Linear.Quaternion:$ccosh from linear-1.19.1.3

Percentage Accurate: 88.8% → 99.9%

Time: 13.7s

Alternatives: 24

Speedup: 0.7×

• 48.9% 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: 88.8% 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: 99.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;\sinh y\_m \leq 2 \cdot 10^{-27}:\\ \;\;\;\;y\_m \cdot \frac{\sin x}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sinh y\_m \cdot \sin x}{x}\\ \end{array} \end{array} \]

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) 2e-27)
    (* y_m (/ (sin x) x))
    (/ (* (sinh y_m) (sin x)) 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) <= 2e-27) {
		tmp = y_m * (sin(x) / x);
	} else {
		tmp = (sinh(y_m) * sin(x)) / 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) <= 2d-27) then
        tmp = y_m * (sin(x) / x)
    else
        tmp = (sinh(y_m) * sin(x)) / 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) <= 2e-27) {
		tmp = y_m * (Math.sin(x) / x);
	} else {
		tmp = (Math.sinh(y_m) * Math.sin(x)) / 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) <= 2e-27:
		tmp = y_m * (math.sin(x) / x)
	else:
		tmp = (math.sinh(y_m) * math.sin(x)) / 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 (sinh(y_m) <= 2e-27)
		tmp = Float64(y_m * Float64(sin(x) / x));
	else
		tmp = Float64(Float64(sinh(y_m) * sin(x)) / 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) <= 2e-27)
		tmp = y_m * (sin(x) / x);
	else
		tmp = (sinh(y_m) * sin(x)) / 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[Sinh[y$95$m], $MachinePrecision], 2e-27], N[(y$95$m * N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sinh[y$95$m], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (sinh.f64 y) < 2.0000000000000001e-27

   1. Initial program 86.8%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sin.f64 67.8

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

   5. Simplified 67.8%

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

   if 2.0000000000000001e-27 < (sinh.f64 y)

   1. Initial program 100.0%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sinh y \leq 2 \cdot 10^{-27}:\\ \;\;\;\;y \cdot \frac{\sin x}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sinh y \cdot \sin x}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 55.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{\sinh y\_m \cdot \sin x}{x}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-117}:\\ \;\;\;\;y\_m \cdot \left(\left(x \cdot x\right) \cdot -0.16666666666666666\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-276}:\\ \;\;\;\;0.5 \cdot \left(1 + -1\right)\\ \mathbf{elif}\;t\_0 \leq 1000:\\ \;\;\;\;y\_m\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \mathsf{fma}\left(y\_m, 0.25, 0.5\right)\\ \end{array} \end{array} \end{array} \]

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-117)
      (* y_m (* (* x x) -0.16666666666666666))
      (if (<= t_0 5e-276)
        (* 0.5 (+ 1.0 -1.0))
        (if (<= t_0 1000.0) y_m (* y_m (fma y_m 0.25 0.5))))))))

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-117) {
		tmp = y_m * ((x * x) * -0.16666666666666666);
	} else if (t_0 <= 5e-276) {
		tmp = 0.5 * (1.0 + -1.0);
	} else if (t_0 <= 1000.0) {
		tmp = y_m;
	} else {
		tmp = y_m * fma(y_m, 0.25, 0.5);
	}
	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-117)
		tmp = Float64(y_m * Float64(Float64(x * x) * -0.16666666666666666));
	elseif (t_0 <= 5e-276)
		tmp = Float64(0.5 * Float64(1.0 + -1.0));
	elseif (t_0 <= 1000.0)
		tmp = y_m;
	else
		tmp = Float64(y_m * fma(y_m, 0.25, 0.5));
	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-117], N[(y$95$m * N[(N[(x * x), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e-276], N[(0.5 * N[(1.0 + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1000.0], y$95$m, N[(y$95$m * N[(y$95$m * 0.25 + 0.5), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -1.00000000000000003e-117

   1. Initial program 99.5%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sin.f64 22.0

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

   5. Simplified 22.0%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 28.2

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

   8. Simplified 28.2%

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

   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. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 15.6

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

   11. Simplified 15.6%

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

   if -1.00000000000000003e-117 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 4.99999999999999967e-276

   1. Initial program 71.2%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. rec-exp N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \]

      5. lower-exp.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \]

      6. lower-neg.f64 43.9

         \[\leadsto 0.5 \cdot \left(e^{y} - e^{\color{blue}{-y}}\right) \]

   5. Simplified 43.9%

      \[\leadsto \color{blue}{0.5 \cdot \left(e^{y} - e^{-y}\right)} \]

   6. Taylor expanded in y around 0

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

   8. Simplified 43.9%

      \[\leadsto 0.5 \cdot \left(e^{y} - \color{blue}{1}\right) \]

   9. Taylor expanded in y around 0

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

   11. Simplified 43.9%

       \[\leadsto 0.5 \cdot \left(\color{blue}{1} - 1\right) \]

   if 4.99999999999999967e-276 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 1e3

   1. Initial program 99.8%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sin.f64 99.9

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

   5. Simplified 99.9%

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

   6. Taylor expanded in x around 0

      \[\leadsto y \cdot \color{blue}{1} \]
   7. Step-by-step derivation

   8. Simplified 62.7%

      \[\leadsto y \cdot \color{blue}{1} \]

   if 1e3 < (/.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

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. rec-exp N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \]

      5. lower-exp.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \]

      6. lower-neg.f64 73.0

         \[\leadsto 0.5 \cdot \left(e^{y} - e^{\color{blue}{-y}}\right) \]

   5. Simplified 73.0%

      \[\leadsto \color{blue}{0.5 \cdot \left(e^{y} - e^{-y}\right)} \]

   6. Taylor expanded in y around 0

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

   8. Simplified 72.3%

      \[\leadsto 0.5 \cdot \left(e^{y} - \color{blue}{1}\right) \]

   9. Taylor expanded in y around 0

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

       1. lower-*.f64 N/A

          \[\leadsto \color{blue}{y \cdot \left(\frac{1}{2} + \frac{1}{4} \cdot y\right)} \]

       2. +-commutative N/A

          \[\leadsto y \cdot \color{blue}{\left(\frac{1}{4} \cdot y + \frac{1}{2}\right)} \]

       3. *-commutative N/A

          \[\leadsto y \cdot \left(\color{blue}{y \cdot \frac{1}{4}} + \frac{1}{2}\right) \]

       4. lower-fma.f64 51.8

          \[\leadsto y \cdot \color{blue}{\mathsf{fma}\left(y, 0.25, 0.5\right)} \]

   11. Simplified 51.8%

       \[\leadsto \color{blue}{y \cdot \mathsf{fma}\left(y, 0.25, 0.5\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 39.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -1 \cdot 10^{-117}:\\ \;\;\;\;y \cdot \left(\left(x \cdot x\right) \cdot -0.16666666666666666\right)\\ \mathbf{elif}\;\frac{\sinh y \cdot \sin x}{x} \leq 5 \cdot 10^{-276}:\\ \;\;\;\;0.5 \cdot \left(1 + -1\right)\\ \mathbf{elif}\;\frac{\sinh y \cdot \sin x}{x} \leq 1000:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(y, 0.25, 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{\sinh y\_m \cdot \sin x}{x}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\frac{\sinh y\_m \cdot \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot -0.16666666666666666, x\right)}{x}\\ \mathbf{elif}\;t\_0 \leq 1000:\\ \;\;\;\;\left(\sin x \cdot \mathsf{fma}\left(y\_m, y\_m \cdot \mathsf{fma}\left(y\_m, y\_m \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)\right) \cdot \frac{y\_m}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{y\_m} - e^{-y\_m}\right)\\ \end{array} \end{array} \end{array} \]

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 (- INFINITY))
      (/ (* (sinh y_m) (fma x (* (* x x) -0.16666666666666666) x)) x)
      (if (<= t_0 1000.0)
        (*
         (*
          (sin x)
          (fma
           y_m
           (* y_m (fma y_m (* y_m 0.008333333333333333) 0.16666666666666666))
           1.0))
         (/ y_m x))
        (* 0.5 (- (exp y_m) (exp (- 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 <= -((double) INFINITY)) {
		tmp = (sinh(y_m) * fma(x, ((x * x) * -0.16666666666666666), x)) / x;
	} else if (t_0 <= 1000.0) {
		tmp = (sin(x) * fma(y_m, (y_m * fma(y_m, (y_m * 0.008333333333333333), 0.16666666666666666)), 1.0)) * (y_m / x);
	} else {
		tmp = 0.5 * (exp(y_m) - exp(-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 <= Float64(-Inf))
		tmp = Float64(Float64(sinh(y_m) * fma(x, Float64(Float64(x * x) * -0.16666666666666666), x)) / x);
	elseif (t_0 <= 1000.0)
		tmp = Float64(Float64(sin(x) * fma(y_m, Float64(y_m * fma(y_m, Float64(y_m * 0.008333333333333333), 0.16666666666666666)), 1.0)) * Float64(y_m / x));
	else
		tmp = Float64(0.5 * Float64(exp(y_m) - exp(Float64(-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, (-Infinity)], N[(N[(N[Sinh[y$95$m], $MachinePrecision] * N[(x * N[(N[(x * x), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[t$95$0, 1000.0], N[(N[(N[Sin[x], $MachinePrecision] * N[(y$95$m * N[(y$95$m * N[(y$95$m * N[(y$95$m * 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[(y$95$m / x), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Exp[y$95$m], $MachinePrecision] - N[Exp[(-y$95$m)], $MachinePrecision]), $MachinePrecision]), $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 x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 72.1

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

   5. Simplified 72.1%

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

   if -inf.0 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 1e3

   1. Initial program 80.4%

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

   3. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. distribute-lft-in N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. distribute-lft-in N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

   5. Simplified 79.7%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

   8. Simplified 99.1%

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

   if 1e3 < (/.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

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. rec-exp N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \]

      5. lower-exp.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \]

      6. lower-neg.f64 73.0

         \[\leadsto 0.5 \cdot \left(e^{y} - e^{\color{blue}{-y}}\right) \]

   5. Simplified 73.0%

      \[\leadsto \color{blue}{0.5 \cdot \left(e^{y} - e^{-y}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -\infty:\\ \;\;\;\;\frac{\sinh y \cdot \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot -0.16666666666666666, x\right)}{x}\\ \mathbf{elif}\;\frac{\sinh y \cdot \sin x}{x} \leq 1000:\\ \;\;\;\;\left(\sin x \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)\right) \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{y} - e^{-y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{\sinh y\_m \cdot \sin x}{x}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\frac{\sinh y\_m \cdot \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot -0.16666666666666666, x\right)}{x}\\ \mathbf{elif}\;t\_0 \leq 1000:\\ \;\;\;\;\left(\sin x \cdot \mathsf{fma}\left(y\_m, y\_m \cdot \mathsf{fma}\left(y\_m, y\_m \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)\right) \cdot \frac{y\_m}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sinh y\_m \cdot \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.008333333333333333, -0.16666666666666666\right), x \cdot \left(x \cdot x\right), x\right)}{x}\\ \end{array} \end{array} \end{array} \]

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 (/ (* (sinh y_m) (sin x)) x)))
   (*
    y_s
    (if (<= t_0 (- INFINITY))
      (/ (* (sinh y_m) (fma x (* (* x x) -0.16666666666666666) x)) x)
      (if (<= t_0 1000.0)
        (*
         (*
          (sin x)
          (fma
           y_m
           (* y_m (fma y_m (* y_m 0.008333333333333333) 0.16666666666666666))
           1.0))
         (/ y_m x))
        (/
         (*
          (sinh y_m)
          (fma
           (fma (* x x) 0.008333333333333333 -0.16666666666666666)
           (* x (* x x))
           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 = (sinh(y_m) * fma(x, ((x * x) * -0.16666666666666666), x)) / x;
	} else if (t_0 <= 1000.0) {
		tmp = (sin(x) * fma(y_m, (y_m * fma(y_m, (y_m * 0.008333333333333333), 0.16666666666666666)), 1.0)) * (y_m / x);
	} else {
		tmp = (sinh(y_m) * fma(fma((x * x), 0.008333333333333333, -0.16666666666666666), (x * (x * x)), 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(Float64(sinh(y_m) * fma(x, Float64(Float64(x * x) * -0.16666666666666666), x)) / x);
	elseif (t_0 <= 1000.0)
		tmp = Float64(Float64(sin(x) * fma(y_m, Float64(y_m * fma(y_m, Float64(y_m * 0.008333333333333333), 0.16666666666666666)), 1.0)) * Float64(y_m / x));
	else
		tmp = Float64(Float64(sinh(y_m) * fma(fma(Float64(x * x), 0.008333333333333333, -0.16666666666666666), Float64(x * Float64(x * x)), 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[(N[Sinh[y$95$m], $MachinePrecision] * N[(x * N[(N[(x * x), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[t$95$0, 1000.0], N[(N[(N[Sin[x], $MachinePrecision] * N[(y$95$m * N[(y$95$m * N[(y$95$m * N[(y$95$m * 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[(y$95$m / x), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sinh[y$95$m], $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision] * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $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 x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 72.1

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

   5. Simplified 72.1%

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

   if -inf.0 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 1e3

   1. Initial program 80.4%

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

   3. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. distribute-lft-in N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. distribute-lft-in N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

   5. Simplified 79.7%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

   8. Simplified 99.1%

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

   if 1e3 < (/.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

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. sub-neg N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 73.0

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

   5. Simplified 73.0%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -\infty:\\ \;\;\;\;\frac{\sinh y \cdot \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot -0.16666666666666666, x\right)}{x}\\ \mathbf{elif}\;\frac{\sinh y \cdot \sin x}{x} \leq 1000:\\ \;\;\;\;\left(\sin x \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)\right) \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sinh y \cdot \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.008333333333333333, -0.16666666666666666\right), x \cdot \left(x \cdot x\right), x\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{\sinh y\_m \cdot \sin x}{x}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\frac{\sinh y\_m \cdot \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot -0.16666666666666666, x\right)}{x}\\ \mathbf{elif}\;t\_0 \leq 1000:\\ \;\;\;\;\left(\sin x \cdot \mathsf{fma}\left(y\_m, y\_m \cdot \mathsf{fma}\left(y\_m, y\_m \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)\right) \cdot \frac{y\_m}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{expm1}\left(y\_m\right)\\ \end{array} \end{array} \end{array} \]

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 (- INFINITY))
      (/ (* (sinh y_m) (fma x (* (* x x) -0.16666666666666666) x)) x)
      (if (<= t_0 1000.0)
        (*
         (*
          (sin x)
          (fma
           y_m
           (* y_m (fma y_m (* y_m 0.008333333333333333) 0.16666666666666666))
           1.0))
         (/ y_m x))
        (* 0.5 (expm1 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 <= -((double) INFINITY)) {
		tmp = (sinh(y_m) * fma(x, ((x * x) * -0.16666666666666666), x)) / x;
	} else if (t_0 <= 1000.0) {
		tmp = (sin(x) * fma(y_m, (y_m * fma(y_m, (y_m * 0.008333333333333333), 0.16666666666666666)), 1.0)) * (y_m / x);
	} else {
		tmp = 0.5 * expm1(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 <= Float64(-Inf))
		tmp = Float64(Float64(sinh(y_m) * fma(x, Float64(Float64(x * x) * -0.16666666666666666), x)) / x);
	elseif (t_0 <= 1000.0)
		tmp = Float64(Float64(sin(x) * fma(y_m, Float64(y_m * fma(y_m, Float64(y_m * 0.008333333333333333), 0.16666666666666666)), 1.0)) * Float64(y_m / x));
	else
		tmp = Float64(0.5 * expm1(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, (-Infinity)], N[(N[(N[Sinh[y$95$m], $MachinePrecision] * N[(x * N[(N[(x * x), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[t$95$0, 1000.0], N[(N[(N[Sin[x], $MachinePrecision] * N[(y$95$m * N[(y$95$m * N[(y$95$m * N[(y$95$m * 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[(y$95$m / x), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(Exp[y$95$m] - 1), $MachinePrecision]), $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 x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 72.1

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

   5. Simplified 72.1%

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

   if -inf.0 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 1e3

   1. Initial program 80.4%

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

   3. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. distribute-lft-in N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. distribute-lft-in N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

   5. Simplified 79.7%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

   8. Simplified 99.1%

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

   if 1e3 < (/.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

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. rec-exp N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \]

      5. lower-exp.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \]

      6. lower-neg.f64 73.0

         \[\leadsto 0.5 \cdot \left(e^{y} - e^{\color{blue}{-y}}\right) \]

   5. Simplified 73.0%

      \[\leadsto \color{blue}{0.5 \cdot \left(e^{y} - e^{-y}\right)} \]

   6. Taylor expanded in y around 0

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

   8. Simplified 72.3%

      \[\leadsto 0.5 \cdot \left(e^{y} - \color{blue}{1}\right) \]

   9. Taylor expanded in y around inf

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

       1. lower-expm1.f64 72.3

          \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(y\right)} \]

   11. Simplified 72.3%

       \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(y\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -\infty:\\ \;\;\;\;\frac{\sinh y \cdot \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot -0.16666666666666666, x\right)}{x}\\ \mathbf{elif}\;\frac{\sinh y \cdot \sin x}{x} \leq 1000:\\ \;\;\;\;\left(\sin x \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)\right) \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{expm1}\left(y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{\sinh y\_m \cdot \sin x}{x}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\frac{\sinh y\_m \cdot \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot -0.16666666666666666, x\right)}{x}\\ \mathbf{elif}\;t\_0 \leq 0.0001:\\ \;\;\;\;y\_m \cdot \frac{\sin x}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{expm1}\left(y\_m\right)\\ \end{array} \end{array} \end{array} \]

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 (- INFINITY))
      (/ (* (sinh y_m) (fma x (* (* x x) -0.16666666666666666) x)) x)
      (if (<= t_0 0.0001) (* y_m (/ (sin x) x)) (* 0.5 (expm1 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 <= -((double) INFINITY)) {
		tmp = (sinh(y_m) * fma(x, ((x * x) * -0.16666666666666666), x)) / x;
	} else if (t_0 <= 0.0001) {
		tmp = y_m * (sin(x) / x);
	} else {
		tmp = 0.5 * expm1(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 <= Float64(-Inf))
		tmp = Float64(Float64(sinh(y_m) * fma(x, Float64(Float64(x * x) * -0.16666666666666666), x)) / x);
	elseif (t_0 <= 0.0001)
		tmp = Float64(y_m * Float64(sin(x) / x));
	else
		tmp = Float64(0.5 * expm1(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, (-Infinity)], N[(N[(N[Sinh[y$95$m], $MachinePrecision] * N[(x * N[(N[(x * x), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[t$95$0, 0.0001], N[(y$95$m * N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(Exp[y$95$m] - 1), $MachinePrecision]), $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 x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 72.1

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

   5. Simplified 72.1%

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

   if -inf.0 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 1.00000000000000005e-4

   1. Initial program 80.4%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sin.f64 98.5

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

   5. Simplified 98.5%

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

   if 1.00000000000000005e-4 < (/.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

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. rec-exp N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \]

      5. lower-exp.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \]

      6. lower-neg.f64 73.0

         \[\leadsto 0.5 \cdot \left(e^{y} - e^{\color{blue}{-y}}\right) \]

   5. Simplified 73.0%

      \[\leadsto \color{blue}{0.5 \cdot \left(e^{y} - e^{-y}\right)} \]

   6. Taylor expanded in y around 0

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

   8. Simplified 72.3%

      \[\leadsto 0.5 \cdot \left(e^{y} - \color{blue}{1}\right) \]

   9. Taylor expanded in y around inf

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

       1. lower-expm1.f64 72.3

          \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(y\right)} \]

   11. Simplified 72.3%

       \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(y\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -\infty:\\ \;\;\;\;\frac{\sinh y \cdot \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot -0.16666666666666666, x\right)}{x}\\ \mathbf{elif}\;\frac{\sinh y \cdot \sin x}{x} \leq 0.0001:\\ \;\;\;\;y \cdot \frac{\sin x}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{expm1}\left(y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{\sinh y\_m \cdot \sin x}{x}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), x\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y\_m \cdot y\_m, \mathsf{fma}\left(y\_m \cdot y\_m, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y\_m \cdot \left(y\_m \cdot y\_m\right), y\_m\right)}{x}\\ \mathbf{elif}\;t\_0 \leq 0.0001:\\ \;\;\;\;y\_m \cdot \frac{\sin x}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{expm1}\left(y\_m\right)\\ \end{array} \end{array} \end{array} \]

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 (- INFINITY))
      (/
       (*
        (fma
         (* x x)
         (*
          x
          (fma
           (* x x)
           (fma (* x x) -0.0001984126984126984 0.008333333333333333)
           -0.16666666666666666))
         x)
        (fma
         (fma
          (* y_m y_m)
          (fma (* y_m y_m) 0.0001984126984126984 0.008333333333333333)
          0.16666666666666666)
         (* y_m (* y_m y_m))
         y_m))
       x)
      (if (<= t_0 0.0001) (* y_m (/ (sin x) x)) (* 0.5 (expm1 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 <= -((double) INFINITY)) {
		tmp = (fma((x * x), (x * fma((x * x), fma((x * x), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666)), x) * fma(fma((y_m * y_m), fma((y_m * y_m), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666), (y_m * (y_m * y_m)), y_m)) / x;
	} else if (t_0 <= 0.0001) {
		tmp = y_m * (sin(x) / x);
	} else {
		tmp = 0.5 * expm1(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 <= Float64(-Inf))
		tmp = Float64(Float64(fma(Float64(x * x), Float64(x * fma(Float64(x * x), fma(Float64(x * x), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666)), x) * fma(fma(Float64(y_m * y_m), fma(Float64(y_m * y_m), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666), Float64(y_m * Float64(y_m * y_m)), y_m)) / x);
	elseif (t_0 <= 0.0001)
		tmp = Float64(y_m * Float64(sin(x) / x));
	else
		tmp = Float64(0.5 * expm1(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, (-Infinity)], N[(N[(N[(N[(x * x), $MachinePrecision] * N[(x * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] * N[(N[(N[(y$95$m * y$95$m), $MachinePrecision] * N[(N[(y$95$m * y$95$m), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y$95$m * N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] + y$95$m), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[t$95$0, 0.0001], N[(y$95$m * N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(Exp[y$95$m] - 1), $MachinePrecision]), $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

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 87.4

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

   5. Simplified 87.4%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

   8. Simplified 64.3%

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

   if -inf.0 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 1.00000000000000005e-4

   1. Initial program 80.4%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sin.f64 98.5

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

   5. Simplified 98.5%

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

   if 1.00000000000000005e-4 < (/.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

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. rec-exp N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \]

      5. lower-exp.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \]

      6. lower-neg.f64 73.0

         \[\leadsto 0.5 \cdot \left(e^{y} - e^{\color{blue}{-y}}\right) \]

   5. Simplified 73.0%

      \[\leadsto \color{blue}{0.5 \cdot \left(e^{y} - e^{-y}\right)} \]

   6. Taylor expanded in y around 0

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

   8. Simplified 72.3%

      \[\leadsto 0.5 \cdot \left(e^{y} - \color{blue}{1}\right) \]

   9. Taylor expanded in y around inf

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

       1. lower-expm1.f64 72.3

          \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(y\right)} \]

   11. Simplified 72.3%

       \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(y\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -\infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), x\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)}{x}\\ \mathbf{elif}\;\frac{\sinh y \cdot \sin x}{x} \leq 0.0001:\\ \;\;\;\;y \cdot \frac{\sin x}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{expm1}\left(y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 85.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{\sinh y\_m \cdot \sin x}{x}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-117}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), x\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y\_m \cdot y\_m, \mathsf{fma}\left(y\_m \cdot y\_m, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y\_m \cdot \left(y\_m \cdot y\_m\right), y\_m\right)}{x}\\ \mathbf{elif}\;t\_0 \leq 1000:\\ \;\;\;\;\frac{y\_m}{x} \cdot \mathsf{fma}\left(y\_m \cdot y\_m, x \cdot \mathsf{fma}\left(y\_m \cdot y\_m, 0.008333333333333333, 0.16666666666666666\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{expm1}\left(y\_m\right)\\ \end{array} \end{array} \end{array} \]

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-117)
      (/
       (*
        (fma
         (* x x)
         (*
          x
          (fma
           (* x x)
           (fma (* x x) -0.0001984126984126984 0.008333333333333333)
           -0.16666666666666666))
         x)
        (fma
         (fma
          (* y_m y_m)
          (fma (* y_m y_m) 0.0001984126984126984 0.008333333333333333)
          0.16666666666666666)
         (* y_m (* y_m y_m))
         y_m))
       x)
      (if (<= t_0 1000.0)
        (*
         (/ y_m x)
         (fma
          (* y_m y_m)
          (* x (fma (* y_m y_m) 0.008333333333333333 0.16666666666666666))
          x))
        (* 0.5 (expm1 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-117) {
		tmp = (fma((x * x), (x * fma((x * x), fma((x * x), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666)), x) * fma(fma((y_m * y_m), fma((y_m * y_m), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666), (y_m * (y_m * y_m)), y_m)) / x;
	} else if (t_0 <= 1000.0) {
		tmp = (y_m / x) * fma((y_m * y_m), (x * fma((y_m * y_m), 0.008333333333333333, 0.16666666666666666)), x);
	} else {
		tmp = 0.5 * expm1(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-117)
		tmp = Float64(Float64(fma(Float64(x * x), Float64(x * fma(Float64(x * x), fma(Float64(x * x), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666)), x) * fma(fma(Float64(y_m * y_m), fma(Float64(y_m * y_m), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666), Float64(y_m * Float64(y_m * y_m)), y_m)) / x);
	elseif (t_0 <= 1000.0)
		tmp = Float64(Float64(y_m / x) * fma(Float64(y_m * y_m), Float64(x * fma(Float64(y_m * y_m), 0.008333333333333333, 0.16666666666666666)), x));
	else
		tmp = Float64(0.5 * expm1(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-117], N[(N[(N[(N[(x * x), $MachinePrecision] * N[(x * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] * N[(N[(N[(y$95$m * y$95$m), $MachinePrecision] * N[(N[(y$95$m * y$95$m), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y$95$m * N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] + y$95$m), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[t$95$0, 1000.0], N[(N[(y$95$m / x), $MachinePrecision] * N[(N[(y$95$m * y$95$m), $MachinePrecision] * N[(x * N[(N[(y$95$m * y$95$m), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(Exp[y$95$m] - 1), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -1.00000000000000003e-117

   1. Initial program 99.5%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 88.3

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

   5. Simplified 88.3%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

   8. Simplified 62.3%

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

   if -1.00000000000000003e-117 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 1e3

   1. Initial program 78.1%

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

   3. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. distribute-lft-in N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. distribute-lft-in N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

   5. Simplified 78.1%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

   8. Simplified 99.8%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. distribute-rgt-in N/A

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

       3. associate-*l* N/A

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

       4. *-lft-identity N/A

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

       5. lower-fma.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 N/A

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

       8. lower-*.f64 N/A

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

       9. +-commutative N/A

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

       10. *-commutative N/A

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

       11. lower-fma.f64 N/A

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

       12. unpow2 N/A

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

       13. lower-*.f64 71.5

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

   11. Simplified 71.5%

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

   if 1e3 < (/.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

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. rec-exp N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \]

      5. lower-exp.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \]

      6. lower-neg.f64 73.0

         \[\leadsto 0.5 \cdot \left(e^{y} - e^{\color{blue}{-y}}\right) \]

   5. Simplified 73.0%

      \[\leadsto \color{blue}{0.5 \cdot \left(e^{y} - e^{-y}\right)} \]

   6. Taylor expanded in y around 0

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

   8. Simplified 72.3%

      \[\leadsto 0.5 \cdot \left(e^{y} - \color{blue}{1}\right) \]

   9. Taylor expanded in y around inf

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

       1. lower-expm1.f64 72.3

          \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(y\right)} \]

   11. Simplified 72.3%

       \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(y\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -1 \cdot 10^{-117}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), x\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)}{x}\\ \mathbf{elif}\;\frac{\sinh y \cdot \sin x}{x} \leq 1000:\\ \;\;\;\;\frac{y}{x} \cdot \mathsf{fma}\left(y \cdot y, x \cdot \mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{expm1}\left(y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 70.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{\sinh y\_m \cdot \sin x}{x}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-107}:\\ \;\;\;\;y\_m \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-276}:\\ \;\;\;\;0.5 \cdot \left(1 + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y\_m, \left(y\_m \cdot y\_m\right) \cdot \mathsf{fma}\left(y\_m, y\_m \cdot \mathsf{fma}\left(y\_m, y\_m \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y\_m\right)\\ \end{array} \end{array} \end{array} \]

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 -2e-107)
      (*
       y_m
       (fma
        (* x x)
        (fma
         (* x x)
         (fma (* x x) -0.0001984126984126984 0.008333333333333333)
         -0.16666666666666666)
        1.0))
      (if (<= t_0 5e-276)
        (* 0.5 (+ 1.0 -1.0))
        (fma
         y_m
         (*
          (* y_m y_m)
          (fma
           y_m
           (* y_m (fma y_m (* y_m 0.0001984126984126984) 0.008333333333333333))
           0.16666666666666666))
         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 <= -2e-107) {
		tmp = y_m * fma((x * x), fma((x * x), fma((x * x), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666), 1.0);
	} else if (t_0 <= 5e-276) {
		tmp = 0.5 * (1.0 + -1.0);
	} else {
		tmp = fma(y_m, ((y_m * y_m) * fma(y_m, (y_m * fma(y_m, (y_m * 0.0001984126984126984), 0.008333333333333333)), 0.16666666666666666)), 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 <= -2e-107)
		tmp = Float64(y_m * fma(Float64(x * x), fma(Float64(x * x), fma(Float64(x * x), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666), 1.0));
	elseif (t_0 <= 5e-276)
		tmp = Float64(0.5 * Float64(1.0 + -1.0));
	else
		tmp = fma(y_m, Float64(Float64(y_m * y_m) * fma(y_m, Float64(y_m * fma(y_m, Float64(y_m * 0.0001984126984126984), 0.008333333333333333)), 0.16666666666666666)), 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, -2e-107], N[(y$95$m * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e-276], N[(0.5 * N[(1.0 + -1.0), $MachinePrecision]), $MachinePrecision], N[(y$95$m * N[(N[(y$95$m * y$95$m), $MachinePrecision] * N[(y$95$m * N[(y$95$m * N[(y$95$m * N[(y$95$m * 0.0001984126984126984), $MachinePrecision] + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $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) < -2e-107

   1. Initial program 99.4%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sin.f64 18.9

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

   5. Simplified 18.9%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 32.9

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

   8. Simplified 32.9%

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

   if -2e-107 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 4.99999999999999967e-276

   1. Initial program 72.2%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. rec-exp N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \]

      5. lower-exp.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \]

      6. lower-neg.f64 42.6

         \[\leadsto 0.5 \cdot \left(e^{y} - e^{\color{blue}{-y}}\right) \]

   5. Simplified 42.6%

      \[\leadsto \color{blue}{0.5 \cdot \left(e^{y} - e^{-y}\right)} \]

   6. Taylor expanded in y around 0

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

   8. Simplified 42.6%

      \[\leadsto 0.5 \cdot \left(e^{y} - \color{blue}{1}\right) \]

   9. Taylor expanded in y around 0

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

   11. Simplified 42.6%

       \[\leadsto 0.5 \cdot \left(\color{blue}{1} - 1\right) \]

   if 4.99999999999999967e-276 < (/.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

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 90.6

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

   5. Simplified 90.6%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. cube-mult N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

   8. Simplified 65.9%

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

4. Final simplification 48.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -2 \cdot 10^{-107}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)\\ \mathbf{elif}\;\frac{\sinh y \cdot \sin x}{x} \leq 5 \cdot 10^{-276}:\\ \;\;\;\;0.5 \cdot \left(1 + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \left(y \cdot y\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 68.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{\sinh y\_m \cdot \sin x}{x}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-165}:\\ \;\;\;\;y\_m \cdot \left(\left(y\_m \cdot y\_m\right) \cdot \mathsf{fma}\left(x \cdot x, -0.027777777777777776, 0.16666666666666666\right)\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-276}:\\ \;\;\;\;0.5 \cdot \left(1 + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y\_m \cdot y\_m, y\_m \cdot \mathsf{fma}\left(y\_m \cdot y\_m, 0.008333333333333333, 0.16666666666666666\right), y\_m\right)\\ \end{array} \end{array} \end{array} \]

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-165)
      (*
       y_m
       (* (* y_m y_m) (fma (* x x) -0.027777777777777776 0.16666666666666666)))
      (if (<= t_0 5e-276)
        (* 0.5 (+ 1.0 -1.0))
        (fma
         (* y_m y_m)
         (* y_m (fma (* y_m y_m) 0.008333333333333333 0.16666666666666666))
         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-165) {
		tmp = y_m * ((y_m * y_m) * fma((x * x), -0.027777777777777776, 0.16666666666666666));
	} else if (t_0 <= 5e-276) {
		tmp = 0.5 * (1.0 + -1.0);
	} else {
		tmp = fma((y_m * y_m), (y_m * fma((y_m * y_m), 0.008333333333333333, 0.16666666666666666)), 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-165)
		tmp = Float64(y_m * Float64(Float64(y_m * y_m) * fma(Float64(x * x), -0.027777777777777776, 0.16666666666666666)));
	elseif (t_0 <= 5e-276)
		tmp = Float64(0.5 * Float64(1.0 + -1.0));
	else
		tmp = fma(Float64(y_m * y_m), Float64(y_m * fma(Float64(y_m * y_m), 0.008333333333333333, 0.16666666666666666)), 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-165], N[(y$95$m * N[(N[(y$95$m * y$95$m), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.027777777777777776 + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e-276], N[(0.5 * N[(1.0 + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m * y$95$m), $MachinePrecision] * N[(y$95$m * N[(N[(y$95$m * y$95$m), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $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) < -1e-165

   1. Initial program 99.5%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-lft-identity N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt-in N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 81.6

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

   5. Simplified 81.6%

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

   6. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. distribute-lft1-in N/A

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

      3. +-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. distribute-lft-in N/A

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

      11. *-rgt-identity N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 57.7

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

   8. Simplified 57.7%

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

   9. Taylor expanded in y around inf

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

       1. metadata-eval N/A

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

       2. distribute-lft-neg-in N/A

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

       3. *-commutative N/A

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

       4. associate-*l* N/A

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

       5. unpow3 N/A

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

       6. unpow2 N/A

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

       7. associate-*r* N/A

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

       8. distribute-lft-neg-in N/A

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

       9. distribute-lft-neg-in N/A

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

       10. distribute-lft-neg-in N/A

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

       11. metadata-eval N/A

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

       12. associate-*r* N/A

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

       13. *-commutative N/A

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

       14. *-commutative N/A

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

       15. associate-*r* N/A

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

   11. Simplified 45.1%

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

   if -1e-165 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 4.99999999999999967e-276

   1. Initial program 69.9%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. rec-exp N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \]

      5. lower-exp.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \]

      6. lower-neg.f64 45.7

         \[\leadsto 0.5 \cdot \left(e^{y} - e^{\color{blue}{-y}}\right) \]

   5. Simplified 45.7%

      \[\leadsto \color{blue}{0.5 \cdot \left(e^{y} - e^{-y}\right)} \]

   6. Taylor expanded in y around 0

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

   8. Simplified 45.7%

      \[\leadsto 0.5 \cdot \left(e^{y} - \color{blue}{1}\right) \]

   9. Taylor expanded in y around 0

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

   11. Simplified 45.7%

       \[\leadsto 0.5 \cdot \left(\color{blue}{1} - 1\right) \]

   if 4.99999999999999967e-276 < (/.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

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

      1. lower-*.f64 N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. distribute-lft-in N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. distribute-lft-in N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

   5. Simplified 86.3%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

   8. Simplified 84.3%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. distribute-rgt-in N/A

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

       3. associate-*l* N/A

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

       4. *-lft-identity N/A

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

       5. lower-fma.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 N/A

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

       8. lower-*.f64 N/A

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

       9. +-commutative N/A

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

       10. *-commutative N/A

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

       11. lower-fma.f64 N/A

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

       12. unpow2 N/A

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

       13. lower-*.f64 65.8

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

   11. Simplified 65.8%

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

   12. Taylor expanded in y around 0

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

       1. distribute-rgt-in N/A

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

       2. *-lft-identity N/A

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

       3. +-commutative N/A

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

       4. associate-*l* N/A

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

       5. lower-fma.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 N/A

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

       10. +-commutative N/A

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

       11. *-commutative N/A

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

       12. lower-fma.f64 N/A

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

       13. unpow2 N/A

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

       14. lower-*.f64 63.8

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

   14. Simplified 63.8%

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

4. Final simplification 51.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -1 \cdot 10^{-165}:\\ \;\;\;\;y \cdot \left(\left(y \cdot y\right) \cdot \mathsf{fma}\left(x \cdot x, -0.027777777777777776, 0.16666666666666666\right)\right)\\ \mathbf{elif}\;\frac{\sinh y \cdot \sin x}{x} \leq 5 \cdot 10^{-276}:\\ \;\;\;\;0.5 \cdot \left(1 + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, y \cdot \mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 63.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{\sinh y\_m \cdot \sin x}{x}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-165}:\\ \;\;\;\;y\_m \cdot \left(\left(y\_m \cdot y\_m\right) \cdot \mathsf{fma}\left(x \cdot x, -0.027777777777777776, 0.16666666666666666\right)\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-276}:\\ \;\;\;\;0.5 \cdot \left(1 + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y\_m, 0.16666666666666666 \cdot \left(y\_m \cdot y\_m\right), y\_m\right)\\ \end{array} \end{array} \end{array} \]

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-165)
      (*
       y_m
       (* (* y_m y_m) (fma (* x x) -0.027777777777777776 0.16666666666666666)))
      (if (<= t_0 5e-276)
        (* 0.5 (+ 1.0 -1.0))
        (fma y_m (* 0.16666666666666666 (* 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-165) {
		tmp = y_m * ((y_m * y_m) * fma((x * x), -0.027777777777777776, 0.16666666666666666));
	} else if (t_0 <= 5e-276) {
		tmp = 0.5 * (1.0 + -1.0);
	} else {
		tmp = fma(y_m, (0.16666666666666666 * (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-165)
		tmp = Float64(y_m * Float64(Float64(y_m * y_m) * fma(Float64(x * x), -0.027777777777777776, 0.16666666666666666)));
	elseif (t_0 <= 5e-276)
		tmp = Float64(0.5 * Float64(1.0 + -1.0));
	else
		tmp = fma(y_m, Float64(0.16666666666666666 * 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-165], N[(y$95$m * N[(N[(y$95$m * y$95$m), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.027777777777777776 + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e-276], N[(0.5 * N[(1.0 + -1.0), $MachinePrecision]), $MachinePrecision], N[(y$95$m * N[(0.16666666666666666 * 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) < -1e-165

   1. Initial program 99.5%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-lft-identity N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt-in N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 81.6

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

   5. Simplified 81.6%

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

   6. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. distribute-lft1-in N/A

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

      3. +-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. distribute-lft-in N/A

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

      11. *-rgt-identity N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 57.7

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

   8. Simplified 57.7%

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

   9. Taylor expanded in y around inf

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

       1. metadata-eval N/A

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

       2. distribute-lft-neg-in N/A

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

       3. *-commutative N/A

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

       4. associate-*l* N/A

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

       5. unpow3 N/A

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

       6. unpow2 N/A

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

       7. associate-*r* N/A

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

       8. distribute-lft-neg-in N/A

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

       9. distribute-lft-neg-in N/A

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

       10. distribute-lft-neg-in N/A

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

       11. metadata-eval N/A

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

       12. associate-*r* N/A

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

       13. *-commutative N/A

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

       14. *-commutative N/A

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

       15. associate-*r* N/A

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

   11. Simplified 45.1%

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

   if -1e-165 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 4.99999999999999967e-276

   1. Initial program 69.9%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. rec-exp N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \]

      5. lower-exp.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \]

      6. lower-neg.f64 45.7

         \[\leadsto 0.5 \cdot \left(e^{y} - e^{\color{blue}{-y}}\right) \]

   5. Simplified 45.7%

      \[\leadsto \color{blue}{0.5 \cdot \left(e^{y} - e^{-y}\right)} \]

   6. Taylor expanded in y around 0

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

   8. Simplified 45.7%

      \[\leadsto 0.5 \cdot \left(e^{y} - \color{blue}{1}\right) \]

   9. Taylor expanded in y around 0

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

   11. Simplified 45.7%

       \[\leadsto 0.5 \cdot \left(\color{blue}{1} - 1\right) \]

   if 4.99999999999999967e-276 < (/.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

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-lft-identity N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt-in N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 77.8

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

   5. Simplified 77.8%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 58.5

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

   8. Simplified 58.5%

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

4. Final simplification 50.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -1 \cdot 10^{-165}:\\ \;\;\;\;y \cdot \left(\left(y \cdot y\right) \cdot \mathsf{fma}\left(x \cdot x, -0.027777777777777776, 0.16666666666666666\right)\right)\\ \mathbf{elif}\;\frac{\sinh y \cdot \sin x}{x} \leq 5 \cdot 10^{-276}:\\ \;\;\;\;0.5 \cdot \left(1 + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 0.16666666666666666 \cdot \left(y \cdot y\right), y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 61.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{\sinh y\_m \cdot \sin x}{x}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-117}:\\ \;\;\;\;y\_m \cdot \left(\left(x \cdot x\right) \cdot -0.16666666666666666\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-276}:\\ \;\;\;\;0.5 \cdot \left(1 + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y\_m, 0.16666666666666666 \cdot \left(y\_m \cdot y\_m\right), y\_m\right)\\ \end{array} \end{array} \end{array} \]

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-117)
      (* y_m (* (* x x) -0.16666666666666666))
      (if (<= t_0 5e-276)
        (* 0.5 (+ 1.0 -1.0))
        (fma y_m (* 0.16666666666666666 (* 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-117) {
		tmp = y_m * ((x * x) * -0.16666666666666666);
	} else if (t_0 <= 5e-276) {
		tmp = 0.5 * (1.0 + -1.0);
	} else {
		tmp = fma(y_m, (0.16666666666666666 * (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-117)
		tmp = Float64(y_m * Float64(Float64(x * x) * -0.16666666666666666));
	elseif (t_0 <= 5e-276)
		tmp = Float64(0.5 * Float64(1.0 + -1.0));
	else
		tmp = fma(y_m, Float64(0.16666666666666666 * 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-117], N[(y$95$m * N[(N[(x * x), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e-276], N[(0.5 * N[(1.0 + -1.0), $MachinePrecision]), $MachinePrecision], N[(y$95$m * N[(0.16666666666666666 * 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) < -1.00000000000000003e-117

   1. Initial program 99.5%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sin.f64 22.0

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

   5. Simplified 22.0%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 28.2

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

   8. Simplified 28.2%

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

   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. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 15.6

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

   11. Simplified 15.6%

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

   if -1.00000000000000003e-117 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 4.99999999999999967e-276

   1. Initial program 71.2%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. rec-exp N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \]

      5. lower-exp.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \]

      6. lower-neg.f64 43.9

         \[\leadsto 0.5 \cdot \left(e^{y} - e^{\color{blue}{-y}}\right) \]

   5. Simplified 43.9%

      \[\leadsto \color{blue}{0.5 \cdot \left(e^{y} - e^{-y}\right)} \]

   6. Taylor expanded in y around 0

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

   8. Simplified 43.9%

      \[\leadsto 0.5 \cdot \left(e^{y} - \color{blue}{1}\right) \]

   9. Taylor expanded in y around 0

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

   11. Simplified 43.9%

       \[\leadsto 0.5 \cdot \left(\color{blue}{1} - 1\right) \]

   if 4.99999999999999967e-276 < (/.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

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-lft-identity N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt-in N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 77.8

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

   5. Simplified 77.8%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 58.5

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

   8. Simplified 58.5%

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

4. Final simplification 40.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -1 \cdot 10^{-117}:\\ \;\;\;\;y \cdot \left(\left(x \cdot x\right) \cdot -0.16666666666666666\right)\\ \mathbf{elif}\;\frac{\sinh y \cdot \sin x}{x} \leq 5 \cdot 10^{-276}:\\ \;\;\;\;0.5 \cdot \left(1 + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 0.16666666666666666 \cdot \left(y \cdot y\right), y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 46.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{\sinh y\_m \cdot \sin x}{x}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-276}:\\ \;\;\;\;0.5 \cdot \left(1 + -1\right)\\ \mathbf{elif}\;t\_0 \leq 1000:\\ \;\;\;\;y\_m\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \mathsf{fma}\left(y\_m, 0.25, 0.5\right)\\ \end{array} \end{array} \end{array} \]

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-276)
      (* 0.5 (+ 1.0 -1.0))
      (if (<= t_0 1000.0) y_m (* y_m (fma y_m 0.25 0.5)))))))

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-276) {
		tmp = 0.5 * (1.0 + -1.0);
	} else if (t_0 <= 1000.0) {
		tmp = y_m;
	} else {
		tmp = y_m * fma(y_m, 0.25, 0.5);
	}
	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-276)
		tmp = Float64(0.5 * Float64(1.0 + -1.0));
	elseif (t_0 <= 1000.0)
		tmp = y_m;
	else
		tmp = Float64(y_m * fma(y_m, 0.25, 0.5));
	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-276], N[(0.5 * N[(1.0 + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1000.0], y$95$m, N[(y$95$m * N[(y$95$m * 0.25 + 0.5), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 4.99999999999999967e-276

   1. Initial program 84.4%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. rec-exp N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \]

      5. lower-exp.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \]

      6. lower-neg.f64 53.1

         \[\leadsto 0.5 \cdot \left(e^{y} - e^{\color{blue}{-y}}\right) \]

   5. Simplified 53.1%

      \[\leadsto \color{blue}{0.5 \cdot \left(e^{y} - e^{-y}\right)} \]

   6. Taylor expanded in y around 0

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

   8. Simplified 24.9%

      \[\leadsto 0.5 \cdot \left(e^{y} - \color{blue}{1}\right) \]

   9. Taylor expanded in y around 0

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

   11. Simplified 24.3%

       \[\leadsto 0.5 \cdot \left(\color{blue}{1} - 1\right) \]

   if 4.99999999999999967e-276 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 1e3

   1. Initial program 99.8%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sin.f64 99.9

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

   5. Simplified 99.9%

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

   6. Taylor expanded in x around 0

      \[\leadsto y \cdot \color{blue}{1} \]
   7. Step-by-step derivation

   8. Simplified 62.7%

      \[\leadsto y \cdot \color{blue}{1} \]

   if 1e3 < (/.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

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. rec-exp N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \]

      5. lower-exp.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \]

      6. lower-neg.f64 73.0

         \[\leadsto 0.5 \cdot \left(e^{y} - e^{\color{blue}{-y}}\right) \]

   5. Simplified 73.0%

      \[\leadsto \color{blue}{0.5 \cdot \left(e^{y} - e^{-y}\right)} \]

   6. Taylor expanded in y around 0

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

   8. Simplified 72.3%

      \[\leadsto 0.5 \cdot \left(e^{y} - \color{blue}{1}\right) \]

   9. Taylor expanded in y around 0

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

       1. lower-*.f64 N/A

          \[\leadsto \color{blue}{y \cdot \left(\frac{1}{2} + \frac{1}{4} \cdot y\right)} \]

       2. +-commutative N/A

          \[\leadsto y \cdot \color{blue}{\left(\frac{1}{4} \cdot y + \frac{1}{2}\right)} \]

       3. *-commutative N/A

          \[\leadsto y \cdot \left(\color{blue}{y \cdot \frac{1}{4}} + \frac{1}{2}\right) \]

       4. lower-fma.f64 51.8

          \[\leadsto y \cdot \color{blue}{\mathsf{fma}\left(y, 0.25, 0.5\right)} \]

   11. Simplified 51.8%

       \[\leadsto \color{blue}{y \cdot \mathsf{fma}\left(y, 0.25, 0.5\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 35.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq 5 \cdot 10^{-276}:\\ \;\;\;\;0.5 \cdot \left(1 + -1\right)\\ \mathbf{elif}\;\frac{\sinh y \cdot \sin x}{x} \leq 1000:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(y, 0.25, 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 79.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\sinh y\_m \cdot \sin x}{x} \leq -1 \cdot 10^{-117}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), x\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y\_m \cdot y\_m, \mathsf{fma}\left(y\_m \cdot y\_m, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y\_m \cdot \left(y\_m \cdot y\_m\right), y\_m\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m}{x} \cdot \mathsf{fma}\left(y\_m \cdot y\_m, x \cdot \mathsf{fma}\left(y\_m \cdot y\_m, 0.008333333333333333, 0.16666666666666666\right), x\right)\\ \end{array} \end{array} \]

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-117)
    (/
     (*
      (fma
       (* x x)
       (*
        x
        (fma
         (* x x)
         (fma (* x x) -0.0001984126984126984 0.008333333333333333)
         -0.16666666666666666))
       x)
      (fma
       (fma
        (* y_m y_m)
        (fma (* y_m y_m) 0.0001984126984126984 0.008333333333333333)
        0.16666666666666666)
       (* y_m (* y_m y_m))
       y_m))
     x)
    (*
     (/ y_m x)
     (fma
      (* y_m y_m)
      (* x (fma (* y_m y_m) 0.008333333333333333 0.16666666666666666))
      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-117) {
		tmp = (fma((x * x), (x * fma((x * x), fma((x * x), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666)), x) * fma(fma((y_m * y_m), fma((y_m * y_m), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666), (y_m * (y_m * y_m)), y_m)) / x;
	} else {
		tmp = (y_m / x) * fma((y_m * y_m), (x * fma((y_m * y_m), 0.008333333333333333, 0.16666666666666666)), 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-117)
		tmp = Float64(Float64(fma(Float64(x * x), Float64(x * fma(Float64(x * x), fma(Float64(x * x), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666)), x) * fma(fma(Float64(y_m * y_m), fma(Float64(y_m * y_m), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666), Float64(y_m * Float64(y_m * y_m)), y_m)) / x);
	else
		tmp = Float64(Float64(y_m / x) * fma(Float64(y_m * y_m), Float64(x * fma(Float64(y_m * y_m), 0.008333333333333333, 0.16666666666666666)), 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_] := N[(y$95$s * If[LessEqual[N[(N[(N[Sinh[y$95$m], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], -1e-117], N[(N[(N[(N[(x * x), $MachinePrecision] * N[(x * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] * N[(N[(N[(y$95$m * y$95$m), $MachinePrecision] * N[(N[(y$95$m * y$95$m), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y$95$m * N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] + y$95$m), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(y$95$m / x), $MachinePrecision] * N[(N[(y$95$m * y$95$m), $MachinePrecision] * N[(x * N[(N[(y$95$m * y$95$m), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -1.00000000000000003e-117

   1. Initial program 99.5%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 88.3

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

   5. Simplified 88.3%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

   8. Simplified 62.3%

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

   if -1.00000000000000003e-117 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 85.8%

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

   3. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. distribute-lft-in N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. distribute-lft-in N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

   5. Simplified 78.9%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

   8. Simplified 91.9%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. distribute-rgt-in N/A

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

       3. associate-*l* N/A

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

       4. *-lft-identity N/A

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

       5. lower-fma.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 N/A

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

       8. lower-*.f64 N/A

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

       9. +-commutative N/A

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

       10. *-commutative N/A

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

       11. lower-fma.f64 N/A

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

       12. unpow2 N/A

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

       13. lower-*.f64 70.0

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

   11. Simplified 70.0%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -1 \cdot 10^{-117}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), x\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x} \cdot \mathsf{fma}\left(y \cdot y, x \cdot \mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 79.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\sinh y\_m \cdot \sin x}{x} \leq -1 \cdot 10^{-117}:\\ \;\;\;\;\frac{y\_m \cdot \left(\mathsf{fma}\left(x \cdot x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), x\right) \cdot \mathsf{fma}\left(y\_m \cdot y\_m, \mathsf{fma}\left(y\_m, y\_m \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m}{x} \cdot \mathsf{fma}\left(y\_m \cdot y\_m, x \cdot \mathsf{fma}\left(y\_m \cdot y\_m, 0.008333333333333333, 0.16666666666666666\right), x\right)\\ \end{array} \end{array} \]

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-117)
    (/
     (*
      y_m
      (*
       (fma
        (* x x)
        (*
         x
         (fma
          (* x x)
          (fma (* x x) -0.0001984126984126984 0.008333333333333333)
          -0.16666666666666666))
        x)
       (fma
        (* y_m y_m)
        (fma y_m (* y_m 0.008333333333333333) 0.16666666666666666)
        1.0)))
     x)
    (*
     (/ y_m x)
     (fma
      (* y_m y_m)
      (* x (fma (* y_m y_m) 0.008333333333333333 0.16666666666666666))
      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-117) {
		tmp = (y_m * (fma((x * x), (x * fma((x * x), fma((x * x), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666)), x) * fma((y_m * y_m), fma(y_m, (y_m * 0.008333333333333333), 0.16666666666666666), 1.0))) / x;
	} else {
		tmp = (y_m / x) * fma((y_m * y_m), (x * fma((y_m * y_m), 0.008333333333333333, 0.16666666666666666)), 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-117)
		tmp = Float64(Float64(y_m * Float64(fma(Float64(x * x), Float64(x * fma(Float64(x * x), fma(Float64(x * x), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666)), x) * fma(Float64(y_m * y_m), fma(y_m, Float64(y_m * 0.008333333333333333), 0.16666666666666666), 1.0))) / x);
	else
		tmp = Float64(Float64(y_m / x) * fma(Float64(y_m * y_m), Float64(x * fma(Float64(y_m * y_m), 0.008333333333333333, 0.16666666666666666)), 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_] := N[(y$95$s * If[LessEqual[N[(N[(N[Sinh[y$95$m], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], -1e-117], N[(N[(y$95$m * N[(N[(N[(x * x), $MachinePrecision] * N[(x * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $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] / x), $MachinePrecision], N[(N[(y$95$m / x), $MachinePrecision] * N[(N[(y$95$m * y$95$m), $MachinePrecision] * N[(x * N[(N[(y$95$m * y$95$m), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -1.00000000000000003e-117

   1. Initial program 99.5%

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

   3. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. distribute-lft-in N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. distribute-lft-in N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

   5. Simplified 85.8%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

   8. Simplified 61.0%

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

   if -1.00000000000000003e-117 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 85.8%

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

   3. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. distribute-lft-in N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. distribute-lft-in N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

   5. Simplified 78.9%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

   8. Simplified 91.9%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. distribute-rgt-in N/A

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

       3. associate-*l* N/A

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

       4. *-lft-identity N/A

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

       5. lower-fma.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 N/A

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

       8. lower-*.f64 N/A

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

       9. +-commutative N/A

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

       10. *-commutative N/A

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

       11. lower-fma.f64 N/A

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

       12. unpow2 N/A

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

       13. lower-*.f64 70.0

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

   11. Simplified 70.0%

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

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -1 \cdot 10^{-117}:\\ \;\;\;\;\frac{y \cdot \left(\mathsf{fma}\left(x \cdot x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), x\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x} \cdot \mathsf{fma}\left(y \cdot y, x \cdot \mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 79.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\sinh y\_m \cdot \sin x}{x} \leq -1 \cdot 10^{-117}:\\ \;\;\;\;\frac{y\_m \cdot \left(\mathsf{fma}\left(x \cdot x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), x\right) \cdot \mathsf{fma}\left(y\_m, y\_m \cdot 0.16666666666666666, 1\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m}{x} \cdot \mathsf{fma}\left(y\_m \cdot y\_m, x \cdot \mathsf{fma}\left(y\_m \cdot y\_m, 0.008333333333333333, 0.16666666666666666\right), x\right)\\ \end{array} \end{array} \]

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-117)
    (/
     (*
      y_m
      (*
       (fma
        (* x x)
        (*
         x
         (fma
          (* x x)
          (fma (* x x) -0.0001984126984126984 0.008333333333333333)
          -0.16666666666666666))
        x)
       (fma y_m (* y_m 0.16666666666666666) 1.0)))
     x)
    (*
     (/ y_m x)
     (fma
      (* y_m y_m)
      (* x (fma (* y_m y_m) 0.008333333333333333 0.16666666666666666))
      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-117) {
		tmp = (y_m * (fma((x * x), (x * fma((x * x), fma((x * x), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666)), x) * fma(y_m, (y_m * 0.16666666666666666), 1.0))) / x;
	} else {
		tmp = (y_m / x) * fma((y_m * y_m), (x * fma((y_m * y_m), 0.008333333333333333, 0.16666666666666666)), 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-117)
		tmp = Float64(Float64(y_m * Float64(fma(Float64(x * x), Float64(x * fma(Float64(x * x), fma(Float64(x * x), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666)), x) * fma(y_m, Float64(y_m * 0.16666666666666666), 1.0))) / x);
	else
		tmp = Float64(Float64(y_m / x) * fma(Float64(y_m * y_m), Float64(x * fma(Float64(y_m * y_m), 0.008333333333333333, 0.16666666666666666)), 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_] := N[(y$95$s * If[LessEqual[N[(N[(N[Sinh[y$95$m], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], -1e-117], N[(N[(y$95$m * N[(N[(N[(x * x), $MachinePrecision] * N[(x * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] * N[(y$95$m * N[(y$95$m * 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(y$95$m / x), $MachinePrecision] * N[(N[(y$95$m * y$95$m), $MachinePrecision] * N[(x * N[(N[(y$95$m * y$95$m), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -1.00000000000000003e-117

   1. Initial program 99.5%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-lft-identity N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt-in N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 80.7

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

   5. Simplified 80.7%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

   8. Simplified 58.7%

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

   if -1.00000000000000003e-117 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 85.8%

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

   3. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. distribute-lft-in N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. distribute-lft-in N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

   5. Simplified 78.9%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

   8. Simplified 91.9%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. distribute-rgt-in N/A

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

       3. associate-*l* N/A

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

       4. *-lft-identity N/A

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

       5. lower-fma.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 N/A

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

       8. lower-*.f64 N/A

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

       9. +-commutative N/A

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

       10. *-commutative N/A

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

       11. lower-fma.f64 N/A

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

       12. unpow2 N/A

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

       13. lower-*.f64 70.0

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

   11. Simplified 70.0%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -1 \cdot 10^{-117}:\\ \;\;\;\;\frac{y \cdot \left(\mathsf{fma}\left(x \cdot x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), x\right) \cdot \mathsf{fma}\left(y, y \cdot 0.16666666666666666, 1\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x} \cdot \mathsf{fma}\left(y \cdot y, x \cdot \mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 79.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\sinh y\_m \cdot \sin x}{x} \leq -5 \cdot 10^{-215}:\\ \;\;\;\;\mathsf{fma}\left(y\_m \cdot y\_m, y\_m \cdot \mathsf{fma}\left(y\_m \cdot y\_m, \mathsf{fma}\left(y\_m \cdot y\_m, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y\_m\right) \cdot \left(\left(x \cdot x\right) \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m}{x} \cdot \mathsf{fma}\left(y\_m \cdot y\_m, x \cdot \mathsf{fma}\left(y\_m \cdot y\_m, 0.008333333333333333, 0.16666666666666666\right), x\right)\\ \end{array} \end{array} \]

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) -5e-215)
    (*
     (fma
      (* y_m y_m)
      (*
       y_m
       (fma
        (* y_m y_m)
        (fma (* y_m y_m) 0.0001984126984126984 0.008333333333333333)
        0.16666666666666666))
      y_m)
     (* (* x x) -0.16666666666666666))
    (*
     (/ y_m x)
     (fma
      (* y_m y_m)
      (* x (fma (* y_m y_m) 0.008333333333333333 0.16666666666666666))
      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) <= -5e-215) {
		tmp = fma((y_m * y_m), (y_m * fma((y_m * y_m), fma((y_m * y_m), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666)), y_m) * ((x * x) * -0.16666666666666666);
	} else {
		tmp = (y_m / x) * fma((y_m * y_m), (x * fma((y_m * y_m), 0.008333333333333333, 0.16666666666666666)), 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) <= -5e-215)
		tmp = Float64(fma(Float64(y_m * y_m), Float64(y_m * fma(Float64(y_m * y_m), fma(Float64(y_m * y_m), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666)), y_m) * Float64(Float64(x * x) * -0.16666666666666666));
	else
		tmp = Float64(Float64(y_m / x) * fma(Float64(y_m * y_m), Float64(x * fma(Float64(y_m * y_m), 0.008333333333333333, 0.16666666666666666)), 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_] := N[(y$95$s * If[LessEqual[N[(N[(N[Sinh[y$95$m], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], -5e-215], N[(N[(N[(y$95$m * y$95$m), $MachinePrecision] * N[(y$95$m * N[(N[(y$95$m * y$95$m), $MachinePrecision] * N[(N[(y$95$m * y$95$m), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + y$95$m), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m / x), $MachinePrecision] * N[(N[(y$95$m * y$95$m), $MachinePrecision] * N[(x * N[(N[(y$95$m * y$95$m), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -4.99999999999999956e-215

   1. Initial program 99.5%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 N/A

          \[\leadsto \frac{\sin x \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), \color{blue}{y \cdot {y}^{2}}, y\right)}{x} \]

      17. unpow2 N/A

          \[\leadsto \frac{\sin x \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), y \cdot \color{blue}{\left(y \cdot y\right)}, y\right)}{x} \]

      18. lower-*.f64 89.5

          \[\leadsto \frac{\sin x \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y \cdot \color{blue}{\left(y \cdot y\right)}, y\right)}{x} \]

   5. Simplified 89.5%

      \[\leadsto \frac{\sin x \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)}}{x} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), y \cdot \left(y \cdot y\right), y\right)}{x} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\left(x \cdot \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2} + 1\right)}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), y \cdot \left(y \cdot y\right), y\right)}{x} \]

      2. distribute-rgt-in N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\frac{-1}{6} \cdot {x}^{2}\right) \cdot x + 1 \cdot x\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), y \cdot \left(y \cdot y\right), y\right)}{x} \]

      3. *-lft-identity N/A

         \[\leadsto \frac{\left(\left(\frac{-1}{6} \cdot {x}^{2}\right) \cdot x + \color{blue}{x}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), y \cdot \left(y \cdot y\right), y\right)}{x} \]

      4. associate-*l* N/A

         \[\leadsto \frac{\left(\color{blue}{\frac{-1}{6} \cdot \left({x}^{2} \cdot x\right)} + x\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), y \cdot \left(y \cdot y\right), y\right)}{x} \]

      5. pow-plus N/A

         \[\leadsto \frac{\left(\frac{-1}{6} \cdot \color{blue}{{x}^{\left(2 + 1\right)}} + x\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), y \cdot \left(y \cdot y\right), y\right)}{x} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\left(\frac{-1}{6} \cdot {x}^{\color{blue}{3}} + x\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), y \cdot \left(y \cdot y\right), y\right)}{x} \]

      7. cube-unmult N/A

         \[\leadsto \frac{\left(\frac{-1}{6} \cdot \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} + x\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), y \cdot \left(y \cdot y\right), y\right)}{x} \]

      8. unpow2 N/A

         \[\leadsto \frac{\left(\frac{-1}{6} \cdot \left(x \cdot \color{blue}{{x}^{2}}\right) + x\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), y \cdot \left(y \cdot y\right), y\right)}{x} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{6}, x \cdot {x}^{2}, x\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), y \cdot \left(y \cdot y\right), y\right)}{x} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{6}, \color{blue}{x \cdot {x}^{2}}, x\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), y \cdot \left(y \cdot y\right), y\right)}{x} \]

      11. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{6}, x \cdot \color{blue}{\left(x \cdot x\right)}, x\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), y \cdot \left(y \cdot y\right), y\right)}{x} \]

      12. lower-*.f64 59.5

          \[\leadsto \frac{\mathsf{fma}\left(-0.16666666666666666, x \cdot \color{blue}{\left(x \cdot x\right)}, x\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)}{x} \]

   8. Simplified 59.5%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.16666666666666666, x \cdot \left(x \cdot x\right), x\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)}{x} \]

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{-1}{6} \cdot \left({x}^{2} \cdot \left(y + {y}^{3} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2}\right) \cdot \left(y + {y}^{3} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)} \]

       2. *-commutative N/A

          \[\leadsto \color{blue}{\left(y + {y}^{3} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right) \cdot \left(\frac{-1}{6} \cdot {x}^{2}\right)} \]

       3. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(y + {y}^{3} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right) \cdot \left(\frac{-1}{6} \cdot {x}^{2}\right)} \]

   11. Simplified 16.1%

       \[\leadsto \color{blue}{\mathsf{fma}\left(y \cdot y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y\right) \cdot \left(-0.16666666666666666 \cdot \left(x \cdot x\right)\right)} \]

   if -4.99999999999999956e-215 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 85.1%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{y \cdot \left(\sin x + {y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{6} \cdot \sin x\right)\right)}}{x} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(\sin x + {y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{6} \cdot \sin x\right)\right)}}{x} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{y \cdot \left(\sin x + \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right)\right) + {y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x\right)\right)}\right)}{x} \]

      3. *-rgt-identity N/A

         \[\leadsto \frac{y \cdot \left(\color{blue}{\sin x \cdot 1} + \left({y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right)\right) + {y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x\right)\right)\right)}{x} \]

      4. distribute-lft-in N/A

         \[\leadsto \frac{y \cdot \left(\sin x \cdot 1 + \color{blue}{{y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{6} \cdot \sin x\right)}\right)}{x} \]

      5. +-commutative N/A

         \[\leadsto \frac{y \cdot \left(\sin x \cdot 1 + {y}^{2} \cdot \color{blue}{\left(\frac{1}{6} \cdot \sin x + \frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right)\right)}\right)}{x} \]

      6. *-commutative N/A

         \[\leadsto \frac{y \cdot \left(\sin x \cdot 1 + {y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x + \color{blue}{\left({y}^{2} \cdot \sin x\right) \cdot \frac{1}{120}}\right)\right)}{x} \]

      7. associate-*r* N/A

         \[\leadsto \frac{y \cdot \left(\sin x \cdot 1 + {y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x + \color{blue}{{y}^{2} \cdot \left(\sin x \cdot \frac{1}{120}\right)}\right)\right)}{x} \]

      8. *-commutative N/A

         \[\leadsto \frac{y \cdot \left(\sin x \cdot 1 + {y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x + {y}^{2} \cdot \color{blue}{\left(\frac{1}{120} \cdot \sin x\right)}\right)\right)}{x} \]

      9. distribute-lft-in N/A

         \[\leadsto \frac{y \cdot \left(\sin x \cdot 1 + \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x\right) + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{120} \cdot \sin x\right)\right)\right)}\right)}{x} \]

      10. associate-*r* N/A

          \[\leadsto \frac{y \cdot \left(\sin x \cdot 1 + \left(\color{blue}{\left({y}^{2} \cdot \frac{1}{6}\right) \cdot \sin x} + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{120} \cdot \sin x\right)\right)\right)\right)}{x} \]

      11. *-commutative N/A

          \[\leadsto \frac{y \cdot \left(\sin x \cdot 1 + \left(\color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right)} \cdot \sin x + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{120} \cdot \sin x\right)\right)\right)\right)}{x} \]

   5. Simplified 77.8%

      \[\leadsto \frac{\color{blue}{y \cdot \left(\sin x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)\right)}}{x} \]

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{y \cdot \left(\sin x \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)}{x}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\sin x \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) \cdot y}}{x} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\left(\sin x \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) \cdot \frac{y}{x}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\sin x \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) \cdot \frac{y}{x}} \]

   8. Simplified 91.5%

      \[\leadsto \color{blue}{\left(\sin x \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)\right) \cdot \frac{y}{x}} \]

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(x \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)} \cdot \frac{y}{x} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \left(x \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) + 1\right)}\right) \cdot \frac{y}{x} \]

       2. distribute-rgt-in N/A

          \[\leadsto \color{blue}{\left(\left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) \cdot x + 1 \cdot x\right)} \cdot \frac{y}{x} \]

       3. associate-*l* N/A

          \[\leadsto \left(\color{blue}{{y}^{2} \cdot \left(\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) \cdot x\right)} + 1 \cdot x\right) \cdot \frac{y}{x} \]

       4. *-lft-identity N/A

          \[\leadsto \left({y}^{2} \cdot \left(\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) \cdot x\right) + \color{blue}{x}\right) \cdot \frac{y}{x} \]

       5. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left({y}^{2}, \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) \cdot x, x\right)} \cdot \frac{y}{x} \]

       6. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{y \cdot y}, \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) \cdot x, x\right) \cdot \frac{y}{x} \]

       7. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{y \cdot y}, \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) \cdot x, x\right) \cdot \frac{y}{x} \]

       8. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(y \cdot y, \color{blue}{\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) \cdot x}, x\right) \cdot \frac{y}{x} \]

       9. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(y \cdot y, \color{blue}{\left(\frac{1}{120} \cdot {y}^{2} + \frac{1}{6}\right)} \cdot x, x\right) \cdot \frac{y}{x} \]

       10. *-commutative N/A

           \[\leadsto \mathsf{fma}\left(y \cdot y, \left(\color{blue}{{y}^{2} \cdot \frac{1}{120}} + \frac{1}{6}\right) \cdot x, x\right) \cdot \frac{y}{x} \]

       11. lower-fma.f64 N/A

           \[\leadsto \mathsf{fma}\left(y \cdot y, \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{1}{120}, \frac{1}{6}\right)} \cdot x, x\right) \cdot \frac{y}{x} \]

       12. unpow2 N/A

           \[\leadsto \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{120}, \frac{1}{6}\right) \cdot x, x\right) \cdot \frac{y}{x} \]

       13. lower-*.f64 71.2

           \[\leadsto \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, 0.008333333333333333, 0.16666666666666666\right) \cdot x, x\right) \cdot \frac{y}{x} \]

   11. Simplified 71.2%

       \[\leadsto \color{blue}{\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right) \cdot x, x\right)} \cdot \frac{y}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 52.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -5 \cdot 10^{-215}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y\right) \cdot \left(\left(x \cdot x\right) \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x} \cdot \mathsf{fma}\left(y \cdot y, x \cdot \mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 79.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y\_m \cdot y\_m, 0.008333333333333333, 0.16666666666666666\right)\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\sinh y\_m \cdot \sin x}{x} \leq -5 \cdot 10^{-215}:\\ \;\;\;\;y\_m \cdot \left(\mathsf{fma}\left(-0.16666666666666666, x \cdot x, 1\right) \cdot \mathsf{fma}\left(y\_m \cdot y\_m, t\_0, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m}{x} \cdot \mathsf{fma}\left(y\_m \cdot y\_m, x \cdot t\_0, x\right)\\ \end{array} \end{array} \end{array} \]

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 (fma (* y_m y_m) 0.008333333333333333 0.16666666666666666)))
   (*
    y_s
    (if (<= (/ (* (sinh y_m) (sin x)) x) -5e-215)
      (*
       y_m
       (* (fma -0.16666666666666666 (* x x) 1.0) (fma (* y_m y_m) t_0 1.0)))
      (* (/ y_m x) (fma (* y_m y_m) (* x t_0) 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 = fma((y_m * y_m), 0.008333333333333333, 0.16666666666666666);
	double tmp;
	if (((sinh(y_m) * sin(x)) / x) <= -5e-215) {
		tmp = y_m * (fma(-0.16666666666666666, (x * x), 1.0) * fma((y_m * y_m), t_0, 1.0));
	} else {
		tmp = (y_m / x) * fma((y_m * y_m), (x * t_0), 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 = fma(Float64(y_m * y_m), 0.008333333333333333, 0.16666666666666666)
	tmp = 0.0
	if (Float64(Float64(sinh(y_m) * sin(x)) / x) <= -5e-215)
		tmp = Float64(y_m * Float64(fma(-0.16666666666666666, Float64(x * x), 1.0) * fma(Float64(y_m * y_m), t_0, 1.0)));
	else
		tmp = Float64(Float64(y_m / x) * fma(Float64(y_m * y_m), Float64(x * t_0), 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[(y$95$m * y$95$m), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision]}, N[(y$95$s * If[LessEqual[N[(N[(N[Sinh[y$95$m], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], -5e-215], N[(y$95$m * N[(N[(-0.16666666666666666 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(y$95$m * y$95$m), $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m / x), $MachinePrecision] * N[(N[(y$95$m * y$95$m), $MachinePrecision] * N[(x * t$95$0), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -4.99999999999999956e-215

   1. Initial program 99.5%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{y \cdot \left(\sin x + {y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{6} \cdot \sin x\right)\right)}}{x} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(\sin x + {y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{6} \cdot \sin x\right)\right)}}{x} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{y \cdot \left(\sin x + \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right)\right) + {y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x\right)\right)}\right)}{x} \]

      3. *-rgt-identity N/A

         \[\leadsto \frac{y \cdot \left(\color{blue}{\sin x \cdot 1} + \left({y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right)\right) + {y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x\right)\right)\right)}{x} \]

      4. distribute-lft-in N/A

         \[\leadsto \frac{y \cdot \left(\sin x \cdot 1 + \color{blue}{{y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{6} \cdot \sin x\right)}\right)}{x} \]

      5. +-commutative N/A

         \[\leadsto \frac{y \cdot \left(\sin x \cdot 1 + {y}^{2} \cdot \color{blue}{\left(\frac{1}{6} \cdot \sin x + \frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right)\right)}\right)}{x} \]

      6. *-commutative N/A

         \[\leadsto \frac{y \cdot \left(\sin x \cdot 1 + {y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x + \color{blue}{\left({y}^{2} \cdot \sin x\right) \cdot \frac{1}{120}}\right)\right)}{x} \]

      7. associate-*r* N/A

         \[\leadsto \frac{y \cdot \left(\sin x \cdot 1 + {y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x + \color{blue}{{y}^{2} \cdot \left(\sin x \cdot \frac{1}{120}\right)}\right)\right)}{x} \]

      8. *-commutative N/A

         \[\leadsto \frac{y \cdot \left(\sin x \cdot 1 + {y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x + {y}^{2} \cdot \color{blue}{\left(\frac{1}{120} \cdot \sin x\right)}\right)\right)}{x} \]

      9. distribute-lft-in N/A

         \[\leadsto \frac{y \cdot \left(\sin x \cdot 1 + \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x\right) + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{120} \cdot \sin x\right)\right)\right)}\right)}{x} \]

      10. associate-*r* N/A

          \[\leadsto \frac{y \cdot \left(\sin x \cdot 1 + \left(\color{blue}{\left({y}^{2} \cdot \frac{1}{6}\right) \cdot \sin x} + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{120} \cdot \sin x\right)\right)\right)\right)}{x} \]

      11. *-commutative N/A

          \[\leadsto \frac{y \cdot \left(\sin x \cdot 1 + \left(\color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right)} \cdot \sin x + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{120} \cdot \sin x\right)\right)\right)\right)}{x} \]

   5. Simplified 87.2%

      \[\leadsto \frac{\color{blue}{y \cdot \left(\sin x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)\right)}}{x} \]

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{y \cdot \left(\sin x \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)}{x}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\sin x \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) \cdot y}}{x} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\left(\sin x \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) \cdot \frac{y}{x}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\sin x \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) \cdot \frac{y}{x}} \]

   8. Simplified 86.6%

      \[\leadsto \color{blue}{\left(\sin x \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)\right) \cdot \frac{y}{x}} \]

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-1}{6} \cdot \left({x}^{2} \cdot \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right) + y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \color{blue}{y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) + \frac{-1}{6} \cdot \left({x}^{2} \cdot \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right)} \]

       2. associate-*r* N/A

          \[\leadsto y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) + \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2}\right) \cdot \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)} \]

       3. *-commutative N/A

          \[\leadsto \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) \cdot y} + \left(\frac{-1}{6} \cdot {x}^{2}\right) \cdot \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) \cdot y + \left(\frac{-1}{6} \cdot {x}^{2}\right) \cdot \color{blue}{\left(\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) \cdot y\right)} \]

       5. associate-*r* N/A

          \[\leadsto \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) \cdot y + \color{blue}{\left(\left(\frac{-1}{6} \cdot {x}^{2}\right) \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) \cdot y} \]

       6. associate-*r* N/A

          \[\leadsto \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) \cdot y + \color{blue}{\left(\frac{-1}{6} \cdot \left({x}^{2} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right)} \cdot y \]

       7. distribute-rgt-out N/A

          \[\leadsto \color{blue}{y \cdot \left(\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) + \frac{-1}{6} \cdot \left({x}^{2} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right)} \]

   11. Simplified 58.9%

       \[\leadsto \color{blue}{y \cdot \left(\mathsf{fma}\left(-0.16666666666666666, x \cdot x, 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), 1\right)\right)} \]

   if -4.99999999999999956e-215 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 85.1%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{y \cdot \left(\sin x + {y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{6} \cdot \sin x\right)\right)}}{x} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(\sin x + {y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{6} \cdot \sin x\right)\right)}}{x} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{y \cdot \left(\sin x + \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right)\right) + {y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x\right)\right)}\right)}{x} \]

      3. *-rgt-identity N/A

         \[\leadsto \frac{y \cdot \left(\color{blue}{\sin x \cdot 1} + \left({y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right)\right) + {y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x\right)\right)\right)}{x} \]

      4. distribute-lft-in N/A

         \[\leadsto \frac{y \cdot \left(\sin x \cdot 1 + \color{blue}{{y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{6} \cdot \sin x\right)}\right)}{x} \]

      5. +-commutative N/A

         \[\leadsto \frac{y \cdot \left(\sin x \cdot 1 + {y}^{2} \cdot \color{blue}{\left(\frac{1}{6} \cdot \sin x + \frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right)\right)}\right)}{x} \]

      6. *-commutative N/A

         \[\leadsto \frac{y \cdot \left(\sin x \cdot 1 + {y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x + \color{blue}{\left({y}^{2} \cdot \sin x\right) \cdot \frac{1}{120}}\right)\right)}{x} \]

      7. associate-*r* N/A

         \[\leadsto \frac{y \cdot \left(\sin x \cdot 1 + {y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x + \color{blue}{{y}^{2} \cdot \left(\sin x \cdot \frac{1}{120}\right)}\right)\right)}{x} \]

      8. *-commutative N/A

         \[\leadsto \frac{y \cdot \left(\sin x \cdot 1 + {y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x + {y}^{2} \cdot \color{blue}{\left(\frac{1}{120} \cdot \sin x\right)}\right)\right)}{x} \]

      9. distribute-lft-in N/A

         \[\leadsto \frac{y \cdot \left(\sin x \cdot 1 + \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x\right) + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{120} \cdot \sin x\right)\right)\right)}\right)}{x} \]

      10. associate-*r* N/A

          \[\leadsto \frac{y \cdot \left(\sin x \cdot 1 + \left(\color{blue}{\left({y}^{2} \cdot \frac{1}{6}\right) \cdot \sin x} + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{120} \cdot \sin x\right)\right)\right)\right)}{x} \]

      11. *-commutative N/A

          \[\leadsto \frac{y \cdot \left(\sin x \cdot 1 + \left(\color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right)} \cdot \sin x + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{120} \cdot \sin x\right)\right)\right)\right)}{x} \]

   5. Simplified 77.8%

      \[\leadsto \frac{\color{blue}{y \cdot \left(\sin x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)\right)}}{x} \]

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{y \cdot \left(\sin x \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)}{x}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\sin x \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) \cdot y}}{x} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\left(\sin x \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) \cdot \frac{y}{x}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\sin x \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) \cdot \frac{y}{x}} \]

   8. Simplified 91.5%

      \[\leadsto \color{blue}{\left(\sin x \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)\right) \cdot \frac{y}{x}} \]

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(x \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)} \cdot \frac{y}{x} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \left(x \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) + 1\right)}\right) \cdot \frac{y}{x} \]

       2. distribute-rgt-in N/A

          \[\leadsto \color{blue}{\left(\left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) \cdot x + 1 \cdot x\right)} \cdot \frac{y}{x} \]

       3. associate-*l* N/A

          \[\leadsto \left(\color{blue}{{y}^{2} \cdot \left(\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) \cdot x\right)} + 1 \cdot x\right) \cdot \frac{y}{x} \]

       4. *-lft-identity N/A

          \[\leadsto \left({y}^{2} \cdot \left(\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) \cdot x\right) + \color{blue}{x}\right) \cdot \frac{y}{x} \]

       5. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left({y}^{2}, \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) \cdot x, x\right)} \cdot \frac{y}{x} \]

       6. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{y \cdot y}, \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) \cdot x, x\right) \cdot \frac{y}{x} \]

       7. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{y \cdot y}, \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) \cdot x, x\right) \cdot \frac{y}{x} \]

       8. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(y \cdot y, \color{blue}{\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) \cdot x}, x\right) \cdot \frac{y}{x} \]

       9. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(y \cdot y, \color{blue}{\left(\frac{1}{120} \cdot {y}^{2} + \frac{1}{6}\right)} \cdot x, x\right) \cdot \frac{y}{x} \]

       10. *-commutative N/A

           \[\leadsto \mathsf{fma}\left(y \cdot y, \left(\color{blue}{{y}^{2} \cdot \frac{1}{120}} + \frac{1}{6}\right) \cdot x, x\right) \cdot \frac{y}{x} \]

       11. lower-fma.f64 N/A

           \[\leadsto \mathsf{fma}\left(y \cdot y, \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{1}{120}, \frac{1}{6}\right)} \cdot x, x\right) \cdot \frac{y}{x} \]

       12. unpow2 N/A

           \[\leadsto \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{120}, \frac{1}{6}\right) \cdot x, x\right) \cdot \frac{y}{x} \]

       13. lower-*.f64 71.2

           \[\leadsto \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, 0.008333333333333333, 0.16666666666666666\right) \cdot x, x\right) \cdot \frac{y}{x} \]

   11. Simplified 71.2%

       \[\leadsto \color{blue}{\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right) \cdot x, x\right)} \cdot \frac{y}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -5 \cdot 10^{-215}:\\ \;\;\;\;y \cdot \left(\mathsf{fma}\left(-0.16666666666666666, x \cdot x, 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x} \cdot \mathsf{fma}\left(y \cdot y, x \cdot \mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 64.0% accurate, 4.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 2.25 \cdot 10^{+57}:\\ \;\;\;\;\mathsf{fma}\left(y\_m, \left(y\_m \cdot y\_m\right) \cdot \mathsf{fma}\left(y\_m, y\_m \cdot \mathsf{fma}\left(y\_m, y\_m \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m}{x} \cdot \left(\left(y\_m \cdot y\_m\right) \cdot \left(0.008333333333333333 \cdot \left(y\_m \cdot \left(y\_m \cdot x\right)\right)\right)\right)\\ \end{array} \end{array} \]

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 (<= x 2.25e+57)
    (fma
     y_m
     (*
      (* y_m y_m)
      (fma
       y_m
       (* y_m (fma y_m (* y_m 0.0001984126984126984) 0.008333333333333333))
       0.16666666666666666))
     y_m)
    (* (/ y_m x) (* (* y_m y_m) (* 0.008333333333333333 (* y_m (* 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 (x <= 2.25e+57) {
		tmp = fma(y_m, ((y_m * y_m) * fma(y_m, (y_m * fma(y_m, (y_m * 0.0001984126984126984), 0.008333333333333333)), 0.16666666666666666)), y_m);
	} else {
		tmp = (y_m / x) * ((y_m * y_m) * (0.008333333333333333 * (y_m * (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 (x <= 2.25e+57)
		tmp = fma(y_m, Float64(Float64(y_m * y_m) * fma(y_m, Float64(y_m * fma(y_m, Float64(y_m * 0.0001984126984126984), 0.008333333333333333)), 0.16666666666666666)), y_m);
	else
		tmp = Float64(Float64(y_m / x) * Float64(Float64(y_m * y_m) * Float64(0.008333333333333333 * Float64(y_m * Float64(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_] := N[(y$95$s * If[LessEqual[x, 2.25e+57], N[(y$95$m * N[(N[(y$95$m * y$95$m), $MachinePrecision] * N[(y$95$m * N[(y$95$m * N[(y$95$m * N[(y$95$m * 0.0001984126984126984), $MachinePrecision] + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + y$95$m), $MachinePrecision], N[(N[(y$95$m / x), $MachinePrecision] * N[(N[(y$95$m * y$95$m), $MachinePrecision] * N[(0.008333333333333333 * N[(y$95$m * N[(y$95$m * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 2.24999999999999998e57

   1. Initial program 87.2%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\sin x \cdot \color{blue}{\left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)}}{x} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\sin x \cdot \left(y \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) + 1\right)}\right)}{x} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\sin x \cdot \color{blue}{\left(y \cdot \left({y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right) + y \cdot 1\right)}}{x} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\sin x \cdot \left(\color{blue}{\left(y \cdot {y}^{2}\right) \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)} + y \cdot 1\right)}{x} \]

      4. *-commutative N/A

         \[\leadsto \frac{\sin x \cdot \left(\color{blue}{\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot \left(y \cdot {y}^{2}\right)} + y \cdot 1\right)}{x} \]

      5. *-rgt-identity N/A

         \[\leadsto \frac{\sin x \cdot \left(\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot \left(y \cdot {y}^{2}\right) + \color{blue}{y}\right)}{x} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\sin x \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right), y \cdot {y}^{2}, y\right)}}{x} \]

      7. +-commutative N/A

         \[\leadsto \frac{\sin x \cdot \mathsf{fma}\left(\color{blue}{{y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right) + \frac{1}{6}}, y \cdot {y}^{2}, y\right)}{x} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\sin x \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left({y}^{2}, \frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}, \frac{1}{6}\right)}, y \cdot {y}^{2}, y\right)}{x} \]

      9. unpow2 N/A

         \[\leadsto \frac{\sin x \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}, \frac{1}{6}\right), y \cdot {y}^{2}, y\right)}{x} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\sin x \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}, \frac{1}{6}\right), y \cdot {y}^{2}, y\right)}{x} \]

      11. +-commutative N/A

          \[\leadsto \frac{\sin x \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \color{blue}{\frac{1}{5040} \cdot {y}^{2} + \frac{1}{120}}, \frac{1}{6}\right), y \cdot {y}^{2}, y\right)}{x} \]

      12. *-commutative N/A

          \[\leadsto \frac{\sin x \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \color{blue}{{y}^{2} \cdot \frac{1}{5040}} + \frac{1}{120}, \frac{1}{6}\right), y \cdot {y}^{2}, y\right)}{x} \]

      13. lower-fma.f64 N/A

          \[\leadsto \frac{\sin x \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{1}{5040}, \frac{1}{120}\right)}, \frac{1}{6}\right), y \cdot {y}^{2}, y\right)}{x} \]

      14. unpow2 N/A

          \[\leadsto \frac{\sin x \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), y \cdot {y}^{2}, y\right)}{x} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{\sin x \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), y \cdot {y}^{2}, y\right)}{x} \]

      16. lower-*.f64 N/A

          \[\leadsto \frac{\sin x \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), \color{blue}{y \cdot {y}^{2}}, y\right)}{x} \]

      17. unpow2 N/A

          \[\leadsto \frac{\sin x \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), y \cdot \color{blue}{\left(y \cdot y\right)}, y\right)}{x} \]

      18. lower-*.f64 80.6

          \[\leadsto \frac{\sin x \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y \cdot \color{blue}{\left(y \cdot y\right)}, y\right)}{x} \]

   5. Simplified 80.6%

      \[\leadsto \frac{\sin x \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)}}{x} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{y + {y}^{3} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{y}^{3} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) + y} \]

      2. cube-mult N/A

         \[\leadsto \color{blue}{\left(y \cdot \left(y \cdot y\right)\right)} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) + y \]

      3. unpow2 N/A

         \[\leadsto \left(y \cdot \color{blue}{{y}^{2}}\right) \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) + y \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{y \cdot \left({y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)} + y \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right), y\right)} \]

   8. Simplified 62.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \left(y \cdot y\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y\right)} \]

   if 2.24999999999999998e57 < x

   1. Initial program 99.9%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{y \cdot \left(\sin x + {y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{6} \cdot \sin x\right)\right)}}{x} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(\sin x + {y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{6} \cdot \sin x\right)\right)}}{x} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{y \cdot \left(\sin x + \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right)\right) + {y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x\right)\right)}\right)}{x} \]

      3. *-rgt-identity N/A

         \[\leadsto \frac{y \cdot \left(\color{blue}{\sin x \cdot 1} + \left({y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right)\right) + {y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x\right)\right)\right)}{x} \]

      4. distribute-lft-in N/A

         \[\leadsto \frac{y \cdot \left(\sin x \cdot 1 + \color{blue}{{y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{6} \cdot \sin x\right)}\right)}{x} \]

      5. +-commutative N/A

         \[\leadsto \frac{y \cdot \left(\sin x \cdot 1 + {y}^{2} \cdot \color{blue}{\left(\frac{1}{6} \cdot \sin x + \frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right)\right)}\right)}{x} \]

      6. *-commutative N/A

         \[\leadsto \frac{y \cdot \left(\sin x \cdot 1 + {y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x + \color{blue}{\left({y}^{2} \cdot \sin x\right) \cdot \frac{1}{120}}\right)\right)}{x} \]

      7. associate-*r* N/A

         \[\leadsto \frac{y \cdot \left(\sin x \cdot 1 + {y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x + \color{blue}{{y}^{2} \cdot \left(\sin x \cdot \frac{1}{120}\right)}\right)\right)}{x} \]

      8. *-commutative N/A

         \[\leadsto \frac{y \cdot \left(\sin x \cdot 1 + {y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x + {y}^{2} \cdot \color{blue}{\left(\frac{1}{120} \cdot \sin x\right)}\right)\right)}{x} \]

      9. distribute-lft-in N/A

         \[\leadsto \frac{y \cdot \left(\sin x \cdot 1 + \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x\right) + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{120} \cdot \sin x\right)\right)\right)}\right)}{x} \]

      10. associate-*r* N/A

          \[\leadsto \frac{y \cdot \left(\sin x \cdot 1 + \left(\color{blue}{\left({y}^{2} \cdot \frac{1}{6}\right) \cdot \sin x} + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{120} \cdot \sin x\right)\right)\right)\right)}{x} \]

      11. *-commutative N/A

          \[\leadsto \frac{y \cdot \left(\sin x \cdot 1 + \left(\color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right)} \cdot \sin x + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{120} \cdot \sin x\right)\right)\right)\right)}{x} \]

   5. Simplified 89.4%

      \[\leadsto \frac{\color{blue}{y \cdot \left(\sin x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)\right)}}{x} \]

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{y \cdot \left(\sin x \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)}{x}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\sin x \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) \cdot y}}{x} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\left(\sin x \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) \cdot \frac{y}{x}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\sin x \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) \cdot \frac{y}{x}} \]

   8. Simplified 85.8%

      \[\leadsto \color{blue}{\left(\sin x \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)\right) \cdot \frac{y}{x}} \]

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(x \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)} \cdot \frac{y}{x} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \left(x \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) + 1\right)}\right) \cdot \frac{y}{x} \]

       2. distribute-rgt-in N/A

          \[\leadsto \color{blue}{\left(\left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) \cdot x + 1 \cdot x\right)} \cdot \frac{y}{x} \]

       3. associate-*l* N/A

          \[\leadsto \left(\color{blue}{{y}^{2} \cdot \left(\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) \cdot x\right)} + 1 \cdot x\right) \cdot \frac{y}{x} \]

       4. *-lft-identity N/A

          \[\leadsto \left({y}^{2} \cdot \left(\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) \cdot x\right) + \color{blue}{x}\right) \cdot \frac{y}{x} \]

       5. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left({y}^{2}, \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) \cdot x, x\right)} \cdot \frac{y}{x} \]

       6. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{y \cdot y}, \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) \cdot x, x\right) \cdot \frac{y}{x} \]

       7. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{y \cdot y}, \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) \cdot x, x\right) \cdot \frac{y}{x} \]

       8. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(y \cdot y, \color{blue}{\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) \cdot x}, x\right) \cdot \frac{y}{x} \]

       9. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(y \cdot y, \color{blue}{\left(\frac{1}{120} \cdot {y}^{2} + \frac{1}{6}\right)} \cdot x, x\right) \cdot \frac{y}{x} \]

       10. *-commutative N/A

           \[\leadsto \mathsf{fma}\left(y \cdot y, \left(\color{blue}{{y}^{2} \cdot \frac{1}{120}} + \frac{1}{6}\right) \cdot x, x\right) \cdot \frac{y}{x} \]

       11. lower-fma.f64 N/A

           \[\leadsto \mathsf{fma}\left(y \cdot y, \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{1}{120}, \frac{1}{6}\right)} \cdot x, x\right) \cdot \frac{y}{x} \]

       12. unpow2 N/A

           \[\leadsto \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{120}, \frac{1}{6}\right) \cdot x, x\right) \cdot \frac{y}{x} \]

       13. lower-*.f64 66.3

           \[\leadsto \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, 0.008333333333333333, 0.16666666666666666\right) \cdot x, x\right) \cdot \frac{y}{x} \]

   11. Simplified 66.3%

       \[\leadsto \color{blue}{\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right) \cdot x, x\right)} \cdot \frac{y}{x} \]

   12. Taylor expanded in y around inf

       \[\leadsto \color{blue}{\left(\frac{1}{120} \cdot \left(x \cdot {y}^{4}\right)\right)} \cdot \frac{y}{x} \]
   13. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\left(\frac{1}{120} \cdot x\right) \cdot {y}^{4}\right)} \cdot \frac{y}{x} \]

       2. metadata-eval N/A

          \[\leadsto \left(\left(\frac{1}{120} \cdot x\right) \cdot {y}^{\color{blue}{\left(2 \cdot 2\right)}}\right) \cdot \frac{y}{x} \]

       3. pow-sqr N/A

          \[\leadsto \left(\left(\frac{1}{120} \cdot x\right) \cdot \color{blue}{\left({y}^{2} \cdot {y}^{2}\right)}\right) \cdot \frac{y}{x} \]

       4. associate-*l* N/A

          \[\leadsto \color{blue}{\left(\left(\left(\frac{1}{120} \cdot x\right) \cdot {y}^{2}\right) \cdot {y}^{2}\right)} \cdot \frac{y}{x} \]

       5. associate-*r* N/A

          \[\leadsto \left(\color{blue}{\left(\frac{1}{120} \cdot \left(x \cdot {y}^{2}\right)\right)} \cdot {y}^{2}\right) \cdot \frac{y}{x} \]

       6. *-commutative N/A

          \[\leadsto \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} \cdot \left(x \cdot {y}^{2}\right)\right)\right)} \cdot \frac{y}{x} \]

       7. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} \cdot \left(x \cdot {y}^{2}\right)\right)\right)} \cdot \frac{y}{x} \]

       8. unpow2 N/A

          \[\leadsto \left(\color{blue}{\left(y \cdot y\right)} \cdot \left(\frac{1}{120} \cdot \left(x \cdot {y}^{2}\right)\right)\right) \cdot \frac{y}{x} \]

       9. lower-*.f64 N/A

          \[\leadsto \left(\color{blue}{\left(y \cdot y\right)} \cdot \left(\frac{1}{120} \cdot \left(x \cdot {y}^{2}\right)\right)\right) \cdot \frac{y}{x} \]

       10. lower-*.f64 N/A

           \[\leadsto \left(\left(y \cdot y\right) \cdot \color{blue}{\left(\frac{1}{120} \cdot \left(x \cdot {y}^{2}\right)\right)}\right) \cdot \frac{y}{x} \]

       11. unpow2 N/A

           \[\leadsto \left(\left(y \cdot y\right) \cdot \left(\frac{1}{120} \cdot \left(x \cdot \color{blue}{\left(y \cdot y\right)}\right)\right)\right) \cdot \frac{y}{x} \]

       12. associate-*r* N/A

           \[\leadsto \left(\left(y \cdot y\right) \cdot \left(\frac{1}{120} \cdot \color{blue}{\left(\left(x \cdot y\right) \cdot y\right)}\right)\right) \cdot \frac{y}{x} \]

       13. *-commutative N/A

           \[\leadsto \left(\left(y \cdot y\right) \cdot \left(\frac{1}{120} \cdot \color{blue}{\left(y \cdot \left(x \cdot y\right)\right)}\right)\right) \cdot \frac{y}{x} \]

       14. lower-*.f64 N/A

           \[\leadsto \left(\left(y \cdot y\right) \cdot \left(\frac{1}{120} \cdot \color{blue}{\left(y \cdot \left(x \cdot y\right)\right)}\right)\right) \cdot \frac{y}{x} \]

       15. *-commutative N/A

           \[\leadsto \left(\left(y \cdot y\right) \cdot \left(\frac{1}{120} \cdot \left(y \cdot \color{blue}{\left(y \cdot x\right)}\right)\right)\right) \cdot \frac{y}{x} \]

       16. lower-*.f64 70.4

           \[\leadsto \left(\left(y \cdot y\right) \cdot \left(0.008333333333333333 \cdot \left(y \cdot \color{blue}{\left(y \cdot x\right)}\right)\right)\right) \cdot \frac{y}{x} \]

   14. Simplified 70.4%

       \[\leadsto \color{blue}{\left(\left(y \cdot y\right) \cdot \left(0.008333333333333333 \cdot \left(y \cdot \left(y \cdot x\right)\right)\right)\right)} \cdot \frac{y}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.25 \cdot 10^{+57}:\\ \;\;\;\;\mathsf{fma}\left(y, \left(y \cdot y\right) \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x} \cdot \left(\left(y \cdot y\right) \cdot \left(0.008333333333333333 \cdot \left(y \cdot \left(y \cdot x\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 61.7% accurate, 5.9× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 4.1 \cdot 10^{+52}:\\ \;\;\;\;\mathsf{fma}\left(y\_m \cdot y\_m, y\_m \cdot \mathsf{fma}\left(y\_m \cdot y\_m, 0.008333333333333333, 0.16666666666666666\right), y\_m\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y\_m \cdot \left(y\_m \cdot \left(y\_m \cdot \left(\left(y\_m \cdot y\_m\right) \cdot 0.016666666666666666\right)\right)\right)\right)\\ \end{array} \end{array} \]

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 (<= x 4.1e+52)
    (fma
     (* y_m y_m)
     (* y_m (fma (* y_m y_m) 0.008333333333333333 0.16666666666666666))
     y_m)
    (* 0.5 (* y_m (* y_m (* y_m (* (* y_m y_m) 0.016666666666666666))))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m) {
	double tmp;
	if (x <= 4.1e+52) {
		tmp = fma((y_m * y_m), (y_m * fma((y_m * y_m), 0.008333333333333333, 0.16666666666666666)), y_m);
	} else {
		tmp = 0.5 * (y_m * (y_m * (y_m * ((y_m * y_m) * 0.016666666666666666))));
	}
	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 (x <= 4.1e+52)
		tmp = fma(Float64(y_m * y_m), Float64(y_m * fma(Float64(y_m * y_m), 0.008333333333333333, 0.16666666666666666)), y_m);
	else
		tmp = Float64(0.5 * Float64(y_m * Float64(y_m * Float64(y_m * Float64(Float64(y_m * y_m) * 0.016666666666666666)))));
	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_] := N[(y$95$s * If[LessEqual[x, 4.1e+52], N[(N[(y$95$m * y$95$m), $MachinePrecision] * N[(y$95$m * N[(N[(y$95$m * y$95$m), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + y$95$m), $MachinePrecision], N[(0.5 * N[(y$95$m * N[(y$95$m * N[(y$95$m * N[(N[(y$95$m * y$95$m), $MachinePrecision] * 0.016666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 4.1e52

   1. Initial program 87.0%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{y \cdot \left(\sin x + {y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{6} \cdot \sin x\right)\right)}}{x} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(\sin x + {y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{6} \cdot \sin x\right)\right)}}{x} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{y \cdot \left(\sin x + \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right)\right) + {y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x\right)\right)}\right)}{x} \]

      3. *-rgt-identity N/A

         \[\leadsto \frac{y \cdot \left(\color{blue}{\sin x \cdot 1} + \left({y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right)\right) + {y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x\right)\right)\right)}{x} \]

      4. distribute-lft-in N/A

         \[\leadsto \frac{y \cdot \left(\sin x \cdot 1 + \color{blue}{{y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{6} \cdot \sin x\right)}\right)}{x} \]

      5. +-commutative N/A

         \[\leadsto \frac{y \cdot \left(\sin x \cdot 1 + {y}^{2} \cdot \color{blue}{\left(\frac{1}{6} \cdot \sin x + \frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right)\right)}\right)}{x} \]

      6. *-commutative N/A

         \[\leadsto \frac{y \cdot \left(\sin x \cdot 1 + {y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x + \color{blue}{\left({y}^{2} \cdot \sin x\right) \cdot \frac{1}{120}}\right)\right)}{x} \]

      7. associate-*r* N/A

         \[\leadsto \frac{y \cdot \left(\sin x \cdot 1 + {y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x + \color{blue}{{y}^{2} \cdot \left(\sin x \cdot \frac{1}{120}\right)}\right)\right)}{x} \]

      8. *-commutative N/A

         \[\leadsto \frac{y \cdot \left(\sin x \cdot 1 + {y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x + {y}^{2} \cdot \color{blue}{\left(\frac{1}{120} \cdot \sin x\right)}\right)\right)}{x} \]

      9. distribute-lft-in N/A

         \[\leadsto \frac{y \cdot \left(\sin x \cdot 1 + \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x\right) + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{120} \cdot \sin x\right)\right)\right)}\right)}{x} \]

      10. associate-*r* N/A

          \[\leadsto \frac{y \cdot \left(\sin x \cdot 1 + \left(\color{blue}{\left({y}^{2} \cdot \frac{1}{6}\right) \cdot \sin x} + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{120} \cdot \sin x\right)\right)\right)\right)}{x} \]

      11. *-commutative N/A

          \[\leadsto \frac{y \cdot \left(\sin x \cdot 1 + \left(\color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right)} \cdot \sin x + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{120} \cdot \sin x\right)\right)\right)\right)}{x} \]

   5. Simplified 78.3%

      \[\leadsto \frac{\color{blue}{y \cdot \left(\sin x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)\right)}}{x} \]

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{y \cdot \left(\sin x \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)}{x}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\sin x \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) \cdot y}}{x} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\left(\sin x \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) \cdot \frac{y}{x}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\sin x \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) \cdot \frac{y}{x}} \]

   8. Simplified 90.8%

      \[\leadsto \color{blue}{\left(\sin x \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)\right) \cdot \frac{y}{x}} \]

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(x \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)} \cdot \frac{y}{x} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \left(x \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) + 1\right)}\right) \cdot \frac{y}{x} \]

       2. distribute-rgt-in N/A

          \[\leadsto \color{blue}{\left(\left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) \cdot x + 1 \cdot x\right)} \cdot \frac{y}{x} \]

       3. associate-*l* N/A

          \[\leadsto \left(\color{blue}{{y}^{2} \cdot \left(\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) \cdot x\right)} + 1 \cdot x\right) \cdot \frac{y}{x} \]

       4. *-lft-identity N/A

          \[\leadsto \left({y}^{2} \cdot \left(\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) \cdot x\right) + \color{blue}{x}\right) \cdot \frac{y}{x} \]

       5. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left({y}^{2}, \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) \cdot x, x\right)} \cdot \frac{y}{x} \]

       6. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{y \cdot y}, \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) \cdot x, x\right) \cdot \frac{y}{x} \]

       7. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{y \cdot y}, \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) \cdot x, x\right) \cdot \frac{y}{x} \]

       8. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(y \cdot y, \color{blue}{\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) \cdot x}, x\right) \cdot \frac{y}{x} \]

       9. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(y \cdot y, \color{blue}{\left(\frac{1}{120} \cdot {y}^{2} + \frac{1}{6}\right)} \cdot x, x\right) \cdot \frac{y}{x} \]

       10. *-commutative N/A

           \[\leadsto \mathsf{fma}\left(y \cdot y, \left(\color{blue}{{y}^{2} \cdot \frac{1}{120}} + \frac{1}{6}\right) \cdot x, x\right) \cdot \frac{y}{x} \]

       11. lower-fma.f64 N/A

           \[\leadsto \mathsf{fma}\left(y \cdot y, \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{1}{120}, \frac{1}{6}\right)} \cdot x, x\right) \cdot \frac{y}{x} \]

       12. unpow2 N/A

           \[\leadsto \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{120}, \frac{1}{6}\right) \cdot x, x\right) \cdot \frac{y}{x} \]

       13. lower-*.f64 69.9

           \[\leadsto \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, 0.008333333333333333, 0.16666666666666666\right) \cdot x, x\right) \cdot \frac{y}{x} \]

   11. Simplified 69.9%

       \[\leadsto \color{blue}{\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right) \cdot x, x\right)} \cdot \frac{y}{x} \]

   12. Taylor expanded in y around 0

       \[\leadsto \color{blue}{y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} \]
   13. Step-by-step derivation

       1. distribute-rgt-in N/A

          \[\leadsto \color{blue}{1 \cdot y + \left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) \cdot y} \]

       2. *-lft-identity N/A

          \[\leadsto \color{blue}{y} + \left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) \cdot y \]

       3. +-commutative N/A

          \[\leadsto \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) \cdot y + y} \]

       4. associate-*l* N/A

          \[\leadsto \color{blue}{{y}^{2} \cdot \left(\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) \cdot y\right)} + y \]

       5. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left({y}^{2}, \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) \cdot y, y\right)} \]

       6. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{y \cdot y}, \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) \cdot y, y\right) \]

       7. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{y \cdot y}, \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) \cdot y, y\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(y \cdot y, \color{blue}{y \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)}, y\right) \]

       9. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(y \cdot y, \color{blue}{y \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)}, y\right) \]

       10. +-commutative N/A

           \[\leadsto \mathsf{fma}\left(y \cdot y, y \cdot \color{blue}{\left(\frac{1}{120} \cdot {y}^{2} + \frac{1}{6}\right)}, y\right) \]

       11. *-commutative N/A

           \[\leadsto \mathsf{fma}\left(y \cdot y, y \cdot \left(\color{blue}{{y}^{2} \cdot \frac{1}{120}} + \frac{1}{6}\right), y\right) \]

       12. lower-fma.f64 N/A

           \[\leadsto \mathsf{fma}\left(y \cdot y, y \cdot \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{1}{120}, \frac{1}{6}\right)}, y\right) \]

       13. unpow2 N/A

           \[\leadsto \mathsf{fma}\left(y \cdot y, y \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{120}, \frac{1}{6}\right), y\right) \]

       14. lower-*.f64 61.8

           \[\leadsto \mathsf{fma}\left(y \cdot y, y \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, 0.008333333333333333, 0.16666666666666666\right), y\right) \]

   14. Simplified 61.8%

       \[\leadsto \color{blue}{\mathsf{fma}\left(y \cdot y, y \cdot \mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), y\right)} \]

   if 4.1e52 < x

   1. Initial program 99.9%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{y} - \frac{1}{e^{y}}\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{y} - \frac{1}{e^{y}}\right)} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(e^{y} - \frac{1}{e^{y}}\right)} \]

      3. lower-exp.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{e^{y}} - \frac{1}{e^{y}}\right) \]

      4. rec-exp N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \]

      5. lower-exp.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \]

      6. lower-neg.f64 68.5

         \[\leadsto 0.5 \cdot \left(e^{y} - e^{\color{blue}{-y}}\right) \]

   5. Simplified 68.5%

      \[\leadsto \color{blue}{0.5 \cdot \left(e^{y} - e^{-y}\right)} \]

   6. Taylor expanded in y around 0

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(y \cdot \left(2 + {y}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {y}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(y \cdot \left(2 + {y}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {y}^{2}\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \left(y \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {y}^{2}\right) + 2\right)}\right) \]

      3. unpow2 N/A

         \[\leadsto \frac{1}{2} \cdot \left(y \cdot \left(\color{blue}{\left(y \cdot y\right)} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {y}^{2}\right) + 2\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \frac{1}{2} \cdot \left(y \cdot \left(\color{blue}{y \cdot \left(y \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {y}^{2}\right)\right)} + 2\right)\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(y \cdot \color{blue}{\mathsf{fma}\left(y, y \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {y}^{2}\right), 2\right)}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(y \cdot \mathsf{fma}\left(y, \color{blue}{y \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {y}^{2}\right)}, 2\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \left(y \cdot \mathsf{fma}\left(y, y \cdot \color{blue}{\left(\frac{1}{60} \cdot {y}^{2} + \frac{1}{3}\right)}, 2\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \left(y \cdot \mathsf{fma}\left(y, y \cdot \left(\color{blue}{{y}^{2} \cdot \frac{1}{60}} + \frac{1}{3}\right), 2\right)\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(y \cdot \mathsf{fma}\left(y, y \cdot \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{1}{60}, \frac{1}{3}\right)}, 2\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \frac{1}{2} \cdot \left(y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{60}, \frac{1}{3}\right), 2\right)\right) \]

      11. lower-*.f64 32.5

          \[\leadsto 0.5 \cdot \left(y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, 0.016666666666666666, 0.3333333333333333\right), 2\right)\right) \]

   8. Simplified 32.5%

      \[\leadsto 0.5 \cdot \color{blue}{\left(y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.016666666666666666, 0.3333333333333333\right), 2\right)\right)} \]

   9. Taylor expanded in y around inf

      \[\leadsto \frac{1}{2} \cdot \left(y \cdot \color{blue}{\left(\frac{1}{60} \cdot {y}^{4}\right)}\right) \]
   10. Step-by-step derivation

       1. metadata-eval N/A

          \[\leadsto \frac{1}{2} \cdot \left(y \cdot \left(\frac{1}{60} \cdot {y}^{\color{blue}{\left(2 \cdot 2\right)}}\right)\right) \]

       2. pow-sqr N/A

          \[\leadsto \frac{1}{2} \cdot \left(y \cdot \left(\frac{1}{60} \cdot \color{blue}{\left({y}^{2} \cdot {y}^{2}\right)}\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \frac{1}{2} \cdot \left(y \cdot \color{blue}{\left(\left(\frac{1}{60} \cdot {y}^{2}\right) \cdot {y}^{2}\right)}\right) \]

       4. *-commutative N/A

          \[\leadsto \frac{1}{2} \cdot \left(y \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{60} \cdot {y}^{2}\right)\right)}\right) \]

       5. unpow2 N/A

          \[\leadsto \frac{1}{2} \cdot \left(y \cdot \left(\color{blue}{\left(y \cdot y\right)} \cdot \left(\frac{1}{60} \cdot {y}^{2}\right)\right)\right) \]

       6. associate-*l* N/A

          \[\leadsto \frac{1}{2} \cdot \left(y \cdot \color{blue}{\left(y \cdot \left(y \cdot \left(\frac{1}{60} \cdot {y}^{2}\right)\right)\right)}\right) \]

       7. *-commutative N/A

          \[\leadsto \frac{1}{2} \cdot \left(y \cdot \left(y \cdot \color{blue}{\left(\left(\frac{1}{60} \cdot {y}^{2}\right) \cdot y\right)}\right)\right) \]

       8. associate-*r* N/A

          \[\leadsto \frac{1}{2} \cdot \left(y \cdot \left(y \cdot \color{blue}{\left(\frac{1}{60} \cdot \left({y}^{2} \cdot y\right)\right)}\right)\right) \]

       9. unpow2 N/A

          \[\leadsto \frac{1}{2} \cdot \left(y \cdot \left(y \cdot \left(\frac{1}{60} \cdot \left(\color{blue}{\left(y \cdot y\right)} \cdot y\right)\right)\right)\right) \]

       10. unpow3 N/A

           \[\leadsto \frac{1}{2} \cdot \left(y \cdot \left(y \cdot \left(\frac{1}{60} \cdot \color{blue}{{y}^{3}}\right)\right)\right) \]

       11. lower-*.f64 N/A

           \[\leadsto \frac{1}{2} \cdot \left(y \cdot \color{blue}{\left(y \cdot \left(\frac{1}{60} \cdot {y}^{3}\right)\right)}\right) \]

       12. unpow3 N/A

           \[\leadsto \frac{1}{2} \cdot \left(y \cdot \left(y \cdot \left(\frac{1}{60} \cdot \color{blue}{\left(\left(y \cdot y\right) \cdot y\right)}\right)\right)\right) \]

       13. unpow2 N/A

           \[\leadsto \frac{1}{2} \cdot \left(y \cdot \left(y \cdot \left(\frac{1}{60} \cdot \left(\color{blue}{{y}^{2}} \cdot y\right)\right)\right)\right) \]

       14. associate-*r* N/A

           \[\leadsto \frac{1}{2} \cdot \left(y \cdot \left(y \cdot \color{blue}{\left(\left(\frac{1}{60} \cdot {y}^{2}\right) \cdot y\right)}\right)\right) \]

       15. *-commutative N/A

           \[\leadsto \frac{1}{2} \cdot \left(y \cdot \left(y \cdot \color{blue}{\left(y \cdot \left(\frac{1}{60} \cdot {y}^{2}\right)\right)}\right)\right) \]

       16. lower-*.f64 N/A

           \[\leadsto \frac{1}{2} \cdot \left(y \cdot \left(y \cdot \color{blue}{\left(y \cdot \left(\frac{1}{60} \cdot {y}^{2}\right)\right)}\right)\right) \]

       17. *-commutative N/A

           \[\leadsto \frac{1}{2} \cdot \left(y \cdot \left(y \cdot \left(y \cdot \color{blue}{\left({y}^{2} \cdot \frac{1}{60}\right)}\right)\right)\right) \]

       18. lower-*.f64 N/A

           \[\leadsto \frac{1}{2} \cdot \left(y \cdot \left(y \cdot \left(y \cdot \color{blue}{\left({y}^{2} \cdot \frac{1}{60}\right)}\right)\right)\right) \]

       19. unpow2 N/A

           \[\leadsto \frac{1}{2} \cdot \left(y \cdot \left(y \cdot \left(y \cdot \left(\color{blue}{\left(y \cdot y\right)} \cdot \frac{1}{60}\right)\right)\right)\right) \]

       20. lower-*.f64 62.1

           \[\leadsto 0.5 \cdot \left(y \cdot \left(y \cdot \left(y \cdot \left(\color{blue}{\left(y \cdot y\right)} \cdot 0.016666666666666666\right)\right)\right)\right) \]

   11. Simplified 62.1%

       \[\leadsto 0.5 \cdot \left(y \cdot \color{blue}{\left(y \cdot \left(y \cdot \left(\left(y \cdot y\right) \cdot 0.016666666666666666\right)\right)\right)}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 34.5% accurate, 12.0× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 4.1 \cdot 10^{+52}:\\ \;\;\;\;y\_m\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(y\_m + 1\right) + -1\right)\\ \end{array} \end{array} \]

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 (<= x 4.1e+52) y_m (* 0.5 (+ (+ y_m 1.0) -1.0)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m) {
	double tmp;
	if (x <= 4.1e+52) {
		tmp = y_m;
	} else {
		tmp = 0.5 * ((y_m + 1.0) + -1.0);
	}
	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 (x <= 4.1d+52) then
        tmp = y_m
    else
        tmp = 0.5d0 * ((y_m + 1.0d0) + (-1.0d0))
    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 (x <= 4.1e+52) {
		tmp = y_m;
	} else {
		tmp = 0.5 * ((y_m + 1.0) + -1.0);
	}
	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 x <= 4.1e+52:
		tmp = y_m
	else:
		tmp = 0.5 * ((y_m + 1.0) + -1.0)
	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 (x <= 4.1e+52)
		tmp = y_m;
	else
		tmp = Float64(0.5 * Float64(Float64(y_m + 1.0) + -1.0));
	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 (x <= 4.1e+52)
		tmp = y_m;
	else
		tmp = 0.5 * ((y_m + 1.0) + -1.0);
	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[x, 4.1e+52], y$95$m, N[(0.5 * N[(N[(y$95$m + 1.0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 4.1e52

   1. Initial program 87.0%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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. lower-*.f64 N/A

         \[\leadsto \color{blue}{y \cdot \frac{\sin x}{x}} \]

      3. lower-/.f64 N/A

         \[\leadsto y \cdot \color{blue}{\frac{\sin x}{x}} \]

      4. lower-sin.f64 53.4

         \[\leadsto y \cdot \frac{\color{blue}{\sin x}}{x} \]

   5. Simplified 53.4%

      \[\leadsto \color{blue}{y \cdot \frac{\sin x}{x}} \]

   6. Taylor expanded in x around 0

      \[\leadsto y \cdot \color{blue}{1} \]
   7. Step-by-step derivation

   8. Simplified 32.2%

      \[\leadsto y \cdot \color{blue}{1} \]

   if 4.1e52 < x

   1. Initial program 99.9%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{y} - \frac{1}{e^{y}}\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{y} - \frac{1}{e^{y}}\right)} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(e^{y} - \frac{1}{e^{y}}\right)} \]

      3. lower-exp.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{e^{y}} - \frac{1}{e^{y}}\right) \]

      4. rec-exp N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \]

      5. lower-exp.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \]

      6. lower-neg.f64 68.5

         \[\leadsto 0.5 \cdot \left(e^{y} - e^{\color{blue}{-y}}\right) \]

   5. Simplified 68.5%

      \[\leadsto \color{blue}{0.5 \cdot \left(e^{y} - e^{-y}\right)} \]

   6. Taylor expanded in y around 0

      \[\leadsto \frac{1}{2} \cdot \left(e^{y} - \color{blue}{1}\right) \]
   7. Step-by-step derivation

   8. Simplified 50.2%

      \[\leadsto 0.5 \cdot \left(e^{y} - \color{blue}{1}\right) \]

   9. Taylor expanded in y around 0

      \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{\left(1 + y\right)} - 1\right) \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{\left(y + 1\right)} - 1\right) \]

       2. lower-+.f64 35.9

          \[\leadsto 0.5 \cdot \left(\color{blue}{\left(y + 1\right)} - 1\right) \]

   11. Simplified 35.9%

       \[\leadsto 0.5 \cdot \left(\color{blue}{\left(y + 1\right)} - 1\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 33.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.1 \cdot 10^{+52}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(y + 1\right) + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 34.3% accurate, 14.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 1.06 \cdot 10^{+64}:\\ \;\;\;\;y\_m\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(1 + -1\right)\\ \end{array} \end{array} \]

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 (<= x 1.06e+64) y_m (* 0.5 (+ 1.0 -1.0)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m) {
	double tmp;
	if (x <= 1.06e+64) {
		tmp = y_m;
	} else {
		tmp = 0.5 * (1.0 + -1.0);
	}
	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 (x <= 1.06d+64) then
        tmp = y_m
    else
        tmp = 0.5d0 * (1.0d0 + (-1.0d0))
    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 (x <= 1.06e+64) {
		tmp = y_m;
	} else {
		tmp = 0.5 * (1.0 + -1.0);
	}
	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 x <= 1.06e+64:
		tmp = y_m
	else:
		tmp = 0.5 * (1.0 + -1.0)
	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 (x <= 1.06e+64)
		tmp = y_m;
	else
		tmp = Float64(0.5 * Float64(1.0 + -1.0));
	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 (x <= 1.06e+64)
		tmp = y_m;
	else
		tmp = 0.5 * (1.0 + -1.0);
	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[x, 1.06e+64], y$95$m, N[(0.5 * N[(1.0 + -1.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.06e64

   1. Initial program 87.2%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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. lower-*.f64 N/A

         \[\leadsto \color{blue}{y \cdot \frac{\sin x}{x}} \]

      3. lower-/.f64 N/A

         \[\leadsto y \cdot \color{blue}{\frac{\sin x}{x}} \]

      4. lower-sin.f64 52.9

         \[\leadsto y \cdot \frac{\color{blue}{\sin x}}{x} \]

   5. Simplified 52.9%

      \[\leadsto \color{blue}{y \cdot \frac{\sin x}{x}} \]

   6. Taylor expanded in x around 0

      \[\leadsto y \cdot \color{blue}{1} \]
   7. Step-by-step derivation

   8. Simplified 31.7%

      \[\leadsto y \cdot \color{blue}{1} \]

   if 1.06e64 < x

   1. Initial program 99.9%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{y} - \frac{1}{e^{y}}\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{y} - \frac{1}{e^{y}}\right)} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(e^{y} - \frac{1}{e^{y}}\right)} \]

      3. lower-exp.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{e^{y}} - \frac{1}{e^{y}}\right) \]

      4. rec-exp N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \]

      5. lower-exp.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \]

      6. lower-neg.f64 71.6

         \[\leadsto 0.5 \cdot \left(e^{y} - e^{\color{blue}{-y}}\right) \]

   5. Simplified 71.6%

      \[\leadsto \color{blue}{0.5 \cdot \left(e^{y} - e^{-y}\right)} \]

   6. Taylor expanded in y around 0

      \[\leadsto \frac{1}{2} \cdot \left(e^{y} - \color{blue}{1}\right) \]
   7. Step-by-step derivation

   8. Simplified 53.7%

      \[\leadsto 0.5 \cdot \left(e^{y} - \color{blue}{1}\right) \]

   9. Taylor expanded in y around 0

      \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{1} - 1\right) \]
   10. Step-by-step derivation

   11. Simplified 37.2%

       \[\leadsto 0.5 \cdot \left(\color{blue}{1} - 1\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 32.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.06 \cdot 10^{+64}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(1 + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 8.2% accurate, 36.2× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \left(y\_m \cdot 0.5\right) \end{array} \]

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 0.5)))

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 * 0.5);
}

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 * 0.5d0)
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 * 0.5);
}

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 * 0.5)

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 * 0.5))
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 * 0.5);
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 * 0.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 89.9%

   \[\frac{\sin x \cdot \sinh y}{x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{y} - \frac{1}{e^{y}}\right)} \]
4. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{y} - \frac{1}{e^{y}}\right)} \]

   2. lower--.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(e^{y} - \frac{1}{e^{y}}\right)} \]

   3. lower-exp.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{e^{y}} - \frac{1}{e^{y}}\right) \]

   4. rec-exp N/A

      \[\leadsto \frac{1}{2} \cdot \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \]

   5. lower-exp.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(e^{y} - \color{blue}{e^{\mathsf{neg}\left(y\right)}}\right) \]

   6. lower-neg.f64 52.7

      \[\leadsto 0.5 \cdot \left(e^{y} - e^{\color{blue}{-y}}\right) \]

5. Simplified 52.7%

   \[\leadsto \color{blue}{0.5 \cdot \left(e^{y} - e^{-y}\right)} \]

6. Taylor expanded in y around 0

   \[\leadsto \frac{1}{2} \cdot \left(e^{y} - \color{blue}{1}\right) \]
7. Step-by-step derivation

8. Simplified 34.4%

   \[\leadsto 0.5 \cdot \left(e^{y} - \color{blue}{1}\right) \]

9. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\frac{1}{2} \cdot y} \]
10. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \color{blue}{y \cdot \frac{1}{2}} \]

    2. lower-*.f64 8.0

       \[\leadsto \color{blue}{y \cdot 0.5} \]

11. Simplified 8.0%

    \[\leadsto \color{blue}{y \cdot 0.5} \]
12. Add Preprocessing

Alternative 24: 28.0% 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 89.9%

   \[\frac{\sin x \cdot \sinh y}{x} \]
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. lower-*.f64 N/A

      \[\leadsto \color{blue}{y \cdot \frac{\sin x}{x}} \]

   3. lower-/.f64 N/A

      \[\leadsto y \cdot \color{blue}{\frac{\sin x}{x}} \]

   4. lower-sin.f64 53.0

      \[\leadsto y \cdot \frac{\color{blue}{\sin x}}{x} \]

5. Simplified 53.0%

   \[\leadsto \color{blue}{y \cdot \frac{\sin x}{x}} \]

6. Taylor expanded in x around 0

   \[\leadsto y \cdot \color{blue}{1} \]
7. Step-by-step derivation

8. Simplified 26.0%

   \[\leadsto y \cdot \color{blue}{1} \]

9. Final simplification 26.0%

   \[\leadsto y \]
10. 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 2024215 
(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))