Linear.Quaternion:$ccos from linear-1.19.1.3

Percentage Accurate: 100.0% → 100.0%

Time: 9.2s

Alternatives: 14

Speedup: 1.0×

• Could not converge on a ground truth

  1. x = 2.593000422638078e-186
  2. y = 6.0163741023195915e+262

• 50.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

Alternative 2: 59.2% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \sin x \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 2.8%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. pow-plus N/A

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

      9. lower-pow.f64 N/A

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

      10. metadata-eval 21.8

          \[\leadsto \mathsf{fma}\left({x}^{\color{blue}{3}}, -0.16666666666666666, x\right) \cdot 1 \]

   8. Applied rewrites 21.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, -0.16666666666666666, x\right)} \cdot 1 \]
   9. Step-by-step derivation

   10. Applied rewrites 21.8%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \sin x \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 99.5%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \sin x \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 2.9%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. pow-plus N/A

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

      8. lower-pow.f64 N/A

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

      9. metadata-eval N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 36.5

          \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(0.008333333333333333, \color{blue}{x \cdot x}, -0.16666666666666666\right), x\right) \cdot 1 \]

   8. Applied rewrites 36.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right), x\right)} \cdot 1 \]
   9. Step-by-step derivation

   10. Applied rewrites 36.5%

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

Alternative 3: 86.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin x \cdot \frac{\sinh y}{y} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, x \cdot x, 0.008333333333333333\right), x \cdot x, -0.16666666666666666\right), {x}^{3}, x\right) \cdot \mathsf{fma}\left(\left(0.008333333333333333 \cdot y\right) \cdot y, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984, y \cdot y, 0.008333333333333333\right) \cdot y, y, 0.16666666666666666\right), y \cdot y, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (sin x) (/ (sinh y) y)) (- INFINITY))
   (*
    (fma
     (fma
      (fma -0.0001984126984126984 (* x x) 0.008333333333333333)
      (* x x)
      -0.16666666666666666)
     (pow x 3.0)
     x)
    (fma (* (* 0.008333333333333333 y) y) (* y y) 1.0))
   (*
    (sin x)
    (fma
     (fma
      (* (fma 0.0001984126984126984 (* y y) 0.008333333333333333) y)
      y
      0.16666666666666666)
     (* y y)
     1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((sin(x) * (sinh(y) / y)) <= -((double) INFINITY)) {
		tmp = fma(fma(fma(-0.0001984126984126984, (x * x), 0.008333333333333333), (x * x), -0.16666666666666666), pow(x, 3.0), x) * fma(((0.008333333333333333 * y) * y), (y * y), 1.0);
	} else {
		tmp = sin(x) * fma(fma((fma(0.0001984126984126984, (y * y), 0.008333333333333333) * y), y, 0.16666666666666666), (y * y), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(sin(x) * Float64(sinh(y) / y)) <= Float64(-Inf))
		tmp = Float64(fma(fma(fma(-0.0001984126984126984, Float64(x * x), 0.008333333333333333), Float64(x * x), -0.16666666666666666), (x ^ 3.0), x) * fma(Float64(Float64(0.008333333333333333 * y) * y), Float64(y * y), 1.0));
	else
		tmp = Float64(sin(x) * fma(fma(Float64(fma(0.0001984126984126984, Float64(y * y), 0.008333333333333333) * y), y, 0.16666666666666666), Float64(y * y), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Sin[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], (-Infinity)], N[(N[(N[(N[(-0.0001984126984126984 * N[(x * x), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[Power[x, 3.0], $MachinePrecision] + x), $MachinePrecision] * N[(N[(N[(0.008333333333333333 * y), $MachinePrecision] * y), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Sin[x], $MachinePrecision] * N[(N[(N[(N[(0.0001984126984126984 * N[(y * y), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] * y), $MachinePrecision] * y + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 86.5

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

   5. Applied rewrites 86.5%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 86.5%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 73.1%

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

   12. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. distribute-rgt-in N/A

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

       3. *-lft-identity N/A

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

       4. *-commutative N/A

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

       5. associate-*l* N/A

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

       6. unpow2 N/A

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

       7. unpow3 N/A

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

       8. lower-fma.f64 N/A

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

   14. Applied rewrites 56.8%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, x \cdot x, 0.008333333333333333\right), x \cdot x, -0.16666666666666666\right), {x}^{3}, x\right)} \cdot \mathsf{fma}\left(\left(0.008333333333333333 \cdot y\right) \cdot y, y \cdot y, 1\right) \]

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 91.4

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

   5. Applied rewrites 91.4%

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

   7. Applied rewrites 91.4%

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

Alternative 4: 88.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 620:\\ \;\;\;\;\sin x \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333 \cdot y, y, 0.16666666666666666\right), y \cdot y, 1\right)\\ \mathbf{elif}\;y \leq 7 \cdot 10^{+72}:\\ \;\;\;\;\mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right), x\right) \cdot \mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \left(\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right) \cdot y\right) \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 620.0)
   (*
    (sin x)
    (fma (fma (* 0.008333333333333333 y) y 0.16666666666666666) (* y y) 1.0))
   (if (<= y 7e+72)
     (*
      (fma
       (pow x 3.0)
       (fma 0.008333333333333333 (* x x) -0.16666666666666666)
       x)
      (fma (* y y) 0.16666666666666666 1.0))
     (*
      (sin x)
      (* (* (fma (* y y) 0.008333333333333333 0.16666666666666666) y) y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 620.0) {
		tmp = sin(x) * fma(fma((0.008333333333333333 * y), y, 0.16666666666666666), (y * y), 1.0);
	} else if (y <= 7e+72) {
		tmp = fma(pow(x, 3.0), fma(0.008333333333333333, (x * x), -0.16666666666666666), x) * fma((y * y), 0.16666666666666666, 1.0);
	} else {
		tmp = sin(x) * ((fma((y * y), 0.008333333333333333, 0.16666666666666666) * y) * y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 620.0)
		tmp = Float64(sin(x) * fma(fma(Float64(0.008333333333333333 * y), y, 0.16666666666666666), Float64(y * y), 1.0));
	elseif (y <= 7e+72)
		tmp = Float64(fma((x ^ 3.0), fma(0.008333333333333333, Float64(x * x), -0.16666666666666666), x) * fma(Float64(y * y), 0.16666666666666666, 1.0));
	else
		tmp = Float64(sin(x) * Float64(Float64(fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666) * y) * y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 620.0], N[(N[Sin[x], $MachinePrecision] * N[(N[(N[(0.008333333333333333 * y), $MachinePrecision] * y + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 7e+72], N[(N[(N[Power[x, 3.0], $MachinePrecision] * N[(0.008333333333333333 * N[(x * x), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] + x), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Sin[x], $MachinePrecision] * N[(N[(N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision] * y), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 620

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 88.1

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

   5. Applied rewrites 88.1%

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

   7. Applied rewrites 88.1%

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

   if 620 < y < 7.0000000000000002e72

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 3.3

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

   5. Applied rewrites 3.3%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. pow-plus N/A

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

      8. lower-pow.f64 N/A

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

      9. metadata-eval N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 45.7

          \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(0.008333333333333333, \color{blue}{x \cdot x}, -0.16666666666666666\right), x\right) \cdot \mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right) \]

   8. Applied rewrites 45.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right), x\right)} \cdot \mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right) \]

   if 7.0000000000000002e72 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 100.0%

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

Alternative 5: 88.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 620:\\ \;\;\;\;\sin x \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333 \cdot y, y, 0.16666666666666666\right), y \cdot y, 1\right)\\ \mathbf{elif}\;y \leq 7 \cdot 10^{+72}:\\ \;\;\;\;\mathsf{fma}\left(\left(x \cdot x\right) \cdot x, \mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right), x\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \left(\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right) \cdot y\right) \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 620.0)
   (*
    (sin x)
    (fma (fma (* 0.008333333333333333 y) y 0.16666666666666666) (* y y) 1.0))
   (if (<= y 7e+72)
     (*
      (fma
       (* (* x x) x)
       (fma 0.008333333333333333 (* x x) -0.16666666666666666)
       x)
      1.0)
     (*
      (sin x)
      (* (* (fma (* y y) 0.008333333333333333 0.16666666666666666) y) y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 620.0) {
		tmp = sin(x) * fma(fma((0.008333333333333333 * y), y, 0.16666666666666666), (y * y), 1.0);
	} else if (y <= 7e+72) {
		tmp = fma(((x * x) * x), fma(0.008333333333333333, (x * x), -0.16666666666666666), x) * 1.0;
	} else {
		tmp = sin(x) * ((fma((y * y), 0.008333333333333333, 0.16666666666666666) * y) * y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 620.0)
		tmp = Float64(sin(x) * fma(fma(Float64(0.008333333333333333 * y), y, 0.16666666666666666), Float64(y * y), 1.0));
	elseif (y <= 7e+72)
		tmp = Float64(fma(Float64(Float64(x * x) * x), fma(0.008333333333333333, Float64(x * x), -0.16666666666666666), x) * 1.0);
	else
		tmp = Float64(sin(x) * Float64(Float64(fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666) * y) * y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 620.0], N[(N[Sin[x], $MachinePrecision] * N[(N[(N[(0.008333333333333333 * y), $MachinePrecision] * y + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 7e+72], N[(N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * N[(0.008333333333333333 * N[(x * x), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] + x), $MachinePrecision] * 1.0), $MachinePrecision], N[(N[Sin[x], $MachinePrecision] * N[(N[(N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision] * y), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 620

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 88.1

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

   5. Applied rewrites 88.1%

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

   7. Applied rewrites 88.1%

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

   if 620 < y < 7.0000000000000002e72

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \sin x \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 2.7%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. pow-plus N/A

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

      8. lower-pow.f64 N/A

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

      9. metadata-eval N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 45.6

          \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(0.008333333333333333, \color{blue}{x \cdot x}, -0.16666666666666666\right), x\right) \cdot 1 \]

   8. Applied rewrites 45.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right), x\right)} \cdot 1 \]
   9. Step-by-step derivation

   10. Applied rewrites 45.6%

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

   if 7.0000000000000002e72 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 100.0%

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

Alternative 6: 82.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 620:\\ \;\;\;\;\sin x \cdot \mathsf{fma}\left(0.16666666666666666 \cdot y, y, 1\right)\\ \mathbf{elif}\;y \leq 7 \cdot 10^{+72}:\\ \;\;\;\;\mathsf{fma}\left(\left(x \cdot x\right) \cdot x, \mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right), x\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \left(\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right) \cdot y\right) \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 620.0)
   (* (sin x) (fma (* 0.16666666666666666 y) y 1.0))
   (if (<= y 7e+72)
     (*
      (fma
       (* (* x x) x)
       (fma 0.008333333333333333 (* x x) -0.16666666666666666)
       x)
      1.0)
     (*
      (sin x)
      (* (* (fma (* y y) 0.008333333333333333 0.16666666666666666) y) y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 620.0) {
		tmp = sin(x) * fma((0.16666666666666666 * y), y, 1.0);
	} else if (y <= 7e+72) {
		tmp = fma(((x * x) * x), fma(0.008333333333333333, (x * x), -0.16666666666666666), x) * 1.0;
	} else {
		tmp = sin(x) * ((fma((y * y), 0.008333333333333333, 0.16666666666666666) * y) * y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 620.0)
		tmp = Float64(sin(x) * fma(Float64(0.16666666666666666 * y), y, 1.0));
	elseif (y <= 7e+72)
		tmp = Float64(fma(Float64(Float64(x * x) * x), fma(0.008333333333333333, Float64(x * x), -0.16666666666666666), x) * 1.0);
	else
		tmp = Float64(sin(x) * Float64(Float64(fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666) * y) * y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 620.0], N[(N[Sin[x], $MachinePrecision] * N[(N[(0.16666666666666666 * y), $MachinePrecision] * y + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 7e+72], N[(N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * N[(0.008333333333333333 * N[(x * x), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] + x), $MachinePrecision] * 1.0), $MachinePrecision], N[(N[Sin[x], $MachinePrecision] * N[(N[(N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision] * y), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 620

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 80.5

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

   5. Applied rewrites 80.5%

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

   7. Applied rewrites 80.5%

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

   if 620 < y < 7.0000000000000002e72

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \sin x \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 2.7%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. pow-plus N/A

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

      8. lower-pow.f64 N/A

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

      9. metadata-eval N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 45.6

          \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(0.008333333333333333, \color{blue}{x \cdot x}, -0.16666666666666666\right), x\right) \cdot 1 \]

   8. Applied rewrites 45.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right), x\right)} \cdot 1 \]
   9. Step-by-step derivation

   10. Applied rewrites 45.6%

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

   if 7.0000000000000002e72 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 100.0%

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

Alternative 7: 34.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin x \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666 \cdot \left(x \cdot x\right), x, x\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(x \cdot x\right) \cdot x, \mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right), x\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sin x) -0.01)
   (* (fma (* -0.16666666666666666 (* x x)) x x) 1.0)
   (*
    (fma
     (* (* x x) x)
     (fma 0.008333333333333333 (* x x) -0.16666666666666666)
     x)
    1.0)))

C

double code(double x, double y) {
	double tmp;
	if (sin(x) <= -0.01) {
		tmp = fma((-0.16666666666666666 * (x * x)), x, x) * 1.0;
	} else {
		tmp = fma(((x * x) * x), fma(0.008333333333333333, (x * x), -0.16666666666666666), x) * 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (sin(x) <= -0.01)
		tmp = Float64(fma(Float64(-0.16666666666666666 * Float64(x * x)), x, x) * 1.0);
	else
		tmp = Float64(fma(Float64(Float64(x * x) * x), fma(0.008333333333333333, Float64(x * x), -0.16666666666666666), x) * 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[Sin[x], $MachinePrecision], -0.01], N[(N[(N[(-0.16666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision] * x + x), $MachinePrecision] * 1.0), $MachinePrecision], N[(N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * N[(0.008333333333333333 * N[(x * x), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] + x), $MachinePrecision] * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 x) < -0.0100000000000000002

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \sin x \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 35.9%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. pow-plus N/A

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

      9. lower-pow.f64 N/A

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

      10. metadata-eval 24.1

          \[\leadsto \mathsf{fma}\left({x}^{\color{blue}{3}}, -0.16666666666666666, x\right) \cdot 1 \]

   8. Applied rewrites 24.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, -0.16666666666666666, x\right)} \cdot 1 \]
   9. Step-by-step derivation

   10. Applied rewrites 24.1%

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

   if -0.0100000000000000002 < (sin.f64 x)

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \sin x \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 53.4%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. pow-plus N/A

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

      8. lower-pow.f64 N/A

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

      9. metadata-eval N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 44.8

          \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(0.008333333333333333, \color{blue}{x \cdot x}, -0.16666666666666666\right), x\right) \cdot 1 \]

   8. Applied rewrites 44.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right), x\right)} \cdot 1 \]
   9. Step-by-step derivation

   10. Applied rewrites 44.8%

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

Alternative 8: 91.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \sin x \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984, y \cdot y, 0.008333333333333333\right) \cdot y, y, 0.16666666666666666\right), y \cdot y, 1\right) \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (*
  (sin x)
  (fma
   (fma
    (* (fma 0.0001984126984126984 (* y y) 0.008333333333333333) y)
    y
    0.16666666666666666)
   (* y y)
   1.0)))

C

double code(double x, double y) {
	return sin(x) * fma(fma((fma(0.0001984126984126984, (y * y), 0.008333333333333333) * y), y, 0.16666666666666666), (y * y), 1.0);
}

Julia

function code(x, y)
	return Float64(sin(x) * fma(fma(Float64(fma(0.0001984126984126984, Float64(y * y), 0.008333333333333333) * y), y, 0.16666666666666666), Float64(y * y), 1.0))
end

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. +-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 89.9

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

5. Applied rewrites 89.9%

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

7. Applied rewrites 89.9%

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

Alternative 9: 91.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \sin x \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right) \cdot y\right) \cdot y, y \cdot y, 1\right) \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (*
  (sin x)
  (fma
   (* (* (fma (* y y) 0.0001984126984126984 0.008333333333333333) y) y)
   (* y y)
   1.0)))

C

double code(double x, double y) {
	return sin(x) * fma(((fma((y * y), 0.0001984126984126984, 0.008333333333333333) * y) * y), (y * y), 1.0);
}

Julia

function code(x, y)
	return Float64(sin(x) * fma(Float64(Float64(fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333) * y) * y), Float64(y * y), 1.0))
end

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. +-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 89.9

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

5. Applied rewrites 89.9%

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

6. Taylor expanded in y around inf

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

8. Applied rewrites 89.7%

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

Alternative 10: 34.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin x \leq 0.002:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666 \cdot \left(x \cdot x\right), x, x\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(x \cdot x\right) \cdot 0.008333333333333333\right) \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sin x) 0.002)
   (* (fma (* -0.16666666666666666 (* x x)) x x) 1.0)
   (* (* (* (* (* x x) 0.008333333333333333) x) (* x x)) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (sin(x) <= 0.002) {
		tmp = fma((-0.16666666666666666 * (x * x)), x, x) * 1.0;
	} else {
		tmp = ((((x * x) * 0.008333333333333333) * x) * (x * x)) * 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (sin(x) <= 0.002)
		tmp = Float64(fma(Float64(-0.16666666666666666 * Float64(x * x)), x, x) * 1.0);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(x * x) * 0.008333333333333333) * x) * Float64(x * x)) * 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[Sin[x], $MachinePrecision], 0.002], N[(N[(N[(-0.16666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision] * x + x), $MachinePrecision] * 1.0), $MachinePrecision], N[(N[(N[(N[(N[(x * x), $MachinePrecision] * 0.008333333333333333), $MachinePrecision] * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 x) < 2e-3

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \sin x \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 46.4%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. pow-plus N/A

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

      9. lower-pow.f64 N/A

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

      10. metadata-eval 41.9

          \[\leadsto \mathsf{fma}\left({x}^{\color{blue}{3}}, -0.16666666666666666, x\right) \cdot 1 \]

   8. Applied rewrites 41.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, -0.16666666666666666, x\right)} \cdot 1 \]
   9. Step-by-step derivation

   10. Applied rewrites 41.9%

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

   if 2e-3 < (sin.f64 x)

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \sin x \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 54.5%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. pow-plus N/A

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

      8. lower-pow.f64 N/A

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

      9. metadata-eval N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 31.2

          \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(0.008333333333333333, \color{blue}{x \cdot x}, -0.16666666666666666\right), x\right) \cdot 1 \]

   8. Applied rewrites 31.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right), x\right)} \cdot 1 \]

   9. Taylor expanded in x around inf

      \[\leadsto \left(\frac{1}{120} \cdot \color{blue}{{x}^{5}}\right) \cdot 1 \]
   10. Step-by-step derivation

   11. Applied rewrites 31.2%

       \[\leadsto \left({x}^{5} \cdot \color{blue}{0.008333333333333333}\right) \cdot 1 \]
   12. Step-by-step derivation

   13. Applied rewrites 31.2%

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

Alternative 11: 91.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \sin x \cdot \mathsf{fma}\left(\left(\left(\left(\left(y \cdot y\right) \cdot 0.0001984126984126984\right) \cdot y\right) \cdot y\right) \cdot y, y, 1\right) \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (* (sin x) (fma (* (* (* (* (* y y) 0.0001984126984126984) y) y) y) y 1.0)))

C

double code(double x, double y) {
	return sin(x) * fma((((((y * y) * 0.0001984126984126984) * y) * y) * y), y, 1.0);
}

Julia

function code(x, y)
	return Float64(sin(x) * fma(Float64(Float64(Float64(Float64(Float64(y * y) * 0.0001984126984126984) * y) * y) * y), y, 1.0))
end

Wolfram

code[x_, y_] := N[(N[Sin[x], $MachinePrecision] * N[(N[(N[(N[(N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984), $MachinePrecision] * y), $MachinePrecision] * y), $MachinePrecision] * y), $MachinePrecision] * y + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. +-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 89.9

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

5. Applied rewrites 89.9%

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

6. Taylor expanded in y around inf

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

8. Applied rewrites 89.7%

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

10. Applied rewrites 89.7%

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

11. Taylor expanded in y around inf

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

13. Applied rewrites 89.7%

    \[\leadsto \sin x \cdot \mathsf{fma}\left(\left(\left(\left(\left(y \cdot y\right) \cdot 0.0001984126984126984\right) \cdot y\right) \cdot y\right) \cdot y, y, 1\right) \]
14. Add Preprocessing

Alternative 12: 76.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 620:\\ \;\;\;\;\sin x \cdot \mathsf{fma}\left(0.16666666666666666 \cdot y, y, 1\right)\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{+150}:\\ \;\;\;\;\mathsf{fma}\left(\left(x \cdot x\right) \cdot x, \mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right), x\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \left(\left(y \cdot y\right) \cdot 0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 620.0)
   (* (sin x) (fma (* 0.16666666666666666 y) y 1.0))
   (if (<= y 6.2e+150)
     (*
      (fma
       (* (* x x) x)
       (fma 0.008333333333333333 (* x x) -0.16666666666666666)
       x)
      1.0)
     (* (sin x) (* (* y y) 0.16666666666666666)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 620.0) {
		tmp = sin(x) * fma((0.16666666666666666 * y), y, 1.0);
	} else if (y <= 6.2e+150) {
		tmp = fma(((x * x) * x), fma(0.008333333333333333, (x * x), -0.16666666666666666), x) * 1.0;
	} else {
		tmp = sin(x) * ((y * y) * 0.16666666666666666);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 620.0)
		tmp = Float64(sin(x) * fma(Float64(0.16666666666666666 * y), y, 1.0));
	elseif (y <= 6.2e+150)
		tmp = Float64(fma(Float64(Float64(x * x) * x), fma(0.008333333333333333, Float64(x * x), -0.16666666666666666), x) * 1.0);
	else
		tmp = Float64(sin(x) * Float64(Float64(y * y) * 0.16666666666666666));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 620.0], N[(N[Sin[x], $MachinePrecision] * N[(N[(0.16666666666666666 * y), $MachinePrecision] * y + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6.2e+150], N[(N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * N[(0.008333333333333333 * N[(x * x), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] + x), $MachinePrecision] * 1.0), $MachinePrecision], N[(N[Sin[x], $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 620

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 80.5

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

   5. Applied rewrites 80.5%

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

   7. Applied rewrites 80.5%

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

   if 620 < y < 6.20000000000000028e150

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \sin x \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 2.8%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. pow-plus N/A

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

      8. lower-pow.f64 N/A

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

      9. metadata-eval N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 43.8

          \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(0.008333333333333333, \color{blue}{x \cdot x}, -0.16666666666666666\right), x\right) \cdot 1 \]

   8. Applied rewrites 43.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right), x\right)} \cdot 1 \]
   9. Step-by-step derivation

   10. Applied rewrites 43.8%

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

   if 6.20000000000000028e150 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 96.7

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

   5. Applied rewrites 96.7%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 96.7%

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

Alternative 13: 64.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 620:\\ \;\;\;\;\sin x \cdot 1\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{+150}:\\ \;\;\;\;\mathsf{fma}\left(\left(x \cdot x\right) \cdot x, \mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right), x\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \left(\left(y \cdot y\right) \cdot 0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 620.0)
   (* (sin x) 1.0)
   (if (<= y 6.2e+150)
     (*
      (fma
       (* (* x x) x)
       (fma 0.008333333333333333 (* x x) -0.16666666666666666)
       x)
      1.0)
     (* (sin x) (* (* y y) 0.16666666666666666)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 620.0) {
		tmp = sin(x) * 1.0;
	} else if (y <= 6.2e+150) {
		tmp = fma(((x * x) * x), fma(0.008333333333333333, (x * x), -0.16666666666666666), x) * 1.0;
	} else {
		tmp = sin(x) * ((y * y) * 0.16666666666666666);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 620.0)
		tmp = Float64(sin(x) * 1.0);
	elseif (y <= 6.2e+150)
		tmp = Float64(fma(Float64(Float64(x * x) * x), fma(0.008333333333333333, Float64(x * x), -0.16666666666666666), x) * 1.0);
	else
		tmp = Float64(sin(x) * Float64(Float64(y * y) * 0.16666666666666666));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 620.0], N[(N[Sin[x], $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[y, 6.2e+150], N[(N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * N[(0.008333333333333333 * N[(x * x), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] + x), $MachinePrecision] * 1.0), $MachinePrecision], N[(N[Sin[x], $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 620

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \sin x \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 63.8%

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

   if 620 < y < 6.20000000000000028e150

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \sin x \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 2.8%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. pow-plus N/A

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

      8. lower-pow.f64 N/A

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

      9. metadata-eval N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 43.8

          \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(0.008333333333333333, \color{blue}{x \cdot x}, -0.16666666666666666\right), x\right) \cdot 1 \]

   8. Applied rewrites 43.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right), x\right)} \cdot 1 \]
   9. Step-by-step derivation

   10. Applied rewrites 43.8%

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

   if 6.20000000000000028e150 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 96.7

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

   5. Applied rewrites 96.7%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 96.7%

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

Alternative 14: 34.2% accurate, 9.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-0.16666666666666666 \cdot \left(x \cdot x\right), x, x\right) \cdot 1 \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (* (fma (* -0.16666666666666666 (* x x)) x x) 1.0))

C

double code(double x, double y) {
	return fma((-0.16666666666666666 * (x * x)), x, x) * 1.0;
}

Julia

function code(x, y)
	return Float64(fma(Float64(-0.16666666666666666 * Float64(x * x)), x, x) * 1.0)
end

Wolfram

code[x_, y_] := N[(N[(N[(-0.16666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision] * x + x), $MachinePrecision] * 1.0), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in y around 0

   \[\leadsto \sin x \cdot \color{blue}{1} \]
4. Step-by-step derivation

5. Applied rewrites 48.5%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. *-rgt-identity N/A

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

   6. lower-fma.f64 N/A

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

   7. *-commutative N/A

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

   8. pow-plus N/A

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

   9. lower-pow.f64 N/A

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

   10. metadata-eval 33.1

       \[\leadsto \mathsf{fma}\left({x}^{\color{blue}{3}}, -0.16666666666666666, x\right) \cdot 1 \]

8. Applied rewrites 33.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, -0.16666666666666666, x\right)} \cdot 1 \]
9. Step-by-step derivation

10. Applied rewrites 33.1%

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

Reproduce

herbie shell --seed 2024313 
(FPCore (x y)
  :name "Linear.Quaternion:$ccos from linear-1.19.1.3"
  :precision binary64
  (* (sin x) (/ (sinh y) y)))