Linear.Quaternion:$ccos from linear-1.19.1.3

Percentage Accurate: 100.0% → 98.7%

Time: 10.1s

Alternatives: 20

Speedup: 1.0×

• 48.9% 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: 98.7% accurate, 0.4× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y)
 :precision binary64
 (let* ((t_0 (/ (sinh y) y)) (t_1 (* t_0 (sin x_m))))
   (*
    x_s
    (if (<= t_1 (- INFINITY))
      (*
       (* (* y y) 0.16666666666666666)
       (* (* (* x_m x_m) -0.16666666666666666) x_m))
      (if (<= t_1 1.0)
        (*
         (fma
          (fma 0.008333333333333333 (* y y) 0.16666666666666666)
          (* y y)
          1.0)
         (sin x_m))
        (* (* 1.0 x_m) t_0))))))

C

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

Julia

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

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_] := Block[{t$95$0 = N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[Sin[x$95$m], $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1.0], N[(N[(N[(0.008333333333333333 * N[(y * y), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sin[x$95$m], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 * x$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \sin x \cdot \color{blue}{\left(1 + \frac{1}{6} \cdot {y}^{2}\right)} \]
   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 66.7

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

   5. Applied rewrites 66.7%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lft-mult-inverse N/A

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

      3. distribute-rgt-in N/A

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

      4. metadata-eval N/A

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

      5. sub-neg N/A

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

      6. lower-*.f64 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. distribute-rgt-in N/A

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

      11. lft-mult-inverse N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 56.7

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

   8. Applied rewrites 56.7%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 56.7%

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

   12. Taylor expanded in x around inf

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

   14. Applied rewrites 16.0%

       \[\leadsto \left(\left(-0.16666666666666666 \cdot \left(x \cdot x\right)\right) \cdot x\right) \cdot \left(\left(y \cdot y\right) \cdot 0.16666666666666666\right) \]

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-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. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 74.6

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

   5. Applied rewrites 74.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 74.6%

      \[\leadsto \left(1 \cdot x\right) \cdot \frac{\sinh y}{y} \]
3. Recombined 3 regimes into one program.

4. Final simplification 75.1%

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

Alternative 2: 98.7% accurate, 0.4× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y)
 :precision binary64
 (let* ((t_0 (/ (sinh y) y)) (t_1 (* t_0 (sin x_m))))
   (*
    x_s
    (if (<= t_1 (- INFINITY))
      (*
       (* (* y y) 0.16666666666666666)
       (* (* (* x_m x_m) -0.16666666666666666) x_m))
      (if (<= t_1 1.0)
        (* (fma (* y y) 0.16666666666666666 1.0) (sin x_m))
        (* (* 1.0 x_m) t_0))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y) {
	double t_0 = sinh(y) / y;
	double t_1 = t_0 * sin(x_m);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = ((y * y) * 0.16666666666666666) * (((x_m * x_m) * -0.16666666666666666) * x_m);
	} else if (t_1 <= 1.0) {
		tmp = fma((y * y), 0.16666666666666666, 1.0) * sin(x_m);
	} else {
		tmp = (1.0 * x_m) * t_0;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y)
	t_0 = Float64(sinh(y) / y)
	t_1 = Float64(t_0 * sin(x_m))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(y * y) * 0.16666666666666666) * Float64(Float64(Float64(x_m * x_m) * -0.16666666666666666) * x_m));
	elseif (t_1 <= 1.0)
		tmp = Float64(fma(Float64(y * y), 0.16666666666666666, 1.0) * sin(x_m));
	else
		tmp = Float64(Float64(1.0 * x_m) * t_0);
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_] := Block[{t$95$0 = N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[Sin[x$95$m], $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1.0], N[(N[(N[(y * y), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision] * N[Sin[x$95$m], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 * x$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \sin x \cdot \color{blue}{\left(1 + \frac{1}{6} \cdot {y}^{2}\right)} \]
   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 66.7

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

   5. Applied rewrites 66.7%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lft-mult-inverse N/A

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

      3. distribute-rgt-in N/A

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

      4. metadata-eval N/A

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

      5. sub-neg N/A

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

      6. lower-*.f64 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. distribute-rgt-in N/A

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

      11. lft-mult-inverse N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 56.7

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

   8. Applied rewrites 56.7%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 56.7%

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

   12. Taylor expanded in x around inf

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

   14. Applied rewrites 16.0%

       \[\leadsto \left(\left(-0.16666666666666666 \cdot \left(x \cdot x\right)\right) \cdot x\right) \cdot \left(\left(y \cdot y\right) \cdot 0.16666666666666666\right) \]

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-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 100.0

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

   5. Applied rewrites 100.0%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 74.6

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

   5. Applied rewrites 74.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 74.6%

      \[\leadsto \left(1 \cdot x\right) \cdot \frac{\sinh y}{y} \]
3. Recombined 3 regimes into one program.

4. Final simplification 75.0%

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

Alternative 3: 94.9% accurate, 0.4× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y)
 :precision binary64
 (let* ((t_0 (* (/ (sinh y) y) (sin x_m))))
   (*
    x_s
    (if (<= t_0 (- INFINITY))
      (*
       (* (* y y) 0.16666666666666666)
       (* (* (* x_m x_m) -0.16666666666666666) x_m))
      (if (<= t_0 1.0)
        (* (fma (* y y) 0.16666666666666666 1.0) (sin x_m))
        (*
         (/
          (*
           (fma
            (fma (* 0.0001984126984126984 (* y y)) (* y y) 0.16666666666666666)
            (* y y)
            1.0)
           y)
          y)
         (* 1.0 x_m)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y) {
	double t_0 = (sinh(y) / y) * sin(x_m);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = ((y * y) * 0.16666666666666666) * (((x_m * x_m) * -0.16666666666666666) * x_m);
	} else if (t_0 <= 1.0) {
		tmp = fma((y * y), 0.16666666666666666, 1.0) * sin(x_m);
	} else {
		tmp = ((fma(fma((0.0001984126984126984 * (y * y)), (y * y), 0.16666666666666666), (y * y), 1.0) * y) / y) * (1.0 * x_m);
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y)
	t_0 = Float64(Float64(sinh(y) / y) * sin(x_m))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(y * y) * 0.16666666666666666) * Float64(Float64(Float64(x_m * x_m) * -0.16666666666666666) * x_m));
	elseif (t_0 <= 1.0)
		tmp = Float64(fma(Float64(y * y), 0.16666666666666666, 1.0) * sin(x_m));
	else
		tmp = Float64(Float64(Float64(fma(fma(Float64(0.0001984126984126984 * Float64(y * y)), Float64(y * y), 0.16666666666666666), Float64(y * y), 1.0) * y) / y) * Float64(1.0 * x_m));
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_] := Block[{t$95$0 = N[(N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision] * N[Sin[x$95$m], $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1.0], N[(N[(N[(y * y), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision] * N[Sin[x$95$m], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(0.0001984126984126984 * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(y * y), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision] / y), $MachinePrecision] * N[(1.0 * x$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \sin x \cdot \color{blue}{\left(1 + \frac{1}{6} \cdot {y}^{2}\right)} \]
   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 66.7

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

   5. Applied rewrites 66.7%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lft-mult-inverse N/A

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

      3. distribute-rgt-in N/A

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

      4. metadata-eval N/A

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

      5. sub-neg N/A

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

      6. lower-*.f64 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. distribute-rgt-in N/A

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

      11. lft-mult-inverse N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 56.7

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

   8. Applied rewrites 56.7%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 56.7%

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

   12. Taylor expanded in x around inf

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

   14. Applied rewrites 16.0%

       \[\leadsto \left(\left(-0.16666666666666666 \cdot \left(x \cdot x\right)\right) \cdot x\right) \cdot \left(\left(y \cdot y\right) \cdot 0.16666666666666666\right) \]

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-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 100.0

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

   5. Applied rewrites 100.0%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 74.6

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

   5. Applied rewrites 74.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 74.6%

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

   9. Taylor expanded in y around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. +-commutative N/A

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

       7. *-commutative N/A

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

       8. lower-fma.f64 N/A

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

       9. +-commutative N/A

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

       10. lower-fma.f64 N/A

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

       11. unpow2 N/A

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

       12. lower-*.f64 N/A

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

       13. unpow2 N/A

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

       14. lower-*.f64 N/A

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

       15. unpow2 N/A

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

       16. lower-*.f64 67.0

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

   11. Applied rewrites 67.0%

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

   12. Taylor expanded in y around inf

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

   14. Applied rewrites 67.0%

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

4. Final simplification 73.2%

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

Alternative 4: 94.7% accurate, 0.4× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y)
 :precision binary64
 (let* ((t_0 (* (/ (sinh y) y) (sin x_m))))
   (*
    x_s
    (if (<= t_0 (- INFINITY))
      (*
       (* (* y y) 0.16666666666666666)
       (* (* (* x_m x_m) -0.16666666666666666) x_m))
      (if (<= t_0 1.0)
        (sin x_m)
        (*
         (/
          (*
           (fma
            (fma (* 0.0001984126984126984 (* y y)) (* y y) 0.16666666666666666)
            (* y y)
            1.0)
           y)
          y)
         (* 1.0 x_m)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y) {
	double t_0 = (sinh(y) / y) * sin(x_m);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = ((y * y) * 0.16666666666666666) * (((x_m * x_m) * -0.16666666666666666) * x_m);
	} else if (t_0 <= 1.0) {
		tmp = sin(x_m);
	} else {
		tmp = ((fma(fma((0.0001984126984126984 * (y * y)), (y * y), 0.16666666666666666), (y * y), 1.0) * y) / y) * (1.0 * x_m);
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y)
	t_0 = Float64(Float64(sinh(y) / y) * sin(x_m))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(y * y) * 0.16666666666666666) * Float64(Float64(Float64(x_m * x_m) * -0.16666666666666666) * x_m));
	elseif (t_0 <= 1.0)
		tmp = sin(x_m);
	else
		tmp = Float64(Float64(Float64(fma(fma(Float64(0.0001984126984126984 * Float64(y * y)), Float64(y * y), 0.16666666666666666), Float64(y * y), 1.0) * y) / y) * Float64(1.0 * x_m));
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_] := Block[{t$95$0 = N[(N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision] * N[Sin[x$95$m], $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1.0], N[Sin[x$95$m], $MachinePrecision], N[(N[(N[(N[(N[(N[(0.0001984126984126984 * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(y * y), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision] / y), $MachinePrecision] * N[(1.0 * x$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \sin x \cdot \color{blue}{\left(1 + \frac{1}{6} \cdot {y}^{2}\right)} \]
   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 66.7

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

   5. Applied rewrites 66.7%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lft-mult-inverse N/A

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

      3. distribute-rgt-in N/A

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

      4. metadata-eval N/A

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

      5. sub-neg N/A

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

      6. lower-*.f64 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. distribute-rgt-in N/A

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

      11. lft-mult-inverse N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 56.7

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

   8. Applied rewrites 56.7%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 56.7%

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

   12. Taylor expanded in x around inf

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

   14. Applied rewrites 16.0%

       \[\leadsto \left(\left(-0.16666666666666666 \cdot \left(x \cdot x\right)\right) \cdot x\right) \cdot \left(\left(y \cdot y\right) \cdot 0.16666666666666666\right) \]

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-sin.f64 99.2

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

   5. Applied rewrites 99.2%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 74.6

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

   5. Applied rewrites 74.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 74.6%

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

   9. Taylor expanded in y around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. +-commutative N/A

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

       7. *-commutative N/A

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

       8. lower-fma.f64 N/A

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

       9. +-commutative N/A

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

       10. lower-fma.f64 N/A

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

       11. unpow2 N/A

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

       12. lower-*.f64 N/A

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

       13. unpow2 N/A

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

       14. lower-*.f64 N/A

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

       15. unpow2 N/A

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

       16. lower-*.f64 67.0

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

   11. Applied rewrites 67.0%

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

   12. Taylor expanded in y around inf

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

   14. Applied rewrites 67.0%

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

4. Final simplification 72.8%

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

Alternative 5: 69.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\sinh y}{y} \cdot \sin x\_m \leq 10^{-7}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.16666666666666666, x\_m \cdot x\_m, 1\right) \cdot x\_m\right) \cdot \mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984 \cdot \left(y \cdot y\right), y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot y}{y} \cdot \left(1 \cdot x\_m\right)\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y)
 :precision binary64
 (*
  x_s
  (if (<= (* (/ (sinh y) y) (sin x_m)) 1e-7)
    (*
     (* (fma -0.16666666666666666 (* x_m x_m) 1.0) x_m)
     (fma (* y y) 0.16666666666666666 1.0))
    (*
     (/
      (*
       (fma
        (fma (* 0.0001984126984126984 (* y y)) (* y y) 0.16666666666666666)
        (* y y)
        1.0)
       y)
      y)
     (* 1.0 x_m)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y) {
	double tmp;
	if (((sinh(y) / y) * sin(x_m)) <= 1e-7) {
		tmp = (fma(-0.16666666666666666, (x_m * x_m), 1.0) * x_m) * fma((y * y), 0.16666666666666666, 1.0);
	} else {
		tmp = ((fma(fma((0.0001984126984126984 * (y * y)), (y * y), 0.16666666666666666), (y * y), 1.0) * y) / y) * (1.0 * x_m);
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y)
	tmp = 0.0
	if (Float64(Float64(sinh(y) / y) * sin(x_m)) <= 1e-7)
		tmp = Float64(Float64(fma(-0.16666666666666666, Float64(x_m * x_m), 1.0) * x_m) * fma(Float64(y * y), 0.16666666666666666, 1.0));
	else
		tmp = Float64(Float64(Float64(fma(fma(Float64(0.0001984126984126984 * Float64(y * y)), Float64(y * y), 0.16666666666666666), Float64(y * y), 1.0) * y) / y) * Float64(1.0 * x_m));
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_] := N[(x$95$s * If[LessEqual[N[(N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision] * N[Sin[x$95$m], $MachinePrecision]), $MachinePrecision], 1e-7], N[(N[(N[(-0.16666666666666666 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * x$95$m), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(0.0001984126984126984 * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(y * y), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision] / y), $MachinePrecision] * N[(1.0 * x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < 9.9999999999999995e-8

   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 88.9

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

   5. Applied rewrites 88.9%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lft-mult-inverse N/A

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

      3. distribute-rgt-in N/A

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

      4. metadata-eval N/A

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

      5. sub-neg N/A

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

      6. lower-*.f64 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. distribute-rgt-in N/A

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

      11. lft-mult-inverse N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 67.4

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

   8. Applied rewrites 67.4%

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

   if 9.9999999999999995e-8 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 56.6

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

   5. Applied rewrites 56.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 56.7%

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

   9. Taylor expanded in y around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. +-commutative N/A

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

       7. *-commutative N/A

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

       8. lower-fma.f64 N/A

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

       9. +-commutative N/A

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

       10. lower-fma.f64 N/A

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

       11. unpow2 N/A

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

       12. lower-*.f64 N/A

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

       13. unpow2 N/A

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

       14. lower-*.f64 N/A

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

       15. unpow2 N/A

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

       16. lower-*.f64 51.0

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

   11. Applied rewrites 51.0%

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

   12. Taylor expanded in y around inf

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

   14. Applied rewrites 51.0%

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

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y}{y} \cdot \sin x \leq 10^{-7}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.16666666666666666, x \cdot x, 1\right) \cdot x\right) \cdot \mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984 \cdot \left(y \cdot y\right), y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot y}{y} \cdot \left(1 \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 68.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\sinh y}{y} \cdot \sin x\_m \leq 10^{-7}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.16666666666666666, x\_m \cdot x\_m, 1\right) \cdot x\_m\right) \cdot \mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot y\right) \cdot \left(1 \cdot x\_m\right)}{y}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y)
 :precision binary64
 (*
  x_s
  (if (<= (* (/ (sinh y) y) (sin x_m)) 1e-7)
    (*
     (* (fma -0.16666666666666666 (* x_m x_m) 1.0) x_m)
     (fma (* y y) 0.16666666666666666 1.0))
    (/
     (*
      (*
       (fma (fma 0.008333333333333333 (* y y) 0.16666666666666666) (* y y) 1.0)
       y)
      (* 1.0 x_m))
     y))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y) {
	double tmp;
	if (((sinh(y) / y) * sin(x_m)) <= 1e-7) {
		tmp = (fma(-0.16666666666666666, (x_m * x_m), 1.0) * x_m) * fma((y * y), 0.16666666666666666, 1.0);
	} else {
		tmp = ((fma(fma(0.008333333333333333, (y * y), 0.16666666666666666), (y * y), 1.0) * y) * (1.0 * x_m)) / y;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y)
	tmp = 0.0
	if (Float64(Float64(sinh(y) / y) * sin(x_m)) <= 1e-7)
		tmp = Float64(Float64(fma(-0.16666666666666666, Float64(x_m * x_m), 1.0) * x_m) * fma(Float64(y * y), 0.16666666666666666, 1.0));
	else
		tmp = Float64(Float64(Float64(fma(fma(0.008333333333333333, Float64(y * y), 0.16666666666666666), Float64(y * y), 1.0) * y) * Float64(1.0 * x_m)) / y);
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_] := N[(x$95$s * If[LessEqual[N[(N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision] * N[Sin[x$95$m], $MachinePrecision]), $MachinePrecision], 1e-7], N[(N[(N[(-0.16666666666666666 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * x$95$m), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(0.008333333333333333 * N[(y * y), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision] * N[(1.0 * x$95$m), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < 9.9999999999999995e-8

   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 88.9

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

   5. Applied rewrites 88.9%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lft-mult-inverse N/A

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

      3. distribute-rgt-in N/A

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

      4. metadata-eval N/A

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

      5. sub-neg N/A

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

      6. lower-*.f64 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. distribute-rgt-in N/A

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

      11. lft-mult-inverse N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 67.4

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

   8. Applied rewrites 67.4%

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

   if 9.9999999999999995e-8 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 56.6

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

   5. Applied rewrites 56.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 56.7%

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

   9. Taylor expanded in y around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. +-commutative N/A

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

       7. lower-fma.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f64 51.0

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

   11. Applied rewrites 51.0%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-*r/ N/A

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

       4. lower-/.f64 N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 51.0

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

   13. Applied rewrites 51.0%

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

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y}{y} \cdot \sin x \leq 10^{-7}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.16666666666666666, x \cdot x, 1\right) \cdot x\right) \cdot \mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot y\right) \cdot \left(1 \cdot x\right)}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\sinh y}{y} \cdot \sin x\_m \leq 10^{-7}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.16666666666666666, x\_m \cdot x\_m, 1\right) \cdot x\_m\right) \cdot \mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.008333333333333333 \cdot \left(y \cdot y\right), y \cdot y, 1\right) \cdot y}{y} \cdot \left(1 \cdot x\_m\right)\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y)
 :precision binary64
 (*
  x_s
  (if (<= (* (/ (sinh y) y) (sin x_m)) 1e-7)
    (*
     (* (fma -0.16666666666666666 (* x_m x_m) 1.0) x_m)
     (fma (* y y) 0.16666666666666666 1.0))
    (*
     (/ (* (fma (* 0.008333333333333333 (* y y)) (* y y) 1.0) y) y)
     (* 1.0 x_m)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y) {
	double tmp;
	if (((sinh(y) / y) * sin(x_m)) <= 1e-7) {
		tmp = (fma(-0.16666666666666666, (x_m * x_m), 1.0) * x_m) * fma((y * y), 0.16666666666666666, 1.0);
	} else {
		tmp = ((fma((0.008333333333333333 * (y * y)), (y * y), 1.0) * y) / y) * (1.0 * x_m);
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y)
	tmp = 0.0
	if (Float64(Float64(sinh(y) / y) * sin(x_m)) <= 1e-7)
		tmp = Float64(Float64(fma(-0.16666666666666666, Float64(x_m * x_m), 1.0) * x_m) * fma(Float64(y * y), 0.16666666666666666, 1.0));
	else
		tmp = Float64(Float64(Float64(fma(Float64(0.008333333333333333 * Float64(y * y)), Float64(y * y), 1.0) * y) / y) * Float64(1.0 * x_m));
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_] := N[(x$95$s * If[LessEqual[N[(N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision] * N[Sin[x$95$m], $MachinePrecision]), $MachinePrecision], 1e-7], N[(N[(N[(-0.16666666666666666 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * x$95$m), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(0.008333333333333333 * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision] / y), $MachinePrecision] * N[(1.0 * x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < 9.9999999999999995e-8

   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 88.9

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

   5. Applied rewrites 88.9%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lft-mult-inverse N/A

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

      3. distribute-rgt-in N/A

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

      4. metadata-eval N/A

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

      5. sub-neg N/A

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

      6. lower-*.f64 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. distribute-rgt-in N/A

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

      11. lft-mult-inverse N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 67.4

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

   8. Applied rewrites 67.4%

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

   if 9.9999999999999995e-8 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 56.6

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

   5. Applied rewrites 56.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 56.7%

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

   9. Taylor expanded in y around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. +-commutative N/A

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

       7. lower-fma.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f64 51.0

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

   11. Applied rewrites 51.0%

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

   12. Taylor expanded in y around inf

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

   14. Applied rewrites 51.0%

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

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y}{y} \cdot \sin x \leq 10^{-7}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.16666666666666666, x \cdot x, 1\right) \cdot x\right) \cdot \mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.008333333333333333 \cdot \left(y \cdot y\right), y \cdot y, 1\right) \cdot y}{y} \cdot \left(1 \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 66.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\sinh y}{y} \cdot \sin x\_m \leq 10^{-7}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.16666666666666666, x\_m \cdot x\_m, 1\right) \cdot x\_m\right) \cdot \mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \cdot x\_m\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right)\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y)
 :precision binary64
 (*
  x_s
  (if (<= (* (/ (sinh y) y) (sin x_m)) 1e-7)
    (*
     (* (fma -0.16666666666666666 (* x_m x_m) 1.0) x_m)
     (fma (* y y) 0.16666666666666666 1.0))
    (*
     (* 1.0 x_m)
     (fma
      (fma 0.008333333333333333 (* y y) 0.16666666666666666)
      (* y y)
      1.0)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y) {
	double tmp;
	if (((sinh(y) / y) * sin(x_m)) <= 1e-7) {
		tmp = (fma(-0.16666666666666666, (x_m * x_m), 1.0) * x_m) * fma((y * y), 0.16666666666666666, 1.0);
	} else {
		tmp = (1.0 * x_m) * fma(fma(0.008333333333333333, (y * y), 0.16666666666666666), (y * y), 1.0);
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y)
	tmp = 0.0
	if (Float64(Float64(sinh(y) / y) * sin(x_m)) <= 1e-7)
		tmp = Float64(Float64(fma(-0.16666666666666666, Float64(x_m * x_m), 1.0) * x_m) * fma(Float64(y * y), 0.16666666666666666, 1.0));
	else
		tmp = Float64(Float64(1.0 * x_m) * fma(fma(0.008333333333333333, Float64(y * y), 0.16666666666666666), Float64(y * y), 1.0));
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_] := N[(x$95$s * If[LessEqual[N[(N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision] * N[Sin[x$95$m], $MachinePrecision]), $MachinePrecision], 1e-7], N[(N[(N[(-0.16666666666666666 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * x$95$m), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 * x$95$m), $MachinePrecision] * N[(N[(0.008333333333333333 * N[(y * y), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < 9.9999999999999995e-8

   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 88.9

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

   5. Applied rewrites 88.9%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lft-mult-inverse N/A

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

      3. distribute-rgt-in N/A

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

      4. metadata-eval N/A

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

      5. sub-neg N/A

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

      6. lower-*.f64 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. distribute-rgt-in N/A

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

      11. lft-mult-inverse N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 67.4

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

   8. Applied rewrites 67.4%

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

   if 9.9999999999999995e-8 < (*.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

      \[\leadsto \sin x \cdot \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} \]
   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. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 85.1

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

   5. Applied rewrites 85.1%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 50.7

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

   8. Applied rewrites 50.7%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 48.7%

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

4. Final simplification 61.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y}{y} \cdot \sin x \leq 10^{-7}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.16666666666666666, x \cdot x, 1\right) \cdot x\right) \cdot \mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \cdot x\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 66.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\sinh y}{y} \cdot \sin x\_m \leq -0.2:\\ \;\;\;\;\left(\left(y \cdot y\right) \cdot 0.16666666666666666\right) \cdot \left(\left(\left(x\_m \cdot x\_m\right) \cdot -0.16666666666666666\right) \cdot x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \cdot x\_m\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right)\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y)
 :precision binary64
 (*
  x_s
  (if (<= (* (/ (sinh y) y) (sin x_m)) -0.2)
    (*
     (* (* y y) 0.16666666666666666)
     (* (* (* x_m x_m) -0.16666666666666666) x_m))
    (*
     (* 1.0 x_m)
     (fma
      (fma 0.008333333333333333 (* y y) 0.16666666666666666)
      (* y y)
      1.0)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y) {
	double tmp;
	if (((sinh(y) / y) * sin(x_m)) <= -0.2) {
		tmp = ((y * y) * 0.16666666666666666) * (((x_m * x_m) * -0.16666666666666666) * x_m);
	} else {
		tmp = (1.0 * x_m) * fma(fma(0.008333333333333333, (y * y), 0.16666666666666666), (y * y), 1.0);
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y)
	tmp = 0.0
	if (Float64(Float64(sinh(y) / y) * sin(x_m)) <= -0.2)
		tmp = Float64(Float64(Float64(y * y) * 0.16666666666666666) * Float64(Float64(Float64(x_m * x_m) * -0.16666666666666666) * x_m));
	else
		tmp = Float64(Float64(1.0 * x_m) * fma(fma(0.008333333333333333, Float64(y * y), 0.16666666666666666), Float64(y * y), 1.0));
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_] := N[(x$95$s * If[LessEqual[N[(N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision] * N[Sin[x$95$m], $MachinePrecision]), $MachinePrecision], -0.2], N[(N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 * x$95$m), $MachinePrecision] * N[(N[(0.008333333333333333 * N[(y * y), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < -0.20000000000000001

   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 77.6

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

   5. Applied rewrites 77.6%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lft-mult-inverse N/A

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

      3. distribute-rgt-in N/A

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

      4. metadata-eval N/A

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

      5. sub-neg N/A

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

      6. lower-*.f64 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. distribute-rgt-in N/A

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

      11. lft-mult-inverse N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 38.7

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

   8. Applied rewrites 38.7%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 38.6%

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

   12. Taylor expanded in x around inf

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

   14. Applied rewrites 11.4%

       \[\leadsto \left(\left(-0.16666666666666666 \cdot \left(x \cdot x\right)\right) \cdot x\right) \cdot \left(\left(y \cdot y\right) \cdot 0.16666666666666666\right) \]

   if -0.20000000000000001 < (*.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

      \[\leadsto \sin x \cdot \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} \]
   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. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 92.7

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

   5. Applied rewrites 92.7%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 73.5

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

   8. Applied rewrites 73.5%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 72.5%

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

4. Final simplification 52.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y}{y} \cdot \sin x \leq -0.2:\\ \;\;\;\;\left(\left(y \cdot y\right) \cdot 0.16666666666666666\right) \cdot \left(\left(\left(x \cdot x\right) \cdot -0.16666666666666666\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \cdot x\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 59.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\sinh y}{y} \cdot \sin x\_m \leq -0.2:\\ \;\;\;\;\left(\left(y \cdot y\right) \cdot 0.16666666666666666\right) \cdot \left(\left(\left(x\_m \cdot x\_m\right) \cdot -0.16666666666666666\right) \cdot x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \cdot x\_m\right) \cdot \mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right)\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y)
 :precision binary64
 (*
  x_s
  (if (<= (* (/ (sinh y) y) (sin x_m)) -0.2)
    (*
     (* (* y y) 0.16666666666666666)
     (* (* (* x_m x_m) -0.16666666666666666) x_m))
    (* (* 1.0 x_m) (fma (* y y) 0.16666666666666666 1.0)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y) {
	double tmp;
	if (((sinh(y) / y) * sin(x_m)) <= -0.2) {
		tmp = ((y * y) * 0.16666666666666666) * (((x_m * x_m) * -0.16666666666666666) * x_m);
	} else {
		tmp = (1.0 * x_m) * fma((y * y), 0.16666666666666666, 1.0);
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y)
	tmp = 0.0
	if (Float64(Float64(sinh(y) / y) * sin(x_m)) <= -0.2)
		tmp = Float64(Float64(Float64(y * y) * 0.16666666666666666) * Float64(Float64(Float64(x_m * x_m) * -0.16666666666666666) * x_m));
	else
		tmp = Float64(Float64(1.0 * x_m) * fma(Float64(y * y), 0.16666666666666666, 1.0));
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_] := N[(x$95$s * If[LessEqual[N[(N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision] * N[Sin[x$95$m], $MachinePrecision]), $MachinePrecision], -0.2], N[(N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 * x$95$m), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < -0.20000000000000001

   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 77.6

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

   5. Applied rewrites 77.6%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lft-mult-inverse N/A

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

      3. distribute-rgt-in N/A

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

      4. metadata-eval N/A

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

      5. sub-neg N/A

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

      6. lower-*.f64 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. distribute-rgt-in N/A

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

      11. lft-mult-inverse N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 38.7

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

   8. Applied rewrites 38.7%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 38.6%

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

   12. Taylor expanded in x around inf

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

   14. Applied rewrites 11.4%

       \[\leadsto \left(\left(-0.16666666666666666 \cdot \left(x \cdot x\right)\right) \cdot x\right) \cdot \left(\left(y \cdot y\right) \cdot 0.16666666666666666\right) \]

   if -0.20000000000000001 < (*.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

      \[\leadsto \sin x \cdot \color{blue}{\left(1 + \frac{1}{6} \cdot {y}^{2}\right)} \]
   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 82.6

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

   5. Applied rewrites 82.6%

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

   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

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 66.8

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

   8. Applied rewrites 66.8%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 64.8%

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

4. Final simplification 47.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y}{y} \cdot \sin x \leq -0.2:\\ \;\;\;\;\left(\left(y \cdot y\right) \cdot 0.16666666666666666\right) \cdot \left(\left(\left(x \cdot x\right) \cdot -0.16666666666666666\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \cdot x\right) \cdot \mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 56.9% accurate, 0.9× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y)
 :precision binary64
 (*
  x_s
  (if (<= (* (/ (sinh y) y) (sin x_m)) -0.05)
    (* (* (* x_m x_m) x_m) -0.16666666666666666)
    (* (* 1.0 x_m) (fma (* y y) 0.16666666666666666 1.0)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y) {
	double tmp;
	if (((sinh(y) / y) * sin(x_m)) <= -0.05) {
		tmp = ((x_m * x_m) * x_m) * -0.16666666666666666;
	} else {
		tmp = (1.0 * x_m) * fma((y * y), 0.16666666666666666, 1.0);
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y)
	tmp = 0.0
	if (Float64(Float64(sinh(y) / y) * sin(x_m)) <= -0.05)
		tmp = Float64(Float64(Float64(x_m * x_m) * x_m) * -0.16666666666666666);
	else
		tmp = Float64(Float64(1.0 * x_m) * fma(Float64(y * y), 0.16666666666666666, 1.0));
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_] := N[(x$95$s * If[LessEqual[N[(N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision] * N[Sin[x$95$m], $MachinePrecision]), $MachinePrecision], -0.05], N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] * -0.16666666666666666), $MachinePrecision], N[(N[(1.0 * x$95$m), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-sin.f64 36.9

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

   5. Applied rewrites 36.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 9.5%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 9.2%

       \[\leadsto \left(\left(x \cdot x\right) \cdot x\right) \cdot -0.16666666666666666 \]

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \sin x \cdot \color{blue}{\left(1 + \frac{1}{6} \cdot {y}^{2}\right)} \]
   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 82.2

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

   5. Applied rewrites 82.2%

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

   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

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 68.3

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

   8. Applied rewrites 68.3%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 66.2%

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

4. Final simplification 46.4%

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

Alternative 12: 56.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\sinh y}{y} \cdot \sin x\_m \leq 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, x\_m \cdot x\_m, 1\right) \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;\left(1 \cdot x\_m\right) \cdot \left(\left(y \cdot y\right) \cdot 0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y)
 :precision binary64
 (*
  x_s
  (if (<= (* (/ (sinh y) y) (sin x_m)) 1e-7)
    (* (fma -0.16666666666666666 (* x_m x_m) 1.0) x_m)
    (* (* 1.0 x_m) (* (* y y) 0.16666666666666666)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y) {
	double tmp;
	if (((sinh(y) / y) * sin(x_m)) <= 1e-7) {
		tmp = fma(-0.16666666666666666, (x_m * x_m), 1.0) * x_m;
	} else {
		tmp = (1.0 * x_m) * ((y * y) * 0.16666666666666666);
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y)
	tmp = 0.0
	if (Float64(Float64(sinh(y) / y) * sin(x_m)) <= 1e-7)
		tmp = Float64(fma(-0.16666666666666666, Float64(x_m * x_m), 1.0) * x_m);
	else
		tmp = Float64(Float64(1.0 * x_m) * Float64(Float64(y * y) * 0.16666666666666666));
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_] := N[(x$95$s * If[LessEqual[N[(N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision] * N[Sin[x$95$m], $MachinePrecision]), $MachinePrecision], 1e-7], N[(N[(-0.16666666666666666 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * x$95$m), $MachinePrecision], N[(N[(1.0 * x$95$m), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < 9.9999999999999995e-8

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-sin.f64 67.1

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

   5. Applied rewrites 67.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 52.9%

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

   if 9.9999999999999995e-8 < (*.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

      \[\leadsto \sin x \cdot \color{blue}{\left(1 + \frac{1}{6} \cdot {y}^{2}\right)} \]
   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 64.6

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

   5. Applied rewrites 64.6%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lft-mult-inverse N/A

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

      3. distribute-rgt-in N/A

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

      4. metadata-eval N/A

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

      5. sub-neg N/A

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

      6. lower-*.f64 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. distribute-rgt-in N/A

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

      11. lft-mult-inverse N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 38.2

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

   8. Applied rewrites 38.2%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 38.1%

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

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 32.9%

       \[\leadsto \left(1 \cdot x\right) \cdot \left(\left(y \cdot y\right) \cdot 0.16666666666666666\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 46.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y}{y} \cdot \sin x \leq 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, x \cdot x, 1\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(1 \cdot x\right) \cdot \left(\left(y \cdot y\right) \cdot 0.16666666666666666\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 35.2% accurate, 0.9× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y)
 :precision binary64
 (*
  x_s
  (if (<= (* (/ (sinh y) y) (sin x_m)) -0.05)
    (* (* (* x_m x_m) x_m) -0.16666666666666666)
    (* 1.0 x_m))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y) {
	double tmp;
	if (((sinh(y) / y) * sin(x_m)) <= -0.05) {
		tmp = ((x_m * x_m) * x_m) * -0.16666666666666666;
	} else {
		tmp = 1.0 * x_m;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8) :: tmp
    if (((sinh(y) / y) * sin(x_m)) <= (-0.05d0)) then
        tmp = ((x_m * x_m) * x_m) * (-0.16666666666666666d0)
    else
        tmp = 1.0d0 * x_m
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y) {
	double tmp;
	if (((Math.sinh(y) / y) * Math.sin(x_m)) <= -0.05) {
		tmp = ((x_m * x_m) * x_m) * -0.16666666666666666;
	} else {
		tmp = 1.0 * x_m;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y):
	tmp = 0
	if ((math.sinh(y) / y) * math.sin(x_m)) <= -0.05:
		tmp = ((x_m * x_m) * x_m) * -0.16666666666666666
	else:
		tmp = 1.0 * x_m
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y)
	tmp = 0.0
	if (Float64(Float64(sinh(y) / y) * sin(x_m)) <= -0.05)
		tmp = Float64(Float64(Float64(x_m * x_m) * x_m) * -0.16666666666666666);
	else
		tmp = Float64(1.0 * x_m);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y)
	tmp = 0.0;
	if (((sinh(y) / y) * sin(x_m)) <= -0.05)
		tmp = ((x_m * x_m) * x_m) * -0.16666666666666666;
	else
		tmp = 1.0 * x_m;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_] := N[(x$95$s * If[LessEqual[N[(N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision] * N[Sin[x$95$m], $MachinePrecision]), $MachinePrecision], -0.05], N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] * -0.16666666666666666), $MachinePrecision], N[(1.0 * x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-sin.f64 36.9

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

   5. Applied rewrites 36.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 9.5%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 9.2%

       \[\leadsto \left(\left(x \cdot x\right) \cdot x\right) \cdot -0.16666666666666666 \]

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-sin.f64 63.0

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

   5. Applied rewrites 63.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 59.2%

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

   9. Taylor expanded in x around 0

      \[\leadsto 1 \cdot x \]
   10. Step-by-step derivation

   11. Applied rewrites 50.7%

       \[\leadsto 1 \cdot x \]
3. Recombined 2 regimes into one program.

4. Final simplification 36.3%

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

Alternative 14: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(\frac{\sinh y}{y} \cdot \sin x\_m\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y) :precision binary64 (* x_s (* (/ (sinh y) y) (sin x_m))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y) {
	return x_s * ((sinh(y) / y) * sin(x_m));
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    code = x_s * ((sinh(y) / y) * sin(x_m))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y) {
	return x_s * ((Math.sinh(y) / y) * Math.sin(x_m));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y):
	return x_s * ((math.sinh(y) / y) * math.sin(x_m))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y)
	return Float64(x_s * Float64(Float64(sinh(y) / y) * sin(x_m)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m, y)
	tmp = x_s * ((sinh(y) / y) * sin(x_m));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_] := N[(x$95$s * N[(N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision] * N[Sin[x$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Final simplification 100.0%

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

Alternative 15: 70.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\sin x\_m \leq 10^{-7}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984, y \cdot y, 0.008333333333333333\right), y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot y}{y} \cdot \left(\mathsf{fma}\left(-0.16666666666666666, x\_m \cdot x\_m, 1\right) \cdot x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(\left(x\_m \cdot x\_m\right) \cdot x\_m\right) \cdot x\_m\right) \cdot 0.008333333333333333\right) \cdot x\_m\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right)\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y)
 :precision binary64
 (*
  x_s
  (if (<= (sin x_m) 1e-7)
    (*
     (/
      (*
       (fma
        (fma
         (fma 0.0001984126984126984 (* y y) 0.008333333333333333)
         (* y y)
         0.16666666666666666)
        (* y y)
        1.0)
       y)
      y)
     (* (fma -0.16666666666666666 (* x_m x_m) 1.0) x_m))
    (*
     (* (* (* (* (* x_m x_m) x_m) x_m) 0.008333333333333333) x_m)
     (fma
      (fma 0.008333333333333333 (* y y) 0.16666666666666666)
      (* y y)
      1.0)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y) {
	double tmp;
	if (sin(x_m) <= 1e-7) {
		tmp = ((fma(fma(fma(0.0001984126984126984, (y * y), 0.008333333333333333), (y * y), 0.16666666666666666), (y * y), 1.0) * y) / y) * (fma(-0.16666666666666666, (x_m * x_m), 1.0) * x_m);
	} else {
		tmp = (((((x_m * x_m) * x_m) * x_m) * 0.008333333333333333) * x_m) * fma(fma(0.008333333333333333, (y * y), 0.16666666666666666), (y * y), 1.0);
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y)
	tmp = 0.0
	if (sin(x_m) <= 1e-7)
		tmp = Float64(Float64(Float64(fma(fma(fma(0.0001984126984126984, Float64(y * y), 0.008333333333333333), Float64(y * y), 0.16666666666666666), Float64(y * y), 1.0) * y) / y) * Float64(fma(-0.16666666666666666, Float64(x_m * x_m), 1.0) * x_m));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(x_m * x_m) * x_m) * x_m) * 0.008333333333333333) * x_m) * fma(fma(0.008333333333333333, Float64(y * y), 0.16666666666666666), Float64(y * y), 1.0));
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_] := N[(x$95$s * If[LessEqual[N[Sin[x$95$m], $MachinePrecision], 1e-7], N[(N[(N[(N[(N[(N[(0.0001984126984126984 * N[(y * y), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] * N[(y * y), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision] / y), $MachinePrecision] * N[(N[(-0.16666666666666666 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] * 0.008333333333333333), $MachinePrecision] * x$95$m), $MachinePrecision] * N[(N[(0.008333333333333333 * N[(y * y), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 x) < 9.9999999999999995e-8

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 77.1

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

   5. Applied rewrites 77.1%

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

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 74.8

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

   8. Applied rewrites 74.8%

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

   if 9.9999999999999995e-8 < (sin.f64 x)

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \sin x \cdot \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} \]
   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. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 85.6

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

   5. Applied rewrites 85.6%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 31.0

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

   8. Applied rewrites 31.0%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 31.0%

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

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin x \leq 10^{-7}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984, y \cdot y, 0.008333333333333333\right), y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot y}{y} \cdot \left(\mathsf{fma}\left(-0.16666666666666666, x \cdot x, 1\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot 0.008333333333333333\right) \cdot x\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 70.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\sin x\_m \leq 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984, y \cdot y, 0.008333333333333333\right), y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot \left(\mathsf{fma}\left(-0.16666666666666666, x\_m \cdot x\_m, 1\right) \cdot x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(\left(x\_m \cdot x\_m\right) \cdot x\_m\right) \cdot x\_m\right) \cdot 0.008333333333333333\right) \cdot x\_m\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right)\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y)
 :precision binary64
 (*
  x_s
  (if (<= (sin x_m) 1e-7)
    (*
     (fma
      (fma
       (fma 0.0001984126984126984 (* y y) 0.008333333333333333)
       (* y y)
       0.16666666666666666)
      (* y y)
      1.0)
     (* (fma -0.16666666666666666 (* x_m x_m) 1.0) x_m))
    (*
     (* (* (* (* (* x_m x_m) x_m) x_m) 0.008333333333333333) x_m)
     (fma
      (fma 0.008333333333333333 (* y y) 0.16666666666666666)
      (* y y)
      1.0)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y) {
	double tmp;
	if (sin(x_m) <= 1e-7) {
		tmp = fma(fma(fma(0.0001984126984126984, (y * y), 0.008333333333333333), (y * y), 0.16666666666666666), (y * y), 1.0) * (fma(-0.16666666666666666, (x_m * x_m), 1.0) * x_m);
	} else {
		tmp = (((((x_m * x_m) * x_m) * x_m) * 0.008333333333333333) * x_m) * fma(fma(0.008333333333333333, (y * y), 0.16666666666666666), (y * y), 1.0);
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y)
	tmp = 0.0
	if (sin(x_m) <= 1e-7)
		tmp = Float64(fma(fma(fma(0.0001984126984126984, Float64(y * y), 0.008333333333333333), Float64(y * y), 0.16666666666666666), Float64(y * y), 1.0) * Float64(fma(-0.16666666666666666, Float64(x_m * x_m), 1.0) * x_m));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(x_m * x_m) * x_m) * x_m) * 0.008333333333333333) * x_m) * fma(fma(0.008333333333333333, Float64(y * y), 0.16666666666666666), Float64(y * y), 1.0));
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_] := N[(x$95$s * If[LessEqual[N[Sin[x$95$m], $MachinePrecision], 1e-7], N[(N[(N[(N[(0.0001984126984126984 * N[(y * y), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] * N[(y * y), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(-0.16666666666666666 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] * 0.008333333333333333), $MachinePrecision] * x$95$m), $MachinePrecision] * N[(N[(0.008333333333333333 * N[(y * y), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 x) < 9.9999999999999995e-8

   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 96.7

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

   5. Applied rewrites 96.7%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lft-mult-inverse N/A

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

      3. distribute-rgt-in N/A

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

      4. metadata-eval N/A

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

      5. sub-neg N/A

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

      6. lower-*.f64 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. distribute-rgt-in N/A

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

      11. lft-mult-inverse N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 73.8

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

   8. Applied rewrites 73.8%

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

   if 9.9999999999999995e-8 < (sin.f64 x)

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \sin x \cdot \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} \]
   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. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 85.6

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

   5. Applied rewrites 85.6%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 31.0

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

   8. Applied rewrites 31.0%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 31.0%

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

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin x \leq 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984, y \cdot y, 0.008333333333333333\right), y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot \left(\mathsf{fma}\left(-0.16666666666666666, x \cdot x, 1\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot 0.008333333333333333\right) \cdot x\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 68.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\sin x\_m \leq 4 \cdot 10^{-11}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.16666666666666666, x\_m \cdot x\_m, 1\right) \cdot x\_m\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(x\_m \cdot x\_m, 0.008333333333333333, -0.16666666666666666\right), x\_m \cdot x\_m, 1\right) \cdot x\_m\right) \cdot \mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right)\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y)
 :precision binary64
 (*
  x_s
  (if (<= (sin x_m) 4e-11)
    (*
     (* (fma -0.16666666666666666 (* x_m x_m) 1.0) x_m)
     (fma (fma 0.008333333333333333 (* y y) 0.16666666666666666) (* y y) 1.0))
    (*
     (*
      (fma
       (fma (* x_m x_m) 0.008333333333333333 -0.16666666666666666)
       (* x_m x_m)
       1.0)
      x_m)
     (fma (* y y) 0.16666666666666666 1.0)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y) {
	double tmp;
	if (sin(x_m) <= 4e-11) {
		tmp = (fma(-0.16666666666666666, (x_m * x_m), 1.0) * x_m) * fma(fma(0.008333333333333333, (y * y), 0.16666666666666666), (y * y), 1.0);
	} else {
		tmp = (fma(fma((x_m * x_m), 0.008333333333333333, -0.16666666666666666), (x_m * x_m), 1.0) * x_m) * fma((y * y), 0.16666666666666666, 1.0);
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y)
	tmp = 0.0
	if (sin(x_m) <= 4e-11)
		tmp = Float64(Float64(fma(-0.16666666666666666, Float64(x_m * x_m), 1.0) * x_m) * fma(fma(0.008333333333333333, Float64(y * y), 0.16666666666666666), Float64(y * y), 1.0));
	else
		tmp = Float64(Float64(fma(fma(Float64(x_m * x_m), 0.008333333333333333, -0.16666666666666666), Float64(x_m * x_m), 1.0) * x_m) * fma(Float64(y * y), 0.16666666666666666, 1.0));
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_] := N[(x$95$s * If[LessEqual[N[Sin[x$95$m], $MachinePrecision], 4e-11], N[(N[(N[(-0.16666666666666666 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * x$95$m), $MachinePrecision] * N[(N[(0.008333333333333333 * N[(y * y), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * x$95$m), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 x) < 3.99999999999999976e-11

   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. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 92.9

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

   5. Applied rewrites 92.9%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lft-mult-inverse N/A

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

      3. distribute-rgt-in N/A

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

      4. metadata-eval N/A

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

      5. sub-neg N/A

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

      6. lower-*.f64 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. distribute-rgt-in N/A

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

      11. lft-mult-inverse N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 71.8

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

   8. Applied rewrites 71.8%

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

   if 3.99999999999999976e-11 < (sin.f64 x)

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-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 73.5

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

   5. Applied rewrites 73.5%

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

   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

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 30.5

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

   8. Applied rewrites 30.5%

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

Alternative 18: 67.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\sin x\_m \leq 10^{-7}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.16666666666666666, x\_m \cdot x\_m, 1\right) \cdot x\_m\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(x\_m \cdot x\_m, 0.008333333333333333, -0.16666666666666666\right), x\_m \cdot x\_m, 1\right) \cdot x\_m\right) \cdot \left(\left(y \cdot y\right) \cdot 0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y)
 :precision binary64
 (*
  x_s
  (if (<= (sin x_m) 1e-7)
    (*
     (* (fma -0.16666666666666666 (* x_m x_m) 1.0) x_m)
     (fma (fma 0.008333333333333333 (* y y) 0.16666666666666666) (* y y) 1.0))
    (*
     (*
      (fma
       (fma (* x_m x_m) 0.008333333333333333 -0.16666666666666666)
       (* x_m x_m)
       1.0)
      x_m)
     (* (* y y) 0.16666666666666666)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y) {
	double tmp;
	if (sin(x_m) <= 1e-7) {
		tmp = (fma(-0.16666666666666666, (x_m * x_m), 1.0) * x_m) * fma(fma(0.008333333333333333, (y * y), 0.16666666666666666), (y * y), 1.0);
	} else {
		tmp = (fma(fma((x_m * x_m), 0.008333333333333333, -0.16666666666666666), (x_m * x_m), 1.0) * x_m) * ((y * y) * 0.16666666666666666);
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y)
	tmp = 0.0
	if (sin(x_m) <= 1e-7)
		tmp = Float64(Float64(fma(-0.16666666666666666, Float64(x_m * x_m), 1.0) * x_m) * fma(fma(0.008333333333333333, Float64(y * y), 0.16666666666666666), Float64(y * y), 1.0));
	else
		tmp = Float64(Float64(fma(fma(Float64(x_m * x_m), 0.008333333333333333, -0.16666666666666666), Float64(x_m * x_m), 1.0) * x_m) * Float64(Float64(y * y) * 0.16666666666666666));
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_] := N[(x$95$s * If[LessEqual[N[Sin[x$95$m], $MachinePrecision], 1e-7], N[(N[(N[(-0.16666666666666666 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * x$95$m), $MachinePrecision] * N[(N[(0.008333333333333333 * N[(y * y), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * x$95$m), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 x) < 9.9999999999999995e-8

   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. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 92.9

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

   5. Applied rewrites 92.9%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lft-mult-inverse N/A

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

      3. distribute-rgt-in N/A

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

      4. metadata-eval N/A

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

      5. sub-neg N/A

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

      6. lower-*.f64 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. distribute-rgt-in N/A

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

      11. lft-mult-inverse N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 71.9

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

   8. Applied rewrites 71.9%

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

   if 9.9999999999999995e-8 < (sin.f64 x)

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-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 73.0

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

   5. Applied rewrites 73.0%

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

   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

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 29.2

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

   8. Applied rewrites 29.2%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 29.0%

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

Alternative 19: 34.0% accurate, 12.8× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y)
 :precision binary64
 (* x_s (* (fma -0.16666666666666666 (* x_m x_m) 1.0) x_m)))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y) {
	return x_s * (fma(-0.16666666666666666, (x_m * x_m), 1.0) * x_m);
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y)
	return Float64(x_s * Float64(fma(-0.16666666666666666, Float64(x_m * x_m), 1.0) * x_m))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_] := N[(x$95$s * N[(N[(-0.16666666666666666 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in y around 0

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

   1. lower-sin.f64 53.9

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

5. Applied rewrites 53.9%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 41.9%

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

Alternative 20: 26.4% accurate, 36.2× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(1 \cdot x\_m\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y) :precision binary64 (* x_s (* 1.0 x_m)))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y) {
	return x_s * (1.0 * x_m);
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    code = x_s * (1.0d0 * x_m)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y) {
	return x_s * (1.0 * x_m);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y):
	return x_s * (1.0 * x_m)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y)
	return Float64(x_s * Float64(1.0 * x_m))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m, y)
	tmp = x_s * (1.0 * x_m);
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_] := N[(x$95$s * N[(1.0 * x$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in y around 0

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

   1. lower-sin.f64 53.9

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

5. Applied rewrites 53.9%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 41.9%

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

9. Taylor expanded in x around 0

   \[\leadsto 1 \cdot x \]
10. Step-by-step derivation

11. Applied rewrites 34.2%

    \[\leadsto 1 \cdot x \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024235 
(FPCore (x y)
  :name "Linear.Quaternion:$ccos from linear-1.19.1.3"
  :precision binary64
  (* (sin x) (/ (sinh y) y)))