1%

    \[\leadsto \color{blue}{x \cdot \left(-0.16666666666666666 \cdot \left(x \cdot x\right)\right)} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024214 
(FPCore (x y)
  :name "Linear.Quaternion:$ccos from linear-1.19.1.3"
  :precision binary64
  (* (sin x) (/ (sinh y) y)))