Linear.Quaternion:$ccos from linear-1.19.1.3

Percentage Accurate: 100.0% → 100.0%

Time: 14.1s

Alternatives: 23

Speedup: 1.0×

• 49.5% 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: 81.9% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sin x) (/ (sinh y) y))))
   (if (<= t_0 (- INFINITY))
     (*
      (fma x (* -0.16666666666666666 (* x x)) x)
      (fma (* y y) (fma (* y y) 0.008333333333333333 0.16666666666666666) 1.0))
     (if (<= t_0 2.0)
       (*
        (sin x)
        (fma
         (* y y)
         (fma y (* y 0.008333333333333333) 0.16666666666666666)
         1.0))
       (/
        (*
         x
         (fma
          (* y (* y y))
          (fma
           y
           (* y (fma (* y y) 0.0001984126984126984 0.008333333333333333))
           0.16666666666666666)
          y))
        y)))))

C

double code(double x, double y) {
	double t_0 = sin(x) * (sinh(y) / y);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma(x, (-0.16666666666666666 * (x * x)), x) * fma((y * y), fma((y * y), 0.008333333333333333, 0.16666666666666666), 1.0);
	} else if (t_0 <= 2.0) {
		tmp = sin(x) * fma((y * y), fma(y, (y * 0.008333333333333333), 0.16666666666666666), 1.0);
	} else {
		tmp = (x * fma((y * (y * y)), fma(y, (y * fma((y * y), 0.0001984126984126984, 0.008333333333333333)), 0.16666666666666666), y)) / y;
	}
	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(x, Float64(-0.16666666666666666 * Float64(x * x)), x) * fma(Float64(y * y), fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666), 1.0));
	elseif (t_0 <= 2.0)
		tmp = Float64(sin(x) * fma(Float64(y * y), fma(y, Float64(y * 0.008333333333333333), 0.16666666666666666), 1.0));
	else
		tmp = Float64(Float64(x * fma(Float64(y * Float64(y * y)), fma(y, Float64(y * fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333)), 0.16666666666666666), y)) / y);
	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[(x * N[(-0.16666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(N[Sin[x], $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision] / y), $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 71.2

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

   5. Simplified 71.2%

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

   6. Taylor expanded in y around 0

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

      1. associate-+r+ N/A

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

      2. distribute-lft-in N/A

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

      3. associate-+r+ N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt1-in N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

   8. Simplified 56.2%

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

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

   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. Simplified 99.1%

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

   if 2 < (*.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. Step-by-step derivation

      1. lift-sin.f64 N/A

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

      2. lift-sinh.f64 N/A

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

      3. clear-num N/A

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

      4. clear-num N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

   7. Simplified 91.9%

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

   8. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   10. Simplified 62.2%

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

Alternative 3: 81.8% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sin x) (/ (sinh y) y))))
   (if (<= t_0 (- INFINITY))
     (*
      (fma x (* -0.16666666666666666 (* x x)) x)
      (fma (* y y) (fma (* y y) 0.008333333333333333 0.16666666666666666) 1.0))
     (if (<= t_0 2.0)
       (* (sin x) (fma 0.16666666666666666 (* y y) 1.0))
       (/
        (*
         x
         (fma
          (* y (* y y))
          (fma
           y
           (* y (fma (* y y) 0.0001984126984126984 0.008333333333333333))
           0.16666666666666666)
          y))
        y)))))

C

double code(double x, double y) {
	double t_0 = sin(x) * (sinh(y) / y);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma(x, (-0.16666666666666666 * (x * x)), x) * fma((y * y), fma((y * y), 0.008333333333333333, 0.16666666666666666), 1.0);
	} else if (t_0 <= 2.0) {
		tmp = sin(x) * fma(0.16666666666666666, (y * y), 1.0);
	} else {
		tmp = (x * fma((y * (y * y)), fma(y, (y * fma((y * y), 0.0001984126984126984, 0.008333333333333333)), 0.16666666666666666), y)) / y;
	}
	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(x, Float64(-0.16666666666666666 * Float64(x * x)), x) * fma(Float64(y * y), fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666), 1.0));
	elseif (t_0 <= 2.0)
		tmp = Float64(sin(x) * fma(0.16666666666666666, Float64(y * y), 1.0));
	else
		tmp = Float64(Float64(x * fma(Float64(y * Float64(y * y)), fma(y, Float64(y * fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333)), 0.16666666666666666), y)) / y);
	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[(x * N[(-0.16666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(N[Sin[x], $MachinePrecision] * N[(0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision] / y), $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 71.2

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

   5. Simplified 71.2%

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

   6. Taylor expanded in y around 0

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

      1. associate-+r+ N/A

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

      2. distribute-lft-in N/A

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

      3. associate-+r+ N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt1-in N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

   8. Simplified 56.2%

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

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

   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 98.7

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

   5. Simplified 98.7%

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

   if 2 < (*.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. Step-by-step derivation

      1. lift-sin.f64 N/A

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

      2. lift-sinh.f64 N/A

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

      3. clear-num N/A

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

      4. clear-num N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

   7. Simplified 91.9%

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

   8. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   10. Simplified 62.2%

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

Alternative 4: 81.6% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sin x) (/ (sinh y) y))))
   (if (<= t_0 (- INFINITY))
     (*
      (fma x (* -0.16666666666666666 (* x x)) x)
      (fma (* y y) (fma (* y y) 0.008333333333333333 0.16666666666666666) 1.0))
     (if (<= t_0 2.0)
       (sin x)
       (/
        (*
         x
         (fma
          (* y (* y y))
          (fma
           y
           (* y (fma (* y y) 0.0001984126984126984 0.008333333333333333))
           0.16666666666666666)
          y))
        y)))))

C

double code(double x, double y) {
	double t_0 = sin(x) * (sinh(y) / y);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma(x, (-0.16666666666666666 * (x * x)), x) * fma((y * y), fma((y * y), 0.008333333333333333, 0.16666666666666666), 1.0);
	} else if (t_0 <= 2.0) {
		tmp = sin(x);
	} else {
		tmp = (x * fma((y * (y * y)), fma(y, (y * fma((y * y), 0.0001984126984126984, 0.008333333333333333)), 0.16666666666666666), y)) / y;
	}
	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(x, Float64(-0.16666666666666666 * Float64(x * x)), x) * fma(Float64(y * y), fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666), 1.0));
	elseif (t_0 <= 2.0)
		tmp = sin(x);
	else
		tmp = Float64(Float64(x * fma(Float64(y * Float64(y * y)), fma(y, Float64(y * fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333)), 0.16666666666666666), y)) / y);
	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[(x * N[(-0.16666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[Sin[x], $MachinePrecision], N[(N[(x * N[(N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision] / y), $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 71.2

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

   5. Simplified 71.2%

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

   6. Taylor expanded in y around 0

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

      1. associate-+r+ N/A

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

      2. distribute-lft-in N/A

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

      3. associate-+r+ N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt1-in N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

   8. Simplified 56.2%

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

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

   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 98.4

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

   5. Simplified 98.4%

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

   if 2 < (*.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. Step-by-step derivation

      1. lift-sin.f64 N/A

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

      2. lift-sinh.f64 N/A

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

      3. clear-num N/A

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

      4. clear-num N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

   7. Simplified 91.9%

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

   8. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   10. Simplified 62.2%

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

Alternative 5: 89.9% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sinh y) y)))
   (if (<= (* (sin x) t_0) 2.0)
     (*
      (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 tmp;
	if ((sin(x) * t_0) <= 2.0) {
		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)
	tmp = 0.0
	if (Float64(sin(x) * t_0) <= 2.0)
		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]}, If[LessEqual[N[(N[Sin[x], $MachinePrecision] * t$95$0), $MachinePrecision], 2.0], 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 2 regimes

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

   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. Simplified 94.4%

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

   if 2 < (*.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 67.6

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

   5. Simplified 67.6%

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

4. Final simplification 87.0%

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

Alternative 6: 59.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right)\\ \mathbf{if}\;\sin x \cdot \frac{\sinh y}{y} \leq 0.05:\\ \;\;\;\;\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, t\_0, 0.16666666666666666\right), 1\right) \cdot \mathsf{fma}\left(x, -0.16666666666666666 \cdot \left(x \cdot x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \mathsf{fma}\left(y \cdot \left(y \cdot y\right), \mathsf{fma}\left(y, y \cdot t\_0, 0.16666666666666666\right), y\right)}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma (* y y) 0.0001984126984126984 0.008333333333333333)))
   (if (<= (* (sin x) (/ (sinh y) y)) 0.05)
     (*
      (fma y (* y (fma (* y y) t_0 0.16666666666666666)) 1.0)
      (fma x (* -0.16666666666666666 (* x x)) x))
     (/ (* x (fma (* y (* y y)) (fma y (* y t_0) 0.16666666666666666) y)) y))))

C

double code(double x, double y) {
	double t_0 = fma((y * y), 0.0001984126984126984, 0.008333333333333333);
	double tmp;
	if ((sin(x) * (sinh(y) / y)) <= 0.05) {
		tmp = fma(y, (y * fma((y * y), t_0, 0.16666666666666666)), 1.0) * fma(x, (-0.16666666666666666 * (x * x)), x);
	} else {
		tmp = (x * fma((y * (y * y)), fma(y, (y * t_0), 0.16666666666666666), y)) / y;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333)
	tmp = 0.0
	if (Float64(sin(x) * Float64(sinh(y) / y)) <= 0.05)
		tmp = Float64(fma(y, Float64(y * fma(Float64(y * y), t_0, 0.16666666666666666)), 1.0) * fma(x, Float64(-0.16666666666666666 * Float64(x * x)), x));
	else
		tmp = Float64(Float64(x * fma(Float64(y * Float64(y * y)), fma(y, Float64(y * t_0), 0.16666666666666666), y)) / y);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision]}, If[LessEqual[N[(N[Sin[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], 0.05], N[(N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * t$95$0 + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x * N[(-0.16666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(y * N[(y * t$95$0), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\sin x + {y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x + {y}^{2} \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{120} \cdot \sin x\right)\right)} \]
   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. Simplified 93.3%

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

   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 63.2

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

   8. Simplified 63.2%

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

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

   1. Initial program 100.0%

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

      1. lift-sin.f64 N/A

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

      2. lift-sinh.f64 N/A

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

      3. clear-num N/A

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

      4. clear-num N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

   7. Simplified 94.3%

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

   8. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   10. Simplified 44.3%

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

4. Final simplification 55.7%

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

Alternative 7: 57.6% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in 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 67.7

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

   5. Simplified 67.7%

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

   6. Taylor expanded in y around 0

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

      1. associate-+r+ N/A

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

      2. distribute-lft-in N/A

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

      3. associate-+r+ N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt1-in N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

   8. Simplified 62.4%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in 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. Simplified 91.6%

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

   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. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   8. Simplified 39.6%

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

   10. Applied egg-rr 42.4%

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

4. Final simplification 54.5%

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

Alternative 8: 54.3% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\sin x + \frac{1}{6} \cdot \left({y}^{2} \cdot \sin x\right)} \]
   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 81.1

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

   5. Simplified 81.1%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt1-in N/A

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

      10. +-commutative N/A

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

   8. Simplified 56.2%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in 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. Simplified 91.6%

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

   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. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   8. Simplified 39.6%

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

   10. Applied egg-rr 42.4%

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

4. Final simplification 50.7%

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

Alternative 9: 54.1% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (sin x) (/ (sinh y) y)) 0.05)
   (*
    x
    (*
     (fma -0.16666666666666666 (* x x) 1.0)
     (fma y (* y 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)) <= 0.05) {
		tmp = x * (fma(-0.16666666666666666, (x * x), 1.0) * fma(y, (y * 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)) <= 0.05)
		tmp = Float64(x * Float64(fma(-0.16666666666666666, Float64(x * x), 1.0) * fma(y, Float64(y * 0.16666666666666666), 1.0)));
	else
		tmp = fma(Float64(x * y), Float64(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], 0.05], N[(x * N[(N[(-0.16666666666666666 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(y * N[(y * 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * y), $MachinePrecision] * N[(y * 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)) < 0.050000000000000003

   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 81.1

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

   5. Simplified 81.1%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt1-in N/A

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

      10. +-commutative N/A

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

   8. Simplified 56.2%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in 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. Simplified 91.6%

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

   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. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   8. Simplified 39.6%

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

   10. Applied egg-rr 41.5%

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

Alternative 10: 53.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(sin(x) * Float64(sinh(y) / y)) <= 0.05)
		tmp = Float64(x * Float64(fma(-0.16666666666666666, Float64(x * x), 1.0) * fma(y, Float64(y * 0.16666666666666666), 1.0)));
	else
		tmp = fma(Float64(y * Float64(y * Float64(Float64(y * y) * 0.0001984126984126984))), 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], 0.05], N[(x * N[(N[(-0.16666666666666666 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(y * N[(y * 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 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)) < 0.050000000000000003

   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 81.1

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

   5. Simplified 81.1%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt1-in N/A

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

      10. +-commutative N/A

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

   8. Simplified 56.2%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in 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. Simplified 91.6%

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

   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. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   8. Simplified 39.6%

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

   9. Taylor expanded in y around inf

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

       1. metadata-eval N/A

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

       2. pow-sqr N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. unpow2 N/A

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

       6. associate-*l* N/A

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

       7. *-commutative N/A

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

       8. associate-*r* N/A

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

       9. unpow2 N/A

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

       10. unpow3 N/A

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

       11. lower-*.f64 N/A

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

       12. unpow3 N/A

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

       13. unpow2 N/A

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

       14. associate-*r* N/A

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

       15. *-commutative N/A

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

       16. lower-*.f64 N/A

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

       17. *-commutative N/A

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

       18. lower-*.f64 N/A

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

       19. unpow2 N/A

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

       20. lower-*.f64 39.6

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

   11. Simplified 39.6%

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

Alternative 11: 51.9% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (sin x) (/ (sinh y) y)) 0.05)
   (*
    x
    (*
     (fma -0.16666666666666666 (* x x) 1.0)
     (fma y (* y 0.16666666666666666) 1.0)))
   (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)) <= 0.05) {
		tmp = x * (fma(-0.16666666666666666, (x * x), 1.0) * fma(y, (y * 0.16666666666666666), 1.0));
	} 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)) <= 0.05)
		tmp = Float64(x * Float64(fma(-0.16666666666666666, Float64(x * x), 1.0) * fma(y, Float64(y * 0.16666666666666666), 1.0)));
	else
		tmp = fma(y, Float64(y * Float64(x * fma(Float64(y * 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], 0.05], N[(x * N[(N[(-0.16666666666666666 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(y * N[(y * 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(y * N[(x * N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\sin x + \frac{1}{6} \cdot \left({y}^{2} \cdot \sin x\right)} \]
   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 81.1

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

   5. Simplified 81.1%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt1-in N/A

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

      10. +-commutative N/A

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

   8. Simplified 56.2%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in 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. Simplified 91.6%

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

   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. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   8. Simplified 39.6%

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

   9. Taylor expanded in y around 0

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

       1. +-commutative N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. lower-fma.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. *-commutative N/A

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

       7. associate-*r* N/A

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

       8. distribute-rgt-out N/A

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

       9. +-commutative N/A

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

       10. lower-*.f64 N/A

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

       11. +-commutative N/A

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

       12. *-commutative N/A

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

       13. lower-fma.f64 N/A

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

       14. unpow2 N/A

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

       15. lower-*.f64 30.5

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

   11. Simplified 30.5%

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

Alternative 12: 49.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin x \leq -0.02:\\ \;\;\;\;x \cdot \left(\left(x \cdot x\right) \cdot \mathsf{fma}\left(y \cdot y, -0.027777777777777776, -0.16666666666666666\right)\right)\\ \mathbf{elif}\;\sin x \leq 0.19:\\ \;\;\;\;\mathsf{fma}\left(0.16666666666666666, x \cdot \left(y \cdot y\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, x \cdot \left(x \cdot \left(x \cdot 0.008333333333333333\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sin x) -0.02)
   (* x (* (* x x) (fma (* y y) -0.027777777777777776 -0.16666666666666666)))
   (if (<= (sin x) 0.19)
     (fma 0.16666666666666666 (* x (* y y)) x)
     (fma (* x x) (* x (* x (* x 0.008333333333333333))) x))))

C

double code(double x, double y) {
	double tmp;
	if (sin(x) <= -0.02) {
		tmp = x * ((x * x) * fma((y * y), -0.027777777777777776, -0.16666666666666666));
	} else if (sin(x) <= 0.19) {
		tmp = fma(0.16666666666666666, (x * (y * y)), x);
	} else {
		tmp = fma((x * x), (x * (x * (x * 0.008333333333333333))), x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (sin(x) <= -0.02)
		tmp = Float64(x * Float64(Float64(x * x) * fma(Float64(y * y), -0.027777777777777776, -0.16666666666666666)));
	elseif (sin(x) <= 0.19)
		tmp = fma(0.16666666666666666, Float64(x * Float64(y * y)), x);
	else
		tmp = fma(Float64(x * x), Float64(x * Float64(x * Float64(x * 0.008333333333333333))), x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[Sin[x], $MachinePrecision], -0.02], N[(x * N[(N[(x * x), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * -0.027777777777777776 + -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[x], $MachinePrecision], 0.19], N[(0.16666666666666666 * N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(x * 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 x) < -0.0200000000000000004

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\sin x + \frac{1}{6} \cdot \left({y}^{2} \cdot \sin x\right)} \]
   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 77.1

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

   5. Simplified 77.1%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt1-in N/A

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

      10. +-commutative N/A

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

   8. Simplified 15.7%

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

   9. Taylor expanded in x around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. lower-*.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 N/A

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

       7. +-commutative N/A

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

       8. distribute-rgt-in N/A

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

       9. *-commutative N/A

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

       10. associate-*l* N/A

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

       11. metadata-eval N/A

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

       12. metadata-eval N/A

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

       13. lower-fma.f64 N/A

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

       14. unpow2 N/A

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

       15. lower-*.f64 14.7

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

   11. Simplified 14.7%

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

   if -0.0200000000000000004 < (sin.f64 x) < 0.19

   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. Simplified 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

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   8. Simplified 84.0%

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

   9. Taylor expanded in y around 0

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

   11. Simplified 69.3%

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

   if 0.19 < (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

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

      1. lower-sin.f64 44.7

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

   5. Simplified 44.7%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-*l* N/A

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

      4. *-lft-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 23.9

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

   8. Simplified 23.9%

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

   9. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 23.9

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

   11. Simplified 23.9%

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

4. Final simplification 44.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin x \leq -0.02:\\ \;\;\;\;x \cdot \left(\left(x \cdot x\right) \cdot \mathsf{fma}\left(y \cdot y, -0.027777777777777776, -0.16666666666666666\right)\right)\\ \mathbf{elif}\;\sin x \leq 0.19:\\ \;\;\;\;\mathsf{fma}\left(0.16666666666666666, x \cdot \left(y \cdot y\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, x \cdot \left(x \cdot \left(x \cdot 0.008333333333333333\right)\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 41.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   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 66.2

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

   5. Simplified 66.2%

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

   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 47.7

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

   8. Simplified 47.7%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in 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. Simplified 91.6%

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

   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. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   8. Simplified 39.6%

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

   9. Taylor expanded in y around 0

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

   11. Simplified 26.4%

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

Alternative 14: 59.4% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (sin.f64 x) < 0.050000000000000003

   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 73.3

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

   5. Simplified 73.3%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. pow-plus N/A

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

      6. metadata-eval N/A

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

      7. cube-unmult N/A

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

      8. unpow2 N/A

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

      9. *-lft-identity N/A

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

      10. lower-fma.f64 N/A

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

   8. Simplified 69.0%

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

   if 0.050000000000000003 < (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. Simplified 94.6%

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

   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. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   8. Simplified 24.4%

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

   9. Taylor expanded in y around inf

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

       1. metadata-eval N/A

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

       2. pow-sqr N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. unpow2 N/A

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

       6. associate-*l* N/A

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

       7. *-commutative N/A

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

       8. associate-*r* N/A

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

       9. unpow2 N/A

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

       10. unpow3 N/A

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

       11. lower-*.f64 N/A

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

       12. unpow3 N/A

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

       13. unpow2 N/A

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

       14. associate-*r* N/A

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

       15. *-commutative N/A

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

       16. lower-*.f64 N/A

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

       17. *-commutative N/A

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

       18. lower-*.f64 N/A

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

       19. unpow2 N/A

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

       20. lower-*.f64 24.4

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

   11. Simplified 24.4%

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

Alternative 15: 58.8% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sin x) 0.05)
   (*
    (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 (* (* y y) 0.0001984126984126984))) (* x (* y y)) x)))

C

double code(double x, double y) {
	double tmp;
	if (sin(x) <= 0.05) {
		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 * ((y * y) * 0.0001984126984126984))), (x * (y * y)), x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (sin(x) <= 0.05)
		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 = fma(Float64(y * Float64(y * Float64(Float64(y * y) * 0.0001984126984126984))), Float64(x * Float64(y * y)), x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[Sin[x], $MachinePrecision], 0.05], 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[(y * N[(y * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 x) < 0.050000000000000003

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\sin x + {y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x + {y}^{2} \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{120} \cdot \sin x\right)\right)} \]
   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. Simplified 91.8%

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

   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 66.9

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

   8. Simplified 66.9%

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

   if 0.050000000000000003 < (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. Simplified 94.6%

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

   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. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   8. Simplified 24.4%

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

   9. Taylor expanded in y around inf

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

       1. metadata-eval N/A

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

       2. pow-sqr N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. unpow2 N/A

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

       6. associate-*l* N/A

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

       7. *-commutative N/A

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

       8. associate-*r* N/A

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

       9. unpow2 N/A

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

       10. unpow3 N/A

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

       11. lower-*.f64 N/A

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

       12. unpow3 N/A

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

       13. unpow2 N/A

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

       14. associate-*r* N/A

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

       15. *-commutative N/A

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

       16. lower-*.f64 N/A

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

       17. *-commutative N/A

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

       18. lower-*.f64 N/A

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

       19. unpow2 N/A

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

       20. lower-*.f64 24.4

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

   11. Simplified 24.4%

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

4. Final simplification 55.3%

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

Alternative 16: 92.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)\\ \mathbf{if}\;y \leq 0.165:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{+77}:\\ \;\;\;\;\frac{\sinh y \cdot \mathsf{fma}\left(x, -0.16666666666666666 \cdot \left(x \cdot x\right), x\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (*
          (sin x)
          (fma
           (* y y)
           (fma y (* y 0.008333333333333333) 0.16666666666666666)
           1.0))))
   (if (<= y 0.165)
     t_0
     (if (<= y 3.8e+77)
       (/ (* (sinh y) (fma x (* -0.16666666666666666 (* x x)) x)) y)
       t_0))))

C

double code(double x, double y) {
	double t_0 = sin(x) * fma((y * y), fma(y, (y * 0.008333333333333333), 0.16666666666666666), 1.0);
	double tmp;
	if (y <= 0.165) {
		tmp = t_0;
	} else if (y <= 3.8e+77) {
		tmp = (sinh(y) * fma(x, (-0.16666666666666666 * (x * x)), x)) / y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(x) * fma(Float64(y * y), fma(y, Float64(y * 0.008333333333333333), 0.16666666666666666), 1.0))
	tmp = 0.0
	if (y <= 0.165)
		tmp = t_0;
	elseif (y <= 3.8e+77)
		tmp = Float64(Float64(sinh(y) * fma(x, Float64(-0.16666666666666666 * Float64(x * x)), x)) / y);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[x], $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 0.165], t$95$0, If[LessEqual[y, 3.8e+77], N[(N[(N[Sinh[y], $MachinePrecision] * N[(x * N[(-0.16666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < 0.165000000000000008 or 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. Simplified 93.2%

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

   if 0.165000000000000008 < y < 3.8000000000000001e77

   1. Initial program 99.8%

      \[\sin x \cdot \frac{\sinh y}{y} \]
   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 81.1

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

   5. Simplified 81.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-sinh.f64 N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 81.3

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

      8. lift-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 81.3

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

   7. Applied egg-rr 81.3%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.165:\\ \;\;\;\;\sin x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{+77}:\\ \;\;\;\;\frac{\sinh y \cdot \mathsf{fma}\left(x, -0.16666666666666666 \cdot \left(x \cdot x\right), x\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 92.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)\\ \mathbf{if}\;y \leq 0.165:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{+77}:\\ \;\;\;\;\frac{\sinh y}{y} \cdot \mathsf{fma}\left(-0.16666666666666666, x \cdot \left(x \cdot x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (*
          (sin x)
          (fma
           (* y y)
           (fma y (* y 0.008333333333333333) 0.16666666666666666)
           1.0))))
   (if (<= y 0.165)
     t_0
     (if (<= y 3.8e+77)
       (* (/ (sinh y) y) (fma -0.16666666666666666 (* x (* x x)) x))
       t_0))))

C

double code(double x, double y) {
	double t_0 = sin(x) * fma((y * y), fma(y, (y * 0.008333333333333333), 0.16666666666666666), 1.0);
	double tmp;
	if (y <= 0.165) {
		tmp = t_0;
	} else if (y <= 3.8e+77) {
		tmp = (sinh(y) / y) * fma(-0.16666666666666666, (x * (x * x)), x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(x) * fma(Float64(y * y), fma(y, Float64(y * 0.008333333333333333), 0.16666666666666666), 1.0))
	tmp = 0.0
	if (y <= 0.165)
		tmp = t_0;
	elseif (y <= 3.8e+77)
		tmp = Float64(Float64(sinh(y) / y) * fma(-0.16666666666666666, Float64(x * Float64(x * x)), x));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[x], $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 0.165], t$95$0, If[LessEqual[y, 3.8e+77], N[(N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision] * N[(-0.16666666666666666 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < 0.165000000000000008 or 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. Simplified 93.2%

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

   if 0.165000000000000008 < y < 3.8000000000000001e77

   1. Initial program 99.8%

      \[\sin x \cdot \frac{\sinh y}{y} \]
   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 81.1

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

   5. Simplified 81.1%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.165:\\ \;\;\;\;\sin x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{+77}:\\ \;\;\;\;\frac{\sinh y}{y} \cdot \mathsf{fma}\left(-0.16666666666666666, x \cdot \left(x \cdot x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 79.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 10^{-33}:\\ \;\;\;\;\frac{\sinh y}{y} \cdot \mathsf{fma}\left(-0.16666666666666666, x \cdot \left(x \cdot x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1e-33)
   (* (/ (sinh y) y) (fma -0.16666666666666666 (* x (* x x)) x))
   (*
    (sin x)
    (fma
     y
     (*
      y
      (fma
       (* y y)
       (fma (* y y) 0.0001984126984126984 0.008333333333333333)
       0.16666666666666666))
     1.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1e-33) {
		tmp = (sinh(y) / y) * fma(-0.16666666666666666, (x * (x * x)), x);
	} else {
		tmp = sin(x) * fma(y, (y * fma((y * y), fma((y * y), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666)), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1e-33)
		tmp = Float64(Float64(sinh(y) / y) * fma(-0.16666666666666666, Float64(x * Float64(x * x)), x));
	else
		tmp = Float64(sin(x) * fma(y, Float64(y * fma(Float64(y * y), fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666)), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1e-33], 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[(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]]

Derivation

1. Split input into 2 regimes

2. if x < 1.0000000000000001e-33

   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 73.5

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

   5. Simplified 73.5%

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

   if 1.0000000000000001e-33 < 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. Simplified 95.9%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 10^{-33}:\\ \;\;\;\;\frac{\sinh y}{y} \cdot \mathsf{fma}\left(-0.16666666666666666, x \cdot \left(x \cdot x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 53.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin x \leq -0.02:\\ \;\;\;\;x \cdot \left(\left(x \cdot x\right) \cdot \mathsf{fma}\left(y \cdot y, -0.027777777777777776, -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, y \cdot \left(x \cdot \mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sin x) -0.02)
   (* 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) <= -0.02) {
		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 (sin(x) <= -0.02)
		tmp = Float64(x * Float64(Float64(x * x) * fma(Float64(y * y), -0.027777777777777776, -0.16666666666666666)));
	else
		tmp = fma(y, Float64(y * Float64(x * fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666))), x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[Sin[x], $MachinePrecision], -0.02], N[(x * N[(N[(x * x), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * -0.027777777777777776 + -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(y * N[(x * N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 x) < -0.0200000000000000004

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\sin x + \frac{1}{6} \cdot \left({y}^{2} \cdot \sin x\right)} \]
   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 77.1

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

   5. Simplified 77.1%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt1-in N/A

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

      10. +-commutative N/A

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

   8. Simplified 15.7%

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

   9. Taylor expanded in x around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. lower-*.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 N/A

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

       7. +-commutative N/A

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

       8. distribute-rgt-in N/A

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

       9. *-commutative N/A

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

       10. associate-*l* N/A

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

       11. metadata-eval N/A

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

       12. metadata-eval N/A

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

       13. lower-fma.f64 N/A

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

       14. unpow2 N/A

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

       15. lower-*.f64 14.7

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

   11. Simplified 14.7%

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

   if -0.0200000000000000004 < (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. Simplified 92.0%

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

   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. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   8. Simplified 63.7%

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

   9. Taylor expanded in y around 0

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

       1. +-commutative N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. lower-fma.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. *-commutative N/A

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

       7. associate-*r* N/A

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

       8. distribute-rgt-out N/A

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

       9. +-commutative N/A

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

       10. lower-*.f64 N/A

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

       11. +-commutative N/A

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

       12. *-commutative N/A

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

       13. lower-fma.f64 N/A

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

       14. unpow2 N/A

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

       15. lower-*.f64 57.4

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

   11. Simplified 57.4%

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

Alternative 20: 49.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin x \leq -0.02:\\ \;\;\;\;x \cdot \left(\left(x \cdot x\right) \cdot \mathsf{fma}\left(y \cdot y, -0.027777777777777776, -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.16666666666666666, x \cdot \left(y \cdot y\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sin x) -0.02)
   (* 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) <= -0.02) {
		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 (sin(x) <= -0.02)
		tmp = Float64(x * Float64(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[Sin[x], $MachinePrecision], -0.02], N[(x * N[(N[(x * x), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * -0.027777777777777776 + -0.16666666666666666), $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) < -0.0200000000000000004

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\sin x + \frac{1}{6} \cdot \left({y}^{2} \cdot \sin x\right)} \]
   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 77.1

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

   5. Simplified 77.1%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt1-in N/A

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

      10. +-commutative N/A

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

   8. Simplified 15.7%

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

   9. Taylor expanded in x around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. lower-*.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 N/A

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

       7. +-commutative N/A

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

       8. distribute-rgt-in N/A

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

       9. *-commutative N/A

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

       10. associate-*l* N/A

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

       11. metadata-eval N/A

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

       12. metadata-eval N/A

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

       13. lower-fma.f64 N/A

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

       14. unpow2 N/A

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

       15. lower-*.f64 14.7

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

   11. Simplified 14.7%

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

   if -0.0200000000000000004 < (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. Simplified 92.0%

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

   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. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   8. Simplified 63.7%

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

   9. Taylor expanded in y around 0

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

   11. Simplified 53.1%

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

Alternative 21: 47.6% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (sin.f64 x) < -0.0200000000000000004

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-sin.f64 57.4

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

   5. Simplified 57.4%

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

   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 12.4

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

   8. Simplified 12.4%

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

   9. Taylor expanded in x around inf

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

       1. lower-*.f64 N/A

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

       2. cube-mult N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 11.4

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

   11. Simplified 11.4%

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

   if -0.0200000000000000004 < (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. Simplified 92.0%

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

   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. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   8. Simplified 63.7%

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

   9. Taylor expanded in y around 0

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

   11. Simplified 53.1%

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

Alternative 22: 47.6% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (sin.f64 x) < -0.0200000000000000004

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-sin.f64 57.4

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

   5. Simplified 57.4%

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

   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 12.4

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

   8. Simplified 12.4%

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

   9. Taylor expanded in x around inf

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

       1. lower-*.f64 N/A

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

       2. cube-mult N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 11.4

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

   11. Simplified 11.4%

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

   if -0.0200000000000000004 < (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

      \[\leadsto \color{blue}{\sin x + \frac{1}{6} \cdot \left({y}^{2} \cdot \sin x\right)} \]
   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 71.5

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

   5. Simplified 71.5%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt1-in N/A

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

      10. +-commutative N/A

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

   8. Simplified 56.3%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. lower-fma.f64 N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 52.6

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

   11. Simplified 52.6%

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

Alternative 23: 10.7% accurate, 13.6× speedup

Math

\[\begin{array}{l} \\ -0.16666666666666666 \cdot \left(x \cdot \left(x \cdot x\right)\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* -0.16666666666666666 (* x (* x x))))

C

double code(double x, double y) {
	return -0.16666666666666666 * (x * (x * x));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (-0.16666666666666666d0) * (x * (x * x))
end function

Java

public static double code(double x, double y) {
	return -0.16666666666666666 * (x * (x * x));
}

Python

def code(x, y):
	return -0.16666666666666666 * (x * (x * x))

Julia

function code(x, y)
	return Float64(-0.16666666666666666 * Float64(x * Float64(x * x)))
end

MATLAB

function tmp = code(x, y)
	tmp = -0.16666666666666666 * (x * (x * x));
end

Wolfram

code[x_, y_] := N[(-0.16666666666666666 * N[(x * 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 52.4

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

5. Simplified 52.4%

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

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 35.7

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

8. Simplified 35.7%

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

9. Taylor expanded in x around inf

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

    1. lower-*.f64 N/A

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

    2. cube-mult N/A

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

    3. unpow2 N/A

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

    4. lower-*.f64 N/A

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

    5. unpow2 N/A

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

    6. lower-*.f64 10.9

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

11. Simplified 10.9%

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

Reproduce

herbie shell --seed 2024207 
(FPCore (x y)
  :name "Linear.Quaternion:$ccos from linear-1.19.1.3"
  :precision binary64
  (* (sin x) (/ (sinh y) y)))