Linear.Quaternion:$ccosh from linear-1.19.1.3

Percentage Accurate: 88.4% → 98.7%

Time: 13.1s

Alternatives: 15

Speedup: 1.0×

• Could not converge on a ground truth

  1. x = 2.0611007246596814e+55
  2. y = 1.1735237125182307e+176

• 49.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 88.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{\sinh y\_m \cdot \sin x}{x}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(0.16666666666666666, y\_m \cdot \left(y\_m \cdot y\_m\right), y\_m\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-14}:\\ \;\;\;\;y\_m \cdot \left(\frac{\sin x}{x} \cdot \mathsf{fma}\left(y\_m \cdot y\_m, \mathsf{fma}\left(y\_m, y\_m \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sinh y\_m \cdot x}{x}\\ \end{array} \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m)
 :precision binary64
 (let* ((t_0 (/ (* (sinh y_m) (sin x)) x)))
   (*
    y_s
    (if (<= t_0 (- INFINITY))
      (*
       (fma x (* x -0.16666666666666666) 1.0)
       (fma 0.16666666666666666 (* y_m (* y_m y_m)) y_m))
      (if (<= t_0 2e-14)
        (*
         y_m
         (*
          (/ (sin x) x)
          (fma
           (* y_m y_m)
           (fma y_m (* y_m 0.008333333333333333) 0.16666666666666666)
           1.0)))
        (/ (* (sinh y_m) x) x))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m) {
	double t_0 = (sinh(y_m) * sin(x)) / x;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma(x, (x * -0.16666666666666666), 1.0) * fma(0.16666666666666666, (y_m * (y_m * y_m)), y_m);
	} else if (t_0 <= 2e-14) {
		tmp = y_m * ((sin(x) / x) * fma((y_m * y_m), fma(y_m, (y_m * 0.008333333333333333), 0.16666666666666666), 1.0));
	} else {
		tmp = (sinh(y_m) * x) / x;
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m)
	t_0 = Float64(Float64(sinh(y_m) * sin(x)) / x)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(fma(x, Float64(x * -0.16666666666666666), 1.0) * fma(0.16666666666666666, Float64(y_m * Float64(y_m * y_m)), y_m));
	elseif (t_0 <= 2e-14)
		tmp = Float64(y_m * Float64(Float64(sin(x) / x) * fma(Float64(y_m * y_m), fma(y_m, Float64(y_m * 0.008333333333333333), 0.16666666666666666), 1.0)));
	else
		tmp = Float64(Float64(sinh(y_m) * x) / x);
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_] := Block[{t$95$0 = N[(N[(N[Sinh[y$95$m], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, N[(y$95$s * If[LessEqual[t$95$0, (-Infinity)], N[(N[(x * N[(x * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * N[(0.16666666666666666 * N[(y$95$m * N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] + y$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e-14], N[(y$95$m * N[(N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision] * N[(N[(y$95$m * y$95$m), $MachinePrecision] * N[(y$95$m * N[(y$95$m * 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sinh[y$95$m], $MachinePrecision] * x), $MachinePrecision] / x), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-lft-identity N/A

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

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

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

      7. unpow2 N/A

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

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

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 70.2

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

   5. Simplified 70.2%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt1-in N/A

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

      5. +-commutative N/A

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

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

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

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

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

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

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

      13. +-commutative N/A

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

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

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

      15. cube-mult N/A

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

      16. unpow2 N/A

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

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

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

      18. unpow2 N/A

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

      19. *-lowering-*.f64 61.9

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

   8. Simplified 61.9%

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

   if -inf.0 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 2e-14

   1. Initial program 75.6%

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

   3. Taylor expanded in y around 0

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

   5. Simplified 98.1%

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

   if 2e-14 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{x} \cdot \sinh y}{x} \]
   4. Step-by-step derivation

   5. Simplified 67.7%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(0.16666666666666666, y \cdot \left(y \cdot y\right), y\right)\\ \mathbf{elif}\;\frac{\sinh y \cdot \sin x}{x} \leq 2 \cdot 10^{-14}:\\ \;\;\;\;y \cdot \left(\frac{\sin x}{x} \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sinh y \cdot x}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{\sinh y\_m \cdot \sin x}{x}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(0.16666666666666666, y\_m \cdot \left(y\_m \cdot y\_m\right), y\_m\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-14}:\\ \;\;\;\;y\_m \cdot \frac{\sin x}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sinh y\_m \cdot x}{x}\\ \end{array} \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m)
 :precision binary64
 (let* ((t_0 (/ (* (sinh y_m) (sin x)) x)))
   (*
    y_s
    (if (<= t_0 (- INFINITY))
      (*
       (fma x (* x -0.16666666666666666) 1.0)
       (fma 0.16666666666666666 (* y_m (* y_m y_m)) y_m))
      (if (<= t_0 2e-14) (* y_m (/ (sin x) x)) (/ (* (sinh y_m) x) x))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m) {
	double t_0 = (sinh(y_m) * sin(x)) / x;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma(x, (x * -0.16666666666666666), 1.0) * fma(0.16666666666666666, (y_m * (y_m * y_m)), y_m);
	} else if (t_0 <= 2e-14) {
		tmp = y_m * (sin(x) / x);
	} else {
		tmp = (sinh(y_m) * x) / x;
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m)
	t_0 = Float64(Float64(sinh(y_m) * sin(x)) / x)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(fma(x, Float64(x * -0.16666666666666666), 1.0) * fma(0.16666666666666666, Float64(y_m * Float64(y_m * y_m)), y_m));
	elseif (t_0 <= 2e-14)
		tmp = Float64(y_m * Float64(sin(x) / x));
	else
		tmp = Float64(Float64(sinh(y_m) * x) / x);
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_] := Block[{t$95$0 = N[(N[(N[Sinh[y$95$m], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, N[(y$95$s * If[LessEqual[t$95$0, (-Infinity)], N[(N[(x * N[(x * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * N[(0.16666666666666666 * N[(y$95$m * N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] + y$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e-14], N[(y$95$m * N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sinh[y$95$m], $MachinePrecision] * x), $MachinePrecision] / x), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-lft-identity N/A

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

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

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

      7. unpow2 N/A

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

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

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 70.2

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

   5. Simplified 70.2%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt1-in N/A

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

      5. +-commutative N/A

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

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

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

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

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

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

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

      13. +-commutative N/A

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

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

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

      15. cube-mult N/A

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

      16. unpow2 N/A

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

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

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

      18. unpow2 N/A

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

      19. *-lowering-*.f64 61.9

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

   8. Simplified 61.9%

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

   if -inf.0 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 2e-14

   1. Initial program 75.6%

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

   3. Taylor expanded in y around 0

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

      1. associate-/l* N/A

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

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

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

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

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

      4. sin-lowering-sin.f64 97.5

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

   5. Simplified 97.5%

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

   if 2e-14 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{x} \cdot \sinh y}{x} \]
   4. Step-by-step derivation

   5. Simplified 67.7%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(0.16666666666666666, y \cdot \left(y \cdot y\right), y\right)\\ \mathbf{elif}\;\frac{\sinh y \cdot \sin x}{x} \leq 2 \cdot 10^{-14}:\\ \;\;\;\;y \cdot \frac{\sin x}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sinh y \cdot x}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := y\_m \cdot \left(y\_m \cdot y\_m\right)\\ t_1 := \frac{\sinh y\_m \cdot \sin x}{x}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(0.16666666666666666, t\_0, y\_m\right)\\ \mathbf{elif}\;t\_1 \leq 10^{-29}:\\ \;\;\;\;y\_m \cdot \frac{\sin x}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right), x \cdot \left(x \cdot x\right), x\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y\_m, y\_m \cdot 0.008333333333333333, 0.16666666666666666\right), t\_0, y\_m\right)}{x}\\ \end{array} \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m)
 :precision binary64
 (let* ((t_0 (* y_m (* y_m y_m))) (t_1 (/ (* (sinh y_m) (sin x)) x)))
   (*
    y_s
    (if (<= t_1 (- INFINITY))
      (*
       (fma x (* x -0.16666666666666666) 1.0)
       (fma 0.16666666666666666 t_0 y_m))
      (if (<= t_1 1e-29)
        (* y_m (/ (sin x) x))
        (/
         (*
          (fma
           (fma 0.008333333333333333 (* x x) -0.16666666666666666)
           (* x (* x x))
           x)
          (fma
           (fma y_m (* y_m 0.008333333333333333) 0.16666666666666666)
           t_0
           y_m))
         x))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m) {
	double t_0 = y_m * (y_m * y_m);
	double t_1 = (sinh(y_m) * sin(x)) / x;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma(x, (x * -0.16666666666666666), 1.0) * fma(0.16666666666666666, t_0, y_m);
	} else if (t_1 <= 1e-29) {
		tmp = y_m * (sin(x) / x);
	} else {
		tmp = (fma(fma(0.008333333333333333, (x * x), -0.16666666666666666), (x * (x * x)), x) * fma(fma(y_m, (y_m * 0.008333333333333333), 0.16666666666666666), t_0, y_m)) / x;
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m)
	t_0 = Float64(y_m * Float64(y_m * y_m))
	t_1 = Float64(Float64(sinh(y_m) * sin(x)) / x)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(fma(x, Float64(x * -0.16666666666666666), 1.0) * fma(0.16666666666666666, t_0, y_m));
	elseif (t_1 <= 1e-29)
		tmp = Float64(y_m * Float64(sin(x) / x));
	else
		tmp = Float64(Float64(fma(fma(0.008333333333333333, Float64(x * x), -0.16666666666666666), Float64(x * Float64(x * x)), x) * fma(fma(y_m, Float64(y_m * 0.008333333333333333), 0.16666666666666666), t_0, y_m)) / x);
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_] := Block[{t$95$0 = N[(y$95$m * N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Sinh[y$95$m], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, N[(y$95$s * If[LessEqual[t$95$1, (-Infinity)], N[(N[(x * N[(x * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * N[(0.16666666666666666 * t$95$0 + y$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e-29], N[(y$95$m * N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.008333333333333333 * N[(x * x), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] * N[(N[(y$95$m * N[(y$95$m * 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * t$95$0 + y$95$m), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-lft-identity N/A

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

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

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

      7. unpow2 N/A

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

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

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 70.2

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

   5. Simplified 70.2%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt1-in N/A

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

      5. +-commutative N/A

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

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

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

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

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

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

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

      13. +-commutative N/A

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

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

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

      15. cube-mult N/A

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

      16. unpow2 N/A

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

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

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

      18. unpow2 N/A

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

      19. *-lowering-*.f64 61.9

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

   8. Simplified 61.9%

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

   if -inf.0 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 9.99999999999999943e-30

   1. Initial program 75.4%

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

   3. Taylor expanded in y around 0

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

      1. associate-/l* N/A

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

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

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

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

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

      4. sin-lowering-sin.f64 97.5

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

   5. Simplified 97.5%

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

   if 9.99999999999999943e-30 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-rgt-identity N/A

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

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

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

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

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

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

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

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

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 83.4

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

   5. Simplified 83.4%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. unpow2 N/A

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

      6. unpow3 N/A

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

      7. *-lft-identity N/A

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

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

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

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

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

      12. unpow2 N/A

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

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

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

      14. cube-mult N/A

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

      15. unpow2 N/A

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

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

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

      17. unpow2 N/A

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

      18. *-lowering-*.f64 59.3

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

   8. Simplified 59.3%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(0.16666666666666666, y \cdot \left(y \cdot y\right), y\right)\\ \mathbf{elif}\;\frac{\sinh y \cdot \sin x}{x} \leq 10^{-29}:\\ \;\;\;\;y \cdot \frac{\sin x}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right), x \cdot \left(x \cdot x\right), x\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{\sinh y\_m \cdot \sin x}{x}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-133}:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(0.16666666666666666, y\_m \cdot \left(y\_m \cdot y\_m\right), y\_m\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-113}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(0.16666666666666666, x \cdot x, 1\right) \cdot \frac{1}{y\_m}}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \left(\mathsf{fma}\left(y\_m \cdot y\_m, \mathsf{fma}\left(y\_m, y\_m \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.008333333333333333, -0.16666666666666666\right), 1\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m)
 :precision binary64
 (let* ((t_0 (/ (* (sinh y_m) (sin x)) x)))
   (*
    y_s
    (if (<= t_0 -5e-133)
      (*
       (fma x (* x -0.16666666666666666) 1.0)
       (fma 0.16666666666666666 (* y_m (* y_m y_m)) y_m))
      (if (<= t_0 2e-113)
        (/ 1.0 (* (fma 0.16666666666666666 (* x x) 1.0) (/ 1.0 y_m)))
        (*
         y_m
         (*
          (fma
           (* y_m y_m)
           (fma y_m (* y_m 0.008333333333333333) 0.16666666666666666)
           1.0)
          (fma
           (* x x)
           (fma x (* x 0.008333333333333333) -0.16666666666666666)
           1.0))))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m) {
	double t_0 = (sinh(y_m) * sin(x)) / x;
	double tmp;
	if (t_0 <= -5e-133) {
		tmp = fma(x, (x * -0.16666666666666666), 1.0) * fma(0.16666666666666666, (y_m * (y_m * y_m)), y_m);
	} else if (t_0 <= 2e-113) {
		tmp = 1.0 / (fma(0.16666666666666666, (x * x), 1.0) * (1.0 / y_m));
	} else {
		tmp = y_m * (fma((y_m * y_m), fma(y_m, (y_m * 0.008333333333333333), 0.16666666666666666), 1.0) * fma((x * x), fma(x, (x * 0.008333333333333333), -0.16666666666666666), 1.0));
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m)
	t_0 = Float64(Float64(sinh(y_m) * sin(x)) / x)
	tmp = 0.0
	if (t_0 <= -5e-133)
		tmp = Float64(fma(x, Float64(x * -0.16666666666666666), 1.0) * fma(0.16666666666666666, Float64(y_m * Float64(y_m * y_m)), y_m));
	elseif (t_0 <= 2e-113)
		tmp = Float64(1.0 / Float64(fma(0.16666666666666666, Float64(x * x), 1.0) * Float64(1.0 / y_m)));
	else
		tmp = Float64(y_m * Float64(fma(Float64(y_m * y_m), fma(y_m, Float64(y_m * 0.008333333333333333), 0.16666666666666666), 1.0) * fma(Float64(x * x), fma(x, Float64(x * 0.008333333333333333), -0.16666666666666666), 1.0)));
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_] := Block[{t$95$0 = N[(N[(N[Sinh[y$95$m], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, N[(y$95$s * If[LessEqual[t$95$0, -5e-133], N[(N[(x * N[(x * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * N[(0.16666666666666666 * N[(y$95$m * N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] + y$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e-113], N[(1.0 / N[(N[(0.16666666666666666 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(1.0 / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y$95$m * N[(N[(N[(y$95$m * y$95$m), $MachinePrecision] * N[(y$95$m * N[(y$95$m * 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -4.9999999999999999e-133

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-lft-identity N/A

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

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

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

      7. unpow2 N/A

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

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

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 76.1

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

   5. Simplified 76.1%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt1-in N/A

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

      5. +-commutative N/A

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

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

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

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

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

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

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

      13. +-commutative N/A

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

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

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

      15. cube-mult N/A

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

      16. unpow2 N/A

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

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

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

      18. unpow2 N/A

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

      19. *-lowering-*.f64 67.7

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

   8. Simplified 67.7%

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

   if -4.9999999999999999e-133 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 1.99999999999999996e-113

   1. Initial program 69.2%

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

   3. Taylor expanded in y around 0

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

      1. associate-/l* N/A

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

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

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

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

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

      4. sin-lowering-sin.f64 99.8

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

   5. Simplified 99.8%

      \[\leadsto \color{blue}{y \cdot \frac{\sin x}{x}} \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

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

      2. clear-num N/A

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

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

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

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

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

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

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

      6. sin-lowering-sin.f64 65.9

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

   7. Applied egg-rr 65.9%

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

   8. Taylor expanded in x around 0

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{6} \cdot \frac{{x}^{2}}{y} + \frac{1}{y}}} \]
   9. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. associate-*r/ N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. distribute-lft1-in N/A

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

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

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

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

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

      11. unpow2 N/A

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

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

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

      13. /-lowering-/.f64 72.3

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

   10. Simplified 72.3%

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

   if 1.99999999999999996e-113 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Simplified 80.3%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

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

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

      3. unpow2 N/A

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

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

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

      5. sub-neg N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. metadata-eval N/A

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

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

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

      11. *-lowering-*.f64 62.5

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

   8. Simplified 62.5%

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

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -5 \cdot 10^{-133}:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(0.16666666666666666, y \cdot \left(y \cdot y\right), y\right)\\ \mathbf{elif}\;\frac{\sinh y \cdot \sin x}{x} \leq 2 \cdot 10^{-113}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(0.16666666666666666, x \cdot x, 1\right) \cdot \frac{1}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.008333333333333333, -0.16666666666666666\right), 1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{\sinh y\_m \cdot \sin x}{x}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-133}:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(0.16666666666666666, y\_m \cdot \left(y\_m \cdot y\_m\right), y\_m\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-14}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(0.16666666666666666, x \cdot x, 1\right) \cdot \frac{1}{y\_m}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y\_m \cdot \left(y\_m \cdot \mathsf{fma}\left(y\_m, y\_m \cdot 0.008333333333333333, 0.16666666666666666\right)\right), y\_m, y\_m\right)\\ \end{array} \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m)
 :precision binary64
 (let* ((t_0 (/ (* (sinh y_m) (sin x)) x)))
   (*
    y_s
    (if (<= t_0 -5e-133)
      (*
       (fma x (* x -0.16666666666666666) 1.0)
       (fma 0.16666666666666666 (* y_m (* y_m y_m)) y_m))
      (if (<= t_0 2e-14)
        (/ 1.0 (* (fma 0.16666666666666666 (* x x) 1.0) (/ 1.0 y_m)))
        (fma
         (*
          y_m
          (* y_m (fma y_m (* y_m 0.008333333333333333) 0.16666666666666666)))
         y_m
         y_m))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m) {
	double t_0 = (sinh(y_m) * sin(x)) / x;
	double tmp;
	if (t_0 <= -5e-133) {
		tmp = fma(x, (x * -0.16666666666666666), 1.0) * fma(0.16666666666666666, (y_m * (y_m * y_m)), y_m);
	} else if (t_0 <= 2e-14) {
		tmp = 1.0 / (fma(0.16666666666666666, (x * x), 1.0) * (1.0 / y_m));
	} else {
		tmp = fma((y_m * (y_m * fma(y_m, (y_m * 0.008333333333333333), 0.16666666666666666))), y_m, y_m);
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m)
	t_0 = Float64(Float64(sinh(y_m) * sin(x)) / x)
	tmp = 0.0
	if (t_0 <= -5e-133)
		tmp = Float64(fma(x, Float64(x * -0.16666666666666666), 1.0) * fma(0.16666666666666666, Float64(y_m * Float64(y_m * y_m)), y_m));
	elseif (t_0 <= 2e-14)
		tmp = Float64(1.0 / Float64(fma(0.16666666666666666, Float64(x * x), 1.0) * Float64(1.0 / y_m)));
	else
		tmp = fma(Float64(y_m * Float64(y_m * fma(y_m, Float64(y_m * 0.008333333333333333), 0.16666666666666666))), y_m, y_m);
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_] := Block[{t$95$0 = N[(N[(N[Sinh[y$95$m], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, N[(y$95$s * If[LessEqual[t$95$0, -5e-133], N[(N[(x * N[(x * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * N[(0.16666666666666666 * N[(y$95$m * N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] + y$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e-14], N[(1.0 / N[(N[(0.16666666666666666 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(1.0 / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m * N[(y$95$m * N[(y$95$m * N[(y$95$m * 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y$95$m + y$95$m), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -4.9999999999999999e-133

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-lft-identity N/A

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

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

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

      7. unpow2 N/A

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

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

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 76.1

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

   5. Simplified 76.1%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt1-in N/A

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

      5. +-commutative N/A

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

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

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

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

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

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

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

      13. +-commutative N/A

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

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

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

      15. cube-mult N/A

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

      16. unpow2 N/A

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

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

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

      18. unpow2 N/A

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

      19. *-lowering-*.f64 67.7

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

   8. Simplified 67.7%

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

   if -4.9999999999999999e-133 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 2e-14

   1. Initial program 71.4%

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

   3. Taylor expanded in y around 0

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

      1. associate-/l* N/A

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

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

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

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

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

      4. sin-lowering-sin.f64 99.2

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

   5. Simplified 99.2%

      \[\leadsto \color{blue}{y \cdot \frac{\sin x}{x}} \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

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

      2. clear-num N/A

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

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

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

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

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

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

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

      6. sin-lowering-sin.f64 67.7

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

   7. Applied egg-rr 67.7%

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

   8. Taylor expanded in x around 0

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{6} \cdot \frac{{x}^{2}}{y} + \frac{1}{y}}} \]
   9. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. associate-*r/ N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. distribute-lft1-in N/A

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

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

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

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

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

      11. unpow2 N/A

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

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

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

      13. /-lowering-/.f64 73.5

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

   10. Simplified 73.5%

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

   if 2e-14 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-rgt-identity N/A

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

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

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

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

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

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

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

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

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 83.1

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

   5. Simplified 83.1%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

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

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

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

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

      7. unpow2 N/A

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

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

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

      9. cube-mult N/A

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

      10. unpow2 N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 55.6

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

   8. Simplified 55.6%

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

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

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

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

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

      5. *-commutative N/A

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

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

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

      7. associate-*l* N/A

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

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

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

      9. *-lowering-*.f64 55.6

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

   10. Applied egg-rr 55.6%

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

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -5 \cdot 10^{-133}:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(0.16666666666666666, y \cdot \left(y \cdot y\right), y\right)\\ \mathbf{elif}\;\frac{\sinh y \cdot \sin x}{x} \leq 2 \cdot 10^{-14}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(0.16666666666666666, x \cdot x, 1\right) \cdot \frac{1}{y}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot \left(y \cdot \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right)\right), y, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{\sinh y\_m \cdot \sin x}{x}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-254}:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(0.16666666666666666, y\_m \cdot \left(y\_m \cdot y\_m\right), y\_m\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;x \cdot \frac{y\_m}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y\_m \cdot \left(y\_m \cdot \mathsf{fma}\left(y\_m, y\_m \cdot 0.008333333333333333, 0.16666666666666666\right)\right), y\_m, y\_m\right)\\ \end{array} \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m)
 :precision binary64
 (let* ((t_0 (/ (* (sinh y_m) (sin x)) x)))
   (*
    y_s
    (if (<= t_0 -1e-254)
      (*
       (fma x (* x -0.16666666666666666) 1.0)
       (fma 0.16666666666666666 (* y_m (* y_m y_m)) y_m))
      (if (<= t_0 0.0)
        (* x (/ y_m x))
        (fma
         (*
          y_m
          (* y_m (fma y_m (* y_m 0.008333333333333333) 0.16666666666666666)))
         y_m
         y_m))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m) {
	double t_0 = (sinh(y_m) * sin(x)) / x;
	double tmp;
	if (t_0 <= -1e-254) {
		tmp = fma(x, (x * -0.16666666666666666), 1.0) * fma(0.16666666666666666, (y_m * (y_m * y_m)), y_m);
	} else if (t_0 <= 0.0) {
		tmp = x * (y_m / x);
	} else {
		tmp = fma((y_m * (y_m * fma(y_m, (y_m * 0.008333333333333333), 0.16666666666666666))), y_m, y_m);
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m)
	t_0 = Float64(Float64(sinh(y_m) * sin(x)) / x)
	tmp = 0.0
	if (t_0 <= -1e-254)
		tmp = Float64(fma(x, Float64(x * -0.16666666666666666), 1.0) * fma(0.16666666666666666, Float64(y_m * Float64(y_m * y_m)), y_m));
	elseif (t_0 <= 0.0)
		tmp = Float64(x * Float64(y_m / x));
	else
		tmp = fma(Float64(y_m * Float64(y_m * fma(y_m, Float64(y_m * 0.008333333333333333), 0.16666666666666666))), y_m, y_m);
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_] := Block[{t$95$0 = N[(N[(N[Sinh[y$95$m], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, N[(y$95$s * If[LessEqual[t$95$0, -1e-254], N[(N[(x * N[(x * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * N[(0.16666666666666666 * N[(y$95$m * N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] + y$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(x * N[(y$95$m / x), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m * N[(y$95$m * N[(y$95$m * N[(y$95$m * 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y$95$m + y$95$m), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -9.9999999999999991e-255

   1. Initial program 99.6%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-lft-identity N/A

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

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

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

      7. unpow2 N/A

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

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

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 78.8

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

   5. Simplified 78.8%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt1-in N/A

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

      5. +-commutative N/A

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

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

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

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

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

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

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

      13. +-commutative N/A

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

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

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

      15. cube-mult N/A

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

      16. unpow2 N/A

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

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

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

      18. unpow2 N/A

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

      19. *-lowering-*.f64 62.2

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

   8. Simplified 62.2%

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

   if -9.9999999999999991e-255 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 0.0

   1. Initial program 54.0%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

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

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

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

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

      6. sin-lowering-sin.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y}{x}} \cdot \sin x \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 99.9

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

   7. Simplified 99.9%

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

   8. Taylor expanded in x around 0

      \[\leadsto \frac{y}{x} \cdot \color{blue}{x} \]
   9. Step-by-step derivation

   10. Simplified 89.8%

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

   if 0.0 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-rgt-identity N/A

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

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

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

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

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

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

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

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

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 88.6

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

   5. Simplified 88.6%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

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

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

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

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

      7. unpow2 N/A

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

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

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

      9. cube-mult N/A

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

      10. unpow2 N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 48.2

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

   8. Simplified 48.2%

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

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

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

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

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

      5. *-commutative N/A

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

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

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

      7. associate-*l* N/A

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

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

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

      9. *-lowering-*.f64 48.2

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

   10. Applied egg-rr 48.2%

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

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -1 \cdot 10^{-254}:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(0.16666666666666666, y \cdot \left(y \cdot y\right), y\right)\\ \mathbf{elif}\;\frac{\sinh y \cdot \sin x}{x} \leq 0:\\ \;\;\;\;x \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot \left(y \cdot \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right)\right), y, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{\sinh y\_m \cdot \sin x}{x}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-254}:\\ \;\;\;\;\mathsf{fma}\left(y\_m \cdot x, x \cdot -0.16666666666666666, y\_m\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;x \cdot \frac{y\_m}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y\_m \cdot \left(y\_m \cdot \mathsf{fma}\left(y\_m, y\_m \cdot 0.008333333333333333, 0.16666666666666666\right)\right), y\_m, y\_m\right)\\ \end{array} \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m)
 :precision binary64
 (let* ((t_0 (/ (* (sinh y_m) (sin x)) x)))
   (*
    y_s
    (if (<= t_0 -1e-254)
      (fma (* y_m x) (* x -0.16666666666666666) y_m)
      (if (<= t_0 0.0)
        (* x (/ y_m x))
        (fma
         (*
          y_m
          (* y_m (fma y_m (* y_m 0.008333333333333333) 0.16666666666666666)))
         y_m
         y_m))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m) {
	double t_0 = (sinh(y_m) * sin(x)) / x;
	double tmp;
	if (t_0 <= -1e-254) {
		tmp = fma((y_m * x), (x * -0.16666666666666666), y_m);
	} else if (t_0 <= 0.0) {
		tmp = x * (y_m / x);
	} else {
		tmp = fma((y_m * (y_m * fma(y_m, (y_m * 0.008333333333333333), 0.16666666666666666))), y_m, y_m);
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m)
	t_0 = Float64(Float64(sinh(y_m) * sin(x)) / x)
	tmp = 0.0
	if (t_0 <= -1e-254)
		tmp = fma(Float64(y_m * x), Float64(x * -0.16666666666666666), y_m);
	elseif (t_0 <= 0.0)
		tmp = Float64(x * Float64(y_m / x));
	else
		tmp = fma(Float64(y_m * Float64(y_m * fma(y_m, Float64(y_m * 0.008333333333333333), 0.16666666666666666))), y_m, y_m);
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_] := Block[{t$95$0 = N[(N[(N[Sinh[y$95$m], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, N[(y$95$s * If[LessEqual[t$95$0, -1e-254], N[(N[(y$95$m * x), $MachinePrecision] * N[(x * -0.16666666666666666), $MachinePrecision] + y$95$m), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(x * N[(y$95$m / x), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m * N[(y$95$m * N[(y$95$m * N[(y$95$m * 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y$95$m + y$95$m), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -9.9999999999999991e-255

   1. Initial program 99.6%

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

   3. Taylor expanded in y around 0

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

      1. associate-/l* N/A

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

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

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

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

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

      4. sin-lowering-sin.f64 37.4

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

   5. Simplified 37.4%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

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

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

      6. *-lowering-*.f64 39.6

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

   8. Simplified 39.6%

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

      1. distribute-lft-in N/A

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

      2. associate-*r* N/A

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

      3. *-rgt-identity N/A

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

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

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

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

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

      6. *-lowering-*.f64 39.6

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

   10. Applied egg-rr 39.6%

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

   if -9.9999999999999991e-255 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 0.0

   1. Initial program 54.0%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

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

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

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

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

      6. sin-lowering-sin.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y}{x}} \cdot \sin x \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 99.9

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

   7. Simplified 99.9%

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

   8. Taylor expanded in x around 0

      \[\leadsto \frac{y}{x} \cdot \color{blue}{x} \]
   9. Step-by-step derivation

   10. Simplified 89.8%

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

   if 0.0 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-rgt-identity N/A

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

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

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

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

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

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

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

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

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 88.6

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

   5. Simplified 88.6%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

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

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

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

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

      7. unpow2 N/A

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

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

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

      9. cube-mult N/A

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

      10. unpow2 N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 48.2

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

   8. Simplified 48.2%

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

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

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

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

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

      5. *-commutative N/A

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

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

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

      7. associate-*l* N/A

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

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

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

      9. *-lowering-*.f64 48.2

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

   10. Applied egg-rr 48.2%

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

4. Final simplification 57.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -1 \cdot 10^{-254}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, x \cdot -0.16666666666666666, y\right)\\ \mathbf{elif}\;\frac{\sinh y \cdot \sin x}{x} \leq 0:\\ \;\;\;\;x \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot \left(y \cdot \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right)\right), y, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 75.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{\sinh y\_m \cdot \sin x}{x}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-254}:\\ \;\;\;\;\mathsf{fma}\left(y\_m \cdot x, x \cdot -0.16666666666666666, y\_m\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;x \cdot \frac{y\_m}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y\_m \cdot \left(y\_m \cdot 0.008333333333333333\right), y\_m \cdot \left(y\_m \cdot y\_m\right), y\_m\right)\\ \end{array} \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m)
 :precision binary64
 (let* ((t_0 (/ (* (sinh y_m) (sin x)) x)))
   (*
    y_s
    (if (<= t_0 -1e-254)
      (fma (* y_m x) (* x -0.16666666666666666) y_m)
      (if (<= t_0 0.0)
        (* x (/ y_m x))
        (fma (* y_m (* y_m 0.008333333333333333)) (* y_m (* y_m y_m)) y_m))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m) {
	double t_0 = (sinh(y_m) * sin(x)) / x;
	double tmp;
	if (t_0 <= -1e-254) {
		tmp = fma((y_m * x), (x * -0.16666666666666666), y_m);
	} else if (t_0 <= 0.0) {
		tmp = x * (y_m / x);
	} else {
		tmp = fma((y_m * (y_m * 0.008333333333333333)), (y_m * (y_m * y_m)), y_m);
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m)
	t_0 = Float64(Float64(sinh(y_m) * sin(x)) / x)
	tmp = 0.0
	if (t_0 <= -1e-254)
		tmp = fma(Float64(y_m * x), Float64(x * -0.16666666666666666), y_m);
	elseif (t_0 <= 0.0)
		tmp = Float64(x * Float64(y_m / x));
	else
		tmp = fma(Float64(y_m * Float64(y_m * 0.008333333333333333)), Float64(y_m * Float64(y_m * y_m)), y_m);
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_] := Block[{t$95$0 = N[(N[(N[Sinh[y$95$m], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, N[(y$95$s * If[LessEqual[t$95$0, -1e-254], N[(N[(y$95$m * x), $MachinePrecision] * N[(x * -0.16666666666666666), $MachinePrecision] + y$95$m), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(x * N[(y$95$m / x), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m * N[(y$95$m * 0.008333333333333333), $MachinePrecision]), $MachinePrecision] * N[(y$95$m * N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] + y$95$m), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -9.9999999999999991e-255

   1. Initial program 99.6%

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

   3. Taylor expanded in y around 0

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

      1. associate-/l* N/A

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

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

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

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

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

      4. sin-lowering-sin.f64 37.4

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

   5. Simplified 37.4%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

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

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

      6. *-lowering-*.f64 39.6

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

   8. Simplified 39.6%

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

      1. distribute-lft-in N/A

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

      2. associate-*r* N/A

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

      3. *-rgt-identity N/A

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

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

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

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

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

      6. *-lowering-*.f64 39.6

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

   10. Applied egg-rr 39.6%

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

   if -9.9999999999999991e-255 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 0.0

   1. Initial program 54.0%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

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

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

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

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

      6. sin-lowering-sin.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y}{x}} \cdot \sin x \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 99.9

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

   7. Simplified 99.9%

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

   8. Taylor expanded in x around 0

      \[\leadsto \frac{y}{x} \cdot \color{blue}{x} \]
   9. Step-by-step derivation

   10. Simplified 89.8%

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

   if 0.0 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-rgt-identity N/A

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

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

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

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

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

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

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

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

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 88.6

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

   5. Simplified 88.6%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

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

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

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

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

      7. unpow2 N/A

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

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

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

      9. cube-mult N/A

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

      10. unpow2 N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 48.2

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

   8. Simplified 48.2%

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

   9. Taylor expanded in y around inf

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

       1. *-commutative N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

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

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

       6. *-commutative N/A

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

       7. *-lowering-*.f64 48.2

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

   11. Simplified 48.2%

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

4. Final simplification 57.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -1 \cdot 10^{-254}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, x \cdot -0.16666666666666666, y\right)\\ \mathbf{elif}\;\frac{\sinh y \cdot \sin x}{x} \leq 0:\\ \;\;\;\;x \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot \left(y \cdot 0.008333333333333333\right), y \cdot \left(y \cdot y\right), y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 70.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{\sinh y\_m \cdot \sin x}{x}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-254}:\\ \;\;\;\;\mathsf{fma}\left(y\_m \cdot x, x \cdot -0.16666666666666666, y\_m\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;x \cdot \frac{y\_m}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.16666666666666666, y\_m \cdot \left(y\_m \cdot y\_m\right), y\_m\right)\\ \end{array} \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m)
 :precision binary64
 (let* ((t_0 (/ (* (sinh y_m) (sin x)) x)))
   (*
    y_s
    (if (<= t_0 -1e-254)
      (fma (* y_m x) (* x -0.16666666666666666) y_m)
      (if (<= t_0 0.0)
        (* x (/ y_m x))
        (fma 0.16666666666666666 (* y_m (* y_m y_m)) y_m))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m) {
	double t_0 = (sinh(y_m) * sin(x)) / x;
	double tmp;
	if (t_0 <= -1e-254) {
		tmp = fma((y_m * x), (x * -0.16666666666666666), y_m);
	} else if (t_0 <= 0.0) {
		tmp = x * (y_m / x);
	} else {
		tmp = fma(0.16666666666666666, (y_m * (y_m * y_m)), y_m);
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m)
	t_0 = Float64(Float64(sinh(y_m) * sin(x)) / x)
	tmp = 0.0
	if (t_0 <= -1e-254)
		tmp = fma(Float64(y_m * x), Float64(x * -0.16666666666666666), y_m);
	elseif (t_0 <= 0.0)
		tmp = Float64(x * Float64(y_m / x));
	else
		tmp = fma(0.16666666666666666, Float64(y_m * Float64(y_m * y_m)), y_m);
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_] := Block[{t$95$0 = N[(N[(N[Sinh[y$95$m], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, N[(y$95$s * If[LessEqual[t$95$0, -1e-254], N[(N[(y$95$m * x), $MachinePrecision] * N[(x * -0.16666666666666666), $MachinePrecision] + y$95$m), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(x * N[(y$95$m / x), $MachinePrecision]), $MachinePrecision], N[(0.16666666666666666 * N[(y$95$m * N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] + y$95$m), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -9.9999999999999991e-255

   1. Initial program 99.6%

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

   3. Taylor expanded in y around 0

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

      1. associate-/l* N/A

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

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

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

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

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

      4. sin-lowering-sin.f64 37.4

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

   5. Simplified 37.4%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

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

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

      6. *-lowering-*.f64 39.6

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

   8. Simplified 39.6%

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

      1. distribute-lft-in N/A

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

      2. associate-*r* N/A

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

      3. *-rgt-identity N/A

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

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

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

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

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

      6. *-lowering-*.f64 39.6

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

   10. Applied egg-rr 39.6%

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

   if -9.9999999999999991e-255 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 0.0

   1. Initial program 54.0%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

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

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

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

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

      6. sin-lowering-sin.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y}{x}} \cdot \sin x \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 99.9

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

   7. Simplified 99.9%

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

   8. Taylor expanded in x around 0

      \[\leadsto \frac{y}{x} \cdot \color{blue}{x} \]
   9. Step-by-step derivation

   10. Simplified 89.8%

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

   if 0.0 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-lft-identity N/A

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

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

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

      7. unpow2 N/A

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

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

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 75.0

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

   5. Simplified 75.0%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

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

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

      3. cube-mult N/A

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

      4. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{6}, y \cdot \color{blue}{{y}^{2}}, y\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{6}, \color{blue}{y \cdot {y}^{2}}, y\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{6}, y \cdot \color{blue}{\left(y \cdot y\right)}, y\right) \]

      7. *-lowering-*.f64 41.5

         \[\leadsto \mathsf{fma}\left(0.16666666666666666, y \cdot \color{blue}{\left(y \cdot y\right)}, y\right) \]

   8. Simplified 41.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.16666666666666666, y \cdot \left(y \cdot y\right), y\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 55.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -1 \cdot 10^{-254}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, x \cdot -0.16666666666666666, y\right)\\ \mathbf{elif}\;\frac{\sinh y \cdot \sin x}{x} \leq 0:\\ \;\;\;\;x \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.16666666666666666, y \cdot \left(y \cdot y\right), y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 70.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{\sinh y\_m \cdot \sin x}{x}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-254}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(y\_m \cdot \left(x \cdot x\right)\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;x \cdot \frac{y\_m}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.16666666666666666, y\_m \cdot \left(y\_m \cdot y\_m\right), y\_m\right)\\ \end{array} \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m)
 :precision binary64
 (let* ((t_0 (/ (* (sinh y_m) (sin x)) x)))
   (*
    y_s
    (if (<= t_0 -1e-254)
      (* -0.16666666666666666 (* y_m (* x x)))
      (if (<= t_0 0.0)
        (* x (/ y_m x))
        (fma 0.16666666666666666 (* y_m (* y_m y_m)) y_m))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m) {
	double t_0 = (sinh(y_m) * sin(x)) / x;
	double tmp;
	if (t_0 <= -1e-254) {
		tmp = -0.16666666666666666 * (y_m * (x * x));
	} else if (t_0 <= 0.0) {
		tmp = x * (y_m / x);
	} else {
		tmp = fma(0.16666666666666666, (y_m * (y_m * y_m)), y_m);
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m)
	t_0 = Float64(Float64(sinh(y_m) * sin(x)) / x)
	tmp = 0.0
	if (t_0 <= -1e-254)
		tmp = Float64(-0.16666666666666666 * Float64(y_m * Float64(x * x)));
	elseif (t_0 <= 0.0)
		tmp = Float64(x * Float64(y_m / x));
	else
		tmp = fma(0.16666666666666666, Float64(y_m * Float64(y_m * y_m)), y_m);
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_] := Block[{t$95$0 = N[(N[(N[Sinh[y$95$m], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, N[(y$95$s * If[LessEqual[t$95$0, -1e-254], N[(-0.16666666666666666 * N[(y$95$m * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(x * N[(y$95$m / x), $MachinePrecision]), $MachinePrecision], N[(0.16666666666666666 * N[(y$95$m * N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] + y$95$m), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -9.9999999999999991e-255

   1. Initial program 99.6%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y \cdot \sin x}{x}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{y \cdot \frac{\sin x}{x}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{y \cdot \frac{\sin x}{x}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto y \cdot \color{blue}{\frac{\sin x}{x}} \]

      4. sin-lowering-sin.f64 37.4

         \[\leadsto y \cdot \frac{\color{blue}{\sin x}}{x} \]

   5. Simplified 37.4%

      \[\leadsto \color{blue}{y \cdot \frac{\sin x}{x}} \]

   6. Taylor expanded in x around 0

      \[\leadsto y \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {x}^{2}\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto y \cdot \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2} + 1\right)} \]

      2. *-commutative N/A

         \[\leadsto y \cdot \left(\color{blue}{{x}^{2} \cdot \frac{-1}{6}} + 1\right) \]

      3. unpow2 N/A

         \[\leadsto y \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{-1}{6} + 1\right) \]

      4. associate-*l* N/A

         \[\leadsto y \cdot \left(\color{blue}{x \cdot \left(x \cdot \frac{-1}{6}\right)} + 1\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto y \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)} \]

      6. *-lowering-*.f64 39.6

         \[\leadsto y \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot -0.16666666666666666}, 1\right) \]

   8. Simplified 39.6%

      \[\leadsto y \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right)} \]

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{-1}{6} \cdot \left({x}^{2} \cdot y\right)} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\frac{-1}{6} \cdot \left({x}^{2} \cdot y\right)} \]

       2. *-commutative N/A

          \[\leadsto \frac{-1}{6} \cdot \color{blue}{\left(y \cdot {x}^{2}\right)} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \frac{-1}{6} \cdot \color{blue}{\left(y \cdot {x}^{2}\right)} \]

       4. unpow2 N/A

          \[\leadsto \frac{-1}{6} \cdot \left(y \cdot \color{blue}{\left(x \cdot x\right)}\right) \]

       5. *-lowering-*.f64 16.6

          \[\leadsto -0.16666666666666666 \cdot \left(y \cdot \color{blue}{\left(x \cdot x\right)}\right) \]

   11. Simplified 16.6%

       \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(y \cdot \left(x \cdot x\right)\right)} \]

   if -9.9999999999999991e-255 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 0.0

   1. Initial program 54.0%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\sinh y}{x} \cdot \sin x} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sinh y}{x} \cdot \sin x} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sinh y}{x}} \cdot \sin x \]

      5. sinh-lowering-sinh.f64 N/A

         \[\leadsto \frac{\color{blue}{\sinh y}}{x} \cdot \sin x \]

      6. sin-lowering-sin.f64 99.9

         \[\leadsto \frac{\sinh y}{x} \cdot \color{blue}{\sin x} \]

   4. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\frac{\sinh y}{x} \cdot \sin x} \]

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y}{x}} \cdot \sin x \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 99.9

         \[\leadsto \color{blue}{\frac{y}{x}} \cdot \sin x \]

   7. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{y}{x}} \cdot \sin x \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{y}{x} \cdot \color{blue}{x} \]
   9. Step-by-step derivation

   10. Simplified 89.8%

       \[\leadsto \frac{y}{x} \cdot \color{blue}{x} \]

   if 0.0 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 99.9%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\sin x \cdot \color{blue}{\left(y \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)}}{x} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\sin x \cdot \left(y \cdot \color{blue}{\left(\frac{1}{6} \cdot {y}^{2} + 1\right)}\right)}{x} \]

      2. distribute-rgt-in N/A

         \[\leadsto \frac{\sin x \cdot \color{blue}{\left(\left(\frac{1}{6} \cdot {y}^{2}\right) \cdot y + 1 \cdot y\right)}}{x} \]

      3. *-commutative N/A

         \[\leadsto \frac{\sin x \cdot \left(\color{blue}{\left({y}^{2} \cdot \frac{1}{6}\right)} \cdot y + 1 \cdot y\right)}{x} \]

      4. associate-*l* N/A

         \[\leadsto \frac{\sin x \cdot \left(\color{blue}{{y}^{2} \cdot \left(\frac{1}{6} \cdot y\right)} + 1 \cdot y\right)}{x} \]

      5. *-lft-identity N/A

         \[\leadsto \frac{\sin x \cdot \left({y}^{2} \cdot \left(\frac{1}{6} \cdot y\right) + \color{blue}{y}\right)}{x} \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\sin x \cdot \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{1}{6} \cdot y, y\right)}}{x} \]

      7. unpow2 N/A

         \[\leadsto \frac{\sin x \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{6} \cdot y, y\right)}{x} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{\sin x \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{6} \cdot y, y\right)}{x} \]

      9. *-commutative N/A

         \[\leadsto \frac{\sin x \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{y \cdot \frac{1}{6}}, y\right)}{x} \]

      10. *-lowering-*.f64 75.0

          \[\leadsto \frac{\sin x \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{y \cdot 0.16666666666666666}, y\right)}{x} \]

   5. Simplified 75.0%

      \[\leadsto \frac{\sin x \cdot \color{blue}{\mathsf{fma}\left(y \cdot y, y \cdot 0.16666666666666666, y\right)}}{x} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{y + \frac{1}{6} \cdot {y}^{3}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{6} \cdot {y}^{3} + y} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{6}, {y}^{3}, y\right)} \]

      3. cube-mult N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{6}, \color{blue}{y \cdot \left(y \cdot y\right)}, y\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{6}, y \cdot \color{blue}{{y}^{2}}, y\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{6}, \color{blue}{y \cdot {y}^{2}}, y\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{6}, y \cdot \color{blue}{\left(y \cdot y\right)}, y\right) \]

      7. *-lowering-*.f64 41.5

         \[\leadsto \mathsf{fma}\left(0.16666666666666666, y \cdot \color{blue}{\left(y \cdot y\right)}, y\right) \]

   8. Simplified 41.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.16666666666666666, y \cdot \left(y \cdot y\right), y\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 47.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -1 \cdot 10^{-254}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(y \cdot \left(x \cdot x\right)\right)\\ \mathbf{elif}\;\frac{\sinh y \cdot \sin x}{x} \leq 0:\\ \;\;\;\;x \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.16666666666666666, y \cdot \left(y \cdot y\right), y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 59.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\sinh y\_m \cdot \sin x}{x} \leq -1 \cdot 10^{-254}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(y\_m \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y\_m}{x}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m)
 :precision binary64
 (*
  y_s
  (if (<= (/ (* (sinh y_m) (sin x)) x) -1e-254)
    (* -0.16666666666666666 (* y_m (* x x)))
    (* x (/ y_m x)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m) {
	double tmp;
	if (((sinh(y_m) * sin(x)) / x) <= -1e-254) {
		tmp = -0.16666666666666666 * (y_m * (x * x));
	} else {
		tmp = x * (y_m / x);
	}
	return y_s * tmp;
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8) :: tmp
    if (((sinh(y_m) * sin(x)) / x) <= (-1d-254)) then
        tmp = (-0.16666666666666666d0) * (y_m * (x * x))
    else
        tmp = x * (y_m / x)
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m) {
	double tmp;
	if (((Math.sinh(y_m) * Math.sin(x)) / x) <= -1e-254) {
		tmp = -0.16666666666666666 * (y_m * (x * x));
	} else {
		tmp = x * (y_m / x);
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m):
	tmp = 0
	if ((math.sinh(y_m) * math.sin(x)) / x) <= -1e-254:
		tmp = -0.16666666666666666 * (y_m * (x * x))
	else:
		tmp = x * (y_m / x)
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m)
	tmp = 0.0
	if (Float64(Float64(sinh(y_m) * sin(x)) / x) <= -1e-254)
		tmp = Float64(-0.16666666666666666 * Float64(y_m * Float64(x * x)));
	else
		tmp = Float64(x * Float64(y_m / x));
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m)
	tmp = 0.0;
	if (((sinh(y_m) * sin(x)) / x) <= -1e-254)
		tmp = -0.16666666666666666 * (y_m * (x * x));
	else
		tmp = x * (y_m / x);
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_] := N[(y$95$s * If[LessEqual[N[(N[(N[Sinh[y$95$m], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], -1e-254], N[(-0.16666666666666666 * N[(y$95$m * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(y$95$m / x), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -9.9999999999999991e-255

   1. Initial program 99.6%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y \cdot \sin x}{x}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{y \cdot \frac{\sin x}{x}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{y \cdot \frac{\sin x}{x}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto y \cdot \color{blue}{\frac{\sin x}{x}} \]

      4. sin-lowering-sin.f64 37.4

         \[\leadsto y \cdot \frac{\color{blue}{\sin x}}{x} \]

   5. Simplified 37.4%

      \[\leadsto \color{blue}{y \cdot \frac{\sin x}{x}} \]

   6. Taylor expanded in x around 0

      \[\leadsto y \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {x}^{2}\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto y \cdot \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2} + 1\right)} \]

      2. *-commutative N/A

         \[\leadsto y \cdot \left(\color{blue}{{x}^{2} \cdot \frac{-1}{6}} + 1\right) \]

      3. unpow2 N/A

         \[\leadsto y \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{-1}{6} + 1\right) \]

      4. associate-*l* N/A

         \[\leadsto y \cdot \left(\color{blue}{x \cdot \left(x \cdot \frac{-1}{6}\right)} + 1\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto y \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)} \]

      6. *-lowering-*.f64 39.6

         \[\leadsto y \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot -0.16666666666666666}, 1\right) \]

   8. Simplified 39.6%

      \[\leadsto y \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right)} \]

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{-1}{6} \cdot \left({x}^{2} \cdot y\right)} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\frac{-1}{6} \cdot \left({x}^{2} \cdot y\right)} \]

       2. *-commutative N/A

          \[\leadsto \frac{-1}{6} \cdot \color{blue}{\left(y \cdot {x}^{2}\right)} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \frac{-1}{6} \cdot \color{blue}{\left(y \cdot {x}^{2}\right)} \]

       4. unpow2 N/A

          \[\leadsto \frac{-1}{6} \cdot \left(y \cdot \color{blue}{\left(x \cdot x\right)}\right) \]

       5. *-lowering-*.f64 16.6

          \[\leadsto -0.16666666666666666 \cdot \left(y \cdot \color{blue}{\left(x \cdot x\right)}\right) \]

   11. Simplified 16.6%

       \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(y \cdot \left(x \cdot x\right)\right)} \]

   if -9.9999999999999991e-255 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 80.1%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\sinh y}{x} \cdot \sin x} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sinh y}{x} \cdot \sin x} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sinh y}{x}} \cdot \sin x \]

      5. sinh-lowering-sinh.f64 N/A

         \[\leadsto \frac{\color{blue}{\sinh y}}{x} \cdot \sin x \]

      6. sin-lowering-sin.f64 99.9

         \[\leadsto \frac{\sinh y}{x} \cdot \color{blue}{\sin x} \]

   4. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\frac{\sinh y}{x} \cdot \sin x} \]

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y}{x}} \cdot \sin x \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 72.3

         \[\leadsto \color{blue}{\frac{y}{x}} \cdot \sin x \]

   7. Simplified 72.3%

      \[\leadsto \color{blue}{\frac{y}{x}} \cdot \sin x \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{y}{x} \cdot \color{blue}{x} \]
   9. Step-by-step derivation

   10. Simplified 54.6%

       \[\leadsto \frac{y}{x} \cdot \color{blue}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -1 \cdot 10^{-254}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(y \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 37.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\sinh y\_m \cdot \sin x}{x} \leq -1 \cdot 10^{-254}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(y\_m \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y\_m\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m)
 :precision binary64
 (*
  y_s
  (if (<= (/ (* (sinh y_m) (sin x)) x) -1e-254)
    (* -0.16666666666666666 (* y_m (* x x)))
    y_m)))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m) {
	double tmp;
	if (((sinh(y_m) * sin(x)) / x) <= -1e-254) {
		tmp = -0.16666666666666666 * (y_m * (x * x));
	} else {
		tmp = y_m;
	}
	return y_s * tmp;
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8) :: tmp
    if (((sinh(y_m) * sin(x)) / x) <= (-1d-254)) then
        tmp = (-0.16666666666666666d0) * (y_m * (x * x))
    else
        tmp = y_m
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m) {
	double tmp;
	if (((Math.sinh(y_m) * Math.sin(x)) / x) <= -1e-254) {
		tmp = -0.16666666666666666 * (y_m * (x * x));
	} else {
		tmp = y_m;
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m):
	tmp = 0
	if ((math.sinh(y_m) * math.sin(x)) / x) <= -1e-254:
		tmp = -0.16666666666666666 * (y_m * (x * x))
	else:
		tmp = y_m
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m)
	tmp = 0.0
	if (Float64(Float64(sinh(y_m) * sin(x)) / x) <= -1e-254)
		tmp = Float64(-0.16666666666666666 * Float64(y_m * Float64(x * x)));
	else
		tmp = y_m;
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m)
	tmp = 0.0;
	if (((sinh(y_m) * sin(x)) / x) <= -1e-254)
		tmp = -0.16666666666666666 * (y_m * (x * x));
	else
		tmp = y_m;
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_] := N[(y$95$s * If[LessEqual[N[(N[(N[Sinh[y$95$m], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], -1e-254], N[(-0.16666666666666666 * N[(y$95$m * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], y$95$m]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -9.9999999999999991e-255

   1. Initial program 99.6%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y \cdot \sin x}{x}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{y \cdot \frac{\sin x}{x}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{y \cdot \frac{\sin x}{x}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto y \cdot \color{blue}{\frac{\sin x}{x}} \]

      4. sin-lowering-sin.f64 37.4

         \[\leadsto y \cdot \frac{\color{blue}{\sin x}}{x} \]

   5. Simplified 37.4%

      \[\leadsto \color{blue}{y \cdot \frac{\sin x}{x}} \]

   6. Taylor expanded in x around 0

      \[\leadsto y \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {x}^{2}\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto y \cdot \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2} + 1\right)} \]

      2. *-commutative N/A

         \[\leadsto y \cdot \left(\color{blue}{{x}^{2} \cdot \frac{-1}{6}} + 1\right) \]

      3. unpow2 N/A

         \[\leadsto y \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{-1}{6} + 1\right) \]

      4. associate-*l* N/A

         \[\leadsto y \cdot \left(\color{blue}{x \cdot \left(x \cdot \frac{-1}{6}\right)} + 1\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto y \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)} \]

      6. *-lowering-*.f64 39.6

         \[\leadsto y \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot -0.16666666666666666}, 1\right) \]

   8. Simplified 39.6%

      \[\leadsto y \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right)} \]

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{-1}{6} \cdot \left({x}^{2} \cdot y\right)} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\frac{-1}{6} \cdot \left({x}^{2} \cdot y\right)} \]

       2. *-commutative N/A

          \[\leadsto \frac{-1}{6} \cdot \color{blue}{\left(y \cdot {x}^{2}\right)} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \frac{-1}{6} \cdot \color{blue}{\left(y \cdot {x}^{2}\right)} \]

       4. unpow2 N/A

          \[\leadsto \frac{-1}{6} \cdot \left(y \cdot \color{blue}{\left(x \cdot x\right)}\right) \]

       5. *-lowering-*.f64 16.6

          \[\leadsto -0.16666666666666666 \cdot \left(y \cdot \color{blue}{\left(x \cdot x\right)}\right) \]

   11. Simplified 16.6%

       \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(y \cdot \left(x \cdot x\right)\right)} \]

   if -9.9999999999999991e-255 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 80.1%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y \cdot \sin x}{x}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{y \cdot \frac{\sin x}{x}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{y \cdot \frac{\sin x}{x}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto y \cdot \color{blue}{\frac{\sin x}{x}} \]

      4. sin-lowering-sin.f64 65.8

         \[\leadsto y \cdot \frac{\color{blue}{\sin x}}{x} \]

   5. Simplified 65.8%

      \[\leadsto \color{blue}{y \cdot \frac{\sin x}{x}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{y} \]
   7. Step-by-step derivation

   8. Simplified 32.2%

      \[\leadsto \color{blue}{y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 27.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -1 \cdot 10^{-254}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(y \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \left(\frac{\sinh y\_m}{x} \cdot \sin x\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m) :precision binary64 (* y_s (* (/ (sinh y_m) x) (sin x))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m) {
	return y_s * ((sinh(y_m) / x) * sin(x));
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    code = y_s * ((sinh(y_m) / x) * sin(x))
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m) {
	return y_s * ((Math.sinh(y_m) / x) * Math.sin(x));
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m):
	return y_s * ((math.sinh(y_m) / x) * math.sin(x))

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m)
	return Float64(y_s * Float64(Float64(sinh(y_m) / x) * sin(x)))
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp = code(y_s, x, y_m)
	tmp = y_s * ((sinh(y_m) / x) * sin(x));
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_] := N[(y$95$s * N[(N[(N[Sinh[y$95$m], $MachinePrecision] / x), $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 86.3%

   \[\frac{\sin x \cdot \sinh y}{x} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. associate-/l* N/A

      \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\frac{\sinh y}{x} \cdot \sin x} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{\frac{\sinh y}{x} \cdot \sin x} \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\sinh y}{x}} \cdot \sin x \]

   5. sinh-lowering-sinh.f64 N/A

      \[\leadsto \frac{\color{blue}{\sinh y}}{x} \cdot \sin x \]

   6. sin-lowering-sin.f64 99.5

      \[\leadsto \frac{\sinh y}{x} \cdot \color{blue}{\sin x} \]

4. Applied egg-rr 99.5%

   \[\leadsto \color{blue}{\frac{\sinh y}{x} \cdot \sin x} \]
5. Add Preprocessing

Alternative 14: 36.0% accurate, 12.8× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \left(y\_m \cdot \mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right)\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m)
 :precision binary64
 (* y_s (* y_m (fma x (* x -0.16666666666666666) 1.0))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m) {
	return y_s * (y_m * fma(x, (x * -0.16666666666666666), 1.0));
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m)
	return Float64(y_s * Float64(y_m * fma(x, Float64(x * -0.16666666666666666), 1.0)))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_] := N[(y$95$s * N[(y$95$m * N[(x * N[(x * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 86.3%

   \[\frac{\sin x \cdot \sinh y}{x} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\frac{y \cdot \sin x}{x}} \]
4. Step-by-step derivation

   1. associate-/l* N/A

      \[\leadsto \color{blue}{y \cdot \frac{\sin x}{x}} \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{y \cdot \frac{\sin x}{x}} \]

   3. /-lowering-/.f64 N/A

      \[\leadsto y \cdot \color{blue}{\frac{\sin x}{x}} \]

   4. sin-lowering-sin.f64 56.7

      \[\leadsto y \cdot \frac{\color{blue}{\sin x}}{x} \]

5. Simplified 56.7%

   \[\leadsto \color{blue}{y \cdot \frac{\sin x}{x}} \]

6. Taylor expanded in x around 0

   \[\leadsto y \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {x}^{2}\right)} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto y \cdot \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2} + 1\right)} \]

   2. *-commutative N/A

      \[\leadsto y \cdot \left(\color{blue}{{x}^{2} \cdot \frac{-1}{6}} + 1\right) \]

   3. unpow2 N/A

      \[\leadsto y \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{-1}{6} + 1\right) \]

   4. associate-*l* N/A

      \[\leadsto y \cdot \left(\color{blue}{x \cdot \left(x \cdot \frac{-1}{6}\right)} + 1\right) \]

   5. accelerator-lowering-fma.f64 N/A

      \[\leadsto y \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{-1}{6}, 1\right)} \]

   6. *-lowering-*.f64 39.5

      \[\leadsto y \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot -0.16666666666666666}, 1\right) \]

8. Simplified 39.5%

   \[\leadsto y \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right)} \]
9. Add Preprocessing

Alternative 15: 27.6% accurate, 217.0× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot y\_m \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m) :precision binary64 (* y_s y_m))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m) {
	return y_s * y_m;
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    code = y_s * y_m
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m) {
	return y_s * y_m;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m):
	return y_s * y_m

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m)
	return Float64(y_s * y_m)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp = code(y_s, x, y_m)
	tmp = y_s * y_m;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_] := N[(y$95$s * y$95$m), $MachinePrecision]

Derivation

1. Initial program 86.3%

   \[\frac{\sin x \cdot \sinh y}{x} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\frac{y \cdot \sin x}{x}} \]
4. Step-by-step derivation

   1. associate-/l* N/A

      \[\leadsto \color{blue}{y \cdot \frac{\sin x}{x}} \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{y \cdot \frac{\sin x}{x}} \]

   3. /-lowering-/.f64 N/A

      \[\leadsto y \cdot \color{blue}{\frac{\sin x}{x}} \]

   4. sin-lowering-sin.f64 56.7

      \[\leadsto y \cdot \frac{\color{blue}{\sin x}}{x} \]

5. Simplified 56.7%

   \[\leadsto \color{blue}{y \cdot \frac{\sin x}{x}} \]

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{y} \]
7. Step-by-step derivation

8. Simplified 29.9%

   \[\leadsto \color{blue}{y} \]
9. Add Preprocessing

Developer Target 1: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sin x \cdot \frac{\sinh y}{x} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* (sin x) (/ (sinh y) x)))

C

double code(double x, double y) {
	return sin(x) * (sinh(y) / x);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = sin(x) * (sinh(y) / x)
end function

Java

public static double code(double x, double y) {
	return Math.sin(x) * (Math.sinh(y) / x);
}

Python

def code(x, y):
	return math.sin(x) * (math.sinh(y) / x)

Julia

function code(x, y)
	return Float64(sin(x) * Float64(sinh(y) / x))
end

MATLAB

function tmp = code(x, y)
	tmp = sin(x) * (sinh(y) / x);
end

Wolfram

code[x_, y_] := N[(N[Sin[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024198 
(FPCore (x y)
  :name "Linear.Quaternion:$ccosh from linear-1.19.1.3"
  :precision binary64

  :alt
  (! :herbie-platform default (* (sin x) (/ (sinh y) x)))

  (/ (* (sin x) (sinh y)) x))