Linear.Quaternion:$ccosh from linear-1.19.1.3

Percentage Accurate: 89.5% → 99.8%

Time: 11.8s

Alternatives: 21

Speedup: 1.0×

• Could not converge on a ground truth

  1. x = 1.8035088267014102e+307
  2. y = 2.6320985979794366e+107

• 51.3% 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: 89.5% 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: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sinh y \cdot \left(\frac{1}{x} \cdot \sin x\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* (sinh y) (* (/ 1.0 x) (sin x))))

C

double code(double x, double y) {
	return sinh(y) * ((1.0 / x) * sin(x));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = sinh(y) * ((1.0d0 / x) * sin(x))
end function

Java

public static double code(double x, double y) {
	return Math.sinh(y) * ((1.0 / x) * Math.sin(x));
}

Python

def code(x, y):
	return math.sinh(y) * ((1.0 / x) * math.sin(x))

Julia

function code(x, y)
	return Float64(sinh(y) * Float64(Float64(1.0 / x) * sin(x)))
end

MATLAB

function tmp = code(x, y)
	tmp = sinh(y) * ((1.0 / x) * sin(x));
end

Wolfram

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

Derivation

1. Initial program 89.9%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. remove-double-neg N/A

      \[\leadsto \frac{1}{x} \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\sin x \cdot \sinh y\right)\right)\right)\right)} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{1}{x} \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\sin x \cdot \sinh y}\right)\right)\right)\right) \]

   6. distribute-rgt-neg-in N/A

      \[\leadsto \frac{1}{x} \cdot \left(\mathsf{neg}\left(\color{blue}{\sin x \cdot \left(\mathsf{neg}\left(\sinh y\right)\right)}\right)\right) \]

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

      \[\leadsto \frac{1}{x} \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\sin x\right)\right) \cdot \left(\mathsf{neg}\left(\sinh y\right)\right)\right)} \]

   8. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\frac{1}{x} \cdot \left(\mathsf{neg}\left(\sin x\right)\right)\right) \cdot \left(\mathsf{neg}\left(\sinh y\right)\right)} \]

   9. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{1}{x} \cdot \left(\mathsf{neg}\left(\sin x\right)\right)\right) \cdot \left(\mathsf{neg}\left(\sinh y\right)\right)} \]

   10. lower-*.f64 N/A

       \[\leadsto \color{blue}{\left(\frac{1}{x} \cdot \left(\mathsf{neg}\left(\sin x\right)\right)\right)} \cdot \left(\mathsf{neg}\left(\sinh y\right)\right) \]

   11. lower-/.f64 N/A

       \[\leadsto \left(\color{blue}{\frac{1}{x}} \cdot \left(\mathsf{neg}\left(\sin x\right)\right)\right) \cdot \left(\mathsf{neg}\left(\sinh y\right)\right) \]

   12. lower-neg.f64 N/A

       \[\leadsto \left(\frac{1}{x} \cdot \color{blue}{\left(\mathsf{neg}\left(\sin x\right)\right)}\right) \cdot \left(\mathsf{neg}\left(\sinh y\right)\right) \]

   13. lower-neg.f64 99.9

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

4. Applied rewrites 99.9%

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

5. Final simplification 99.9%

   \[\leadsto \sinh y \cdot \left(\frac{1}{x} \cdot \sin x\right) \]
6. Add Preprocessing

Alternative 2: 74.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin x \cdot \sinh y}{x}\\ t_1 := -\sinh y\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;t\_1 \cdot \left(0.16666666666666666 \cdot \left(x \cdot x\right)\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sin x}{x} \cdot \mathsf{fma}\left(y \cdot y, y \cdot 0.16666666666666666, y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot -0.008333333333333333, 0.16666666666666666\right), -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sin x) (sinh y)) x)) (t_1 (- (sinh y))))
   (if (<= t_0 (- INFINITY))
     (* t_1 (* 0.16666666666666666 (* x x)))
     (if (<= t_0 5e-7)
       (* (/ (sin x) x) (fma (* y y) (* y 0.16666666666666666) y))
       (*
        t_1
        (fma
         (* x x)
         (fma x (* x -0.008333333333333333) 0.16666666666666666)
         -1.0))))))

C

double code(double x, double y) {
	double t_0 = (sin(x) * sinh(y)) / x;
	double t_1 = -sinh(y);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = t_1 * (0.16666666666666666 * (x * x));
	} else if (t_0 <= 5e-7) {
		tmp = (sin(x) / x) * fma((y * y), (y * 0.16666666666666666), y);
	} else {
		tmp = t_1 * fma((x * x), fma(x, (x * -0.008333333333333333), 0.16666666666666666), -1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(sin(x) * sinh(y)) / x)
	t_1 = Float64(-sinh(y))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(t_1 * Float64(0.16666666666666666 * Float64(x * x)));
	elseif (t_0 <= 5e-7)
		tmp = Float64(Float64(sin(x) / x) * fma(Float64(y * y), Float64(y * 0.16666666666666666), y));
	else
		tmp = Float64(t_1 * fma(Float64(x * x), fma(x, Float64(x * -0.008333333333333333), 0.16666666666666666), -1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sin[x], $MachinePrecision] * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, Block[{t$95$1 = (-N[Sinh[y], $MachinePrecision])}, If[LessEqual[t$95$0, (-Infinity)], N[(t$95$1 * N[(0.16666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e-7], N[(N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(y * 0.16666666666666666), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * -0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + -1.0), $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. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. remove-double-neg N/A

         \[\leadsto \frac{1}{x} \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\sin x \cdot \sinh y\right)\right)\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{1}{x} \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\sin x \cdot \sinh y}\right)\right)\right)\right) \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto \frac{1}{x} \cdot \left(\mathsf{neg}\left(\color{blue}{\sin x \cdot \left(\mathsf{neg}\left(\sinh y\right)\right)}\right)\right) \]

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

         \[\leadsto \frac{1}{x} \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\sin x\right)\right) \cdot \left(\mathsf{neg}\left(\sinh y\right)\right)\right)} \]

      8. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{x} \cdot \left(\mathsf{neg}\left(\sin x\right)\right)\right) \cdot \left(\mathsf{neg}\left(\sinh y\right)\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{x} \cdot \left(\mathsf{neg}\left(\sin x\right)\right)\right) \cdot \left(\mathsf{neg}\left(\sinh y\right)\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\frac{1}{x} \cdot \left(\mathsf{neg}\left(\sin x\right)\right)\right)} \cdot \left(\mathsf{neg}\left(\sinh y\right)\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \left(\color{blue}{\frac{1}{x}} \cdot \left(\mathsf{neg}\left(\sin x\right)\right)\right) \cdot \left(\mathsf{neg}\left(\sinh y\right)\right) \]

      12. lower-neg.f64 N/A

          \[\leadsto \left(\frac{1}{x} \cdot \color{blue}{\left(\mathsf{neg}\left(\sin x\right)\right)}\right) \cdot \left(\mathsf{neg}\left(\sinh y\right)\right) \]

      13. lower-neg.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 62.7

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

   7. Applied rewrites 62.7%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 26.9%

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

   if -inf.0 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 4.99999999999999977e-7

   1. Initial program 79.6%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*r/ N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. associate-*r* N/A

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

      11. associate-/l* N/A

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

      12. distribute-rgt-out N/A

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

      13. *-rgt-identity N/A

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

      14. distribute-lft-in N/A

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

   5. Applied rewrites 98.5%

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

   if 4.99999999999999977e-7 < (/.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. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. remove-double-neg N/A

         \[\leadsto \frac{1}{x} \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\sin x \cdot \sinh y\right)\right)\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{1}{x} \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\sin x \cdot \sinh y}\right)\right)\right)\right) \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto \frac{1}{x} \cdot \left(\mathsf{neg}\left(\color{blue}{\sin x \cdot \left(\mathsf{neg}\left(\sinh y\right)\right)}\right)\right) \]

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

         \[\leadsto \frac{1}{x} \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\sin x\right)\right) \cdot \left(\mathsf{neg}\left(\sinh y\right)\right)\right)} \]

      8. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{x} \cdot \left(\mathsf{neg}\left(\sin x\right)\right)\right) \cdot \left(\mathsf{neg}\left(\sinh y\right)\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{x} \cdot \left(\mathsf{neg}\left(\sin x\right)\right)\right) \cdot \left(\mathsf{neg}\left(\sinh y\right)\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\frac{1}{x} \cdot \left(\mathsf{neg}\left(\sin x\right)\right)\right)} \cdot \left(\mathsf{neg}\left(\sinh y\right)\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \left(\color{blue}{\frac{1}{x}} \cdot \left(\mathsf{neg}\left(\sin x\right)\right)\right) \cdot \left(\mathsf{neg}\left(\sinh y\right)\right) \]

      12. lower-neg.f64 N/A

          \[\leadsto \left(\frac{1}{x} \cdot \color{blue}{\left(\mathsf{neg}\left(\sin x\right)\right)}\right) \cdot \left(\mathsf{neg}\left(\sinh y\right)\right) \]

      13. lower-neg.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 77.8

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

   7. Applied rewrites 77.8%

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

4. Final simplification 74.6%

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

Alternative 3: 74.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin x \cdot \sinh y}{x}\\ t_1 := -\sinh y\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;t\_1 \cdot \left(0.16666666666666666 \cdot \left(x \cdot x\right)\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-41}:\\ \;\;\;\;y \cdot \frac{\sin x}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot -0.008333333333333333, 0.16666666666666666\right), -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sin x) (sinh y)) x)) (t_1 (- (sinh y))))
   (if (<= t_0 (- INFINITY))
     (* t_1 (* 0.16666666666666666 (* x x)))
     (if (<= t_0 5e-41)
       (* y (/ (sin x) x))
       (*
        t_1
        (fma
         (* x x)
         (fma x (* x -0.008333333333333333) 0.16666666666666666)
         -1.0))))))

C

double code(double x, double y) {
	double t_0 = (sin(x) * sinh(y)) / x;
	double t_1 = -sinh(y);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = t_1 * (0.16666666666666666 * (x * x));
	} else if (t_0 <= 5e-41) {
		tmp = y * (sin(x) / x);
	} else {
		tmp = t_1 * fma((x * x), fma(x, (x * -0.008333333333333333), 0.16666666666666666), -1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(sin(x) * sinh(y)) / x)
	t_1 = Float64(-sinh(y))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(t_1 * Float64(0.16666666666666666 * Float64(x * x)));
	elseif (t_0 <= 5e-41)
		tmp = Float64(y * Float64(sin(x) / x));
	else
		tmp = Float64(t_1 * fma(Float64(x * x), fma(x, Float64(x * -0.008333333333333333), 0.16666666666666666), -1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sin[x], $MachinePrecision] * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, Block[{t$95$1 = (-N[Sinh[y], $MachinePrecision])}, If[LessEqual[t$95$0, (-Infinity)], N[(t$95$1 * N[(0.16666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e-41], N[(y * N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * -0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + -1.0), $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. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. remove-double-neg N/A

         \[\leadsto \frac{1}{x} \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\sin x \cdot \sinh y\right)\right)\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{1}{x} \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\sin x \cdot \sinh y}\right)\right)\right)\right) \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto \frac{1}{x} \cdot \left(\mathsf{neg}\left(\color{blue}{\sin x \cdot \left(\mathsf{neg}\left(\sinh y\right)\right)}\right)\right) \]

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

         \[\leadsto \frac{1}{x} \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\sin x\right)\right) \cdot \left(\mathsf{neg}\left(\sinh y\right)\right)\right)} \]

      8. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{x} \cdot \left(\mathsf{neg}\left(\sin x\right)\right)\right) \cdot \left(\mathsf{neg}\left(\sinh y\right)\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{x} \cdot \left(\mathsf{neg}\left(\sin x\right)\right)\right) \cdot \left(\mathsf{neg}\left(\sinh y\right)\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\frac{1}{x} \cdot \left(\mathsf{neg}\left(\sin x\right)\right)\right)} \cdot \left(\mathsf{neg}\left(\sinh y\right)\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \left(\color{blue}{\frac{1}{x}} \cdot \left(\mathsf{neg}\left(\sin x\right)\right)\right) \cdot \left(\mathsf{neg}\left(\sinh y\right)\right) \]

      12. lower-neg.f64 N/A

          \[\leadsto \left(\frac{1}{x} \cdot \color{blue}{\left(\mathsf{neg}\left(\sin x\right)\right)}\right) \cdot \left(\mathsf{neg}\left(\sinh y\right)\right) \]

      13. lower-neg.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 62.7

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

   7. Applied rewrites 62.7%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 26.9%

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

   if -inf.0 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 4.9999999999999996e-41

   1. Initial program 78.9%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 98.3

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

   5. Applied rewrites 98.3%

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

   if 4.9999999999999996e-41 < (/.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. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. remove-double-neg N/A

         \[\leadsto \frac{1}{x} \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\sin x \cdot \sinh y\right)\right)\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{1}{x} \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\sin x \cdot \sinh y}\right)\right)\right)\right) \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto \frac{1}{x} \cdot \left(\mathsf{neg}\left(\color{blue}{\sin x \cdot \left(\mathsf{neg}\left(\sinh y\right)\right)}\right)\right) \]

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

         \[\leadsto \frac{1}{x} \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\sin x\right)\right) \cdot \left(\mathsf{neg}\left(\sinh y\right)\right)\right)} \]

      8. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{x} \cdot \left(\mathsf{neg}\left(\sin x\right)\right)\right) \cdot \left(\mathsf{neg}\left(\sinh y\right)\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{x} \cdot \left(\mathsf{neg}\left(\sin x\right)\right)\right) \cdot \left(\mathsf{neg}\left(\sinh y\right)\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\frac{1}{x} \cdot \left(\mathsf{neg}\left(\sin x\right)\right)\right)} \cdot \left(\mathsf{neg}\left(\sinh y\right)\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \left(\color{blue}{\frac{1}{x}} \cdot \left(\mathsf{neg}\left(\sin x\right)\right)\right) \cdot \left(\mathsf{neg}\left(\sinh y\right)\right) \]

      12. lower-neg.f64 N/A

          \[\leadsto \left(\frac{1}{x} \cdot \color{blue}{\left(\mathsf{neg}\left(\sin x\right)\right)}\right) \cdot \left(\mathsf{neg}\left(\sinh y\right)\right) \]

      13. lower-neg.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 79.1

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

   7. Applied rewrites 79.1%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin x \cdot \sinh y}{x} \leq -\infty:\\ \;\;\;\;\left(-\sinh y\right) \cdot \left(0.16666666666666666 \cdot \left(x \cdot x\right)\right)\\ \mathbf{elif}\;\frac{\sin x \cdot \sinh y}{x} \leq 5 \cdot 10^{-41}:\\ \;\;\;\;y \cdot \frac{\sin x}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(-\sinh y\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot -0.008333333333333333, 0.16666666666666666\right), -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin x \cdot \sinh y}{x}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(-\sinh y\right) \cdot \left(0.16666666666666666 \cdot \left(x \cdot x\right)\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-41}:\\ \;\;\;\;y \cdot \frac{\sin x}{x}\\ \mathbf{else}:\\ \;\;\;\;\sinh y \cdot \left(--1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sin x) (sinh y)) x)))
   (if (<= t_0 (- INFINITY))
     (* (- (sinh y)) (* 0.16666666666666666 (* x x)))
     (if (<= t_0 5e-41) (* y (/ (sin x) x)) (* (sinh y) (- -1.0))))))

C

double code(double x, double y) {
	double t_0 = (sin(x) * sinh(y)) / x;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = -sinh(y) * (0.16666666666666666 * (x * x));
	} else if (t_0 <= 5e-41) {
		tmp = y * (sin(x) / x);
	} else {
		tmp = sinh(y) * -(-1.0);
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double t_0 = (Math.sin(x) * Math.sinh(y)) / x;
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = -Math.sinh(y) * (0.16666666666666666 * (x * x));
	} else if (t_0 <= 5e-41) {
		tmp = y * (Math.sin(x) / x);
	} else {
		tmp = Math.sinh(y) * -(-1.0);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (math.sin(x) * math.sinh(y)) / x
	tmp = 0
	if t_0 <= -math.inf:
		tmp = -math.sinh(y) * (0.16666666666666666 * (x * x))
	elif t_0 <= 5e-41:
		tmp = y * (math.sin(x) / x)
	else:
		tmp = math.sinh(y) * -(-1.0)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(sin(x) * sinh(y)) / x)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(-sinh(y)) * Float64(0.16666666666666666 * Float64(x * x)));
	elseif (t_0 <= 5e-41)
		tmp = Float64(y * Float64(sin(x) / x));
	else
		tmp = Float64(sinh(y) * Float64(-(-1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (sin(x) * sinh(y)) / x;
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = -sinh(y) * (0.16666666666666666 * (x * x));
	elseif (t_0 <= 5e-41)
		tmp = y * (sin(x) / x);
	else
		tmp = sinh(y) * -(-1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sin[x], $MachinePrecision] * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[((-N[Sinh[y], $MachinePrecision]) * N[(0.16666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e-41], N[(y * N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[Sinh[y], $MachinePrecision] * (--1.0)), $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. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. remove-double-neg N/A

         \[\leadsto \frac{1}{x} \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\sin x \cdot \sinh y\right)\right)\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{1}{x} \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\sin x \cdot \sinh y}\right)\right)\right)\right) \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto \frac{1}{x} \cdot \left(\mathsf{neg}\left(\color{blue}{\sin x \cdot \left(\mathsf{neg}\left(\sinh y\right)\right)}\right)\right) \]

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

         \[\leadsto \frac{1}{x} \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\sin x\right)\right) \cdot \left(\mathsf{neg}\left(\sinh y\right)\right)\right)} \]

      8. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{x} \cdot \left(\mathsf{neg}\left(\sin x\right)\right)\right) \cdot \left(\mathsf{neg}\left(\sinh y\right)\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{x} \cdot \left(\mathsf{neg}\left(\sin x\right)\right)\right) \cdot \left(\mathsf{neg}\left(\sinh y\right)\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\frac{1}{x} \cdot \left(\mathsf{neg}\left(\sin x\right)\right)\right)} \cdot \left(\mathsf{neg}\left(\sinh y\right)\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \left(\color{blue}{\frac{1}{x}} \cdot \left(\mathsf{neg}\left(\sin x\right)\right)\right) \cdot \left(\mathsf{neg}\left(\sinh y\right)\right) \]

      12. lower-neg.f64 N/A

          \[\leadsto \left(\frac{1}{x} \cdot \color{blue}{\left(\mathsf{neg}\left(\sin x\right)\right)}\right) \cdot \left(\mathsf{neg}\left(\sinh y\right)\right) \]

      13. lower-neg.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 62.7

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

   7. Applied rewrites 62.7%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 26.9%

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

   if -inf.0 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 4.9999999999999996e-41

   1. Initial program 78.9%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 98.3

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

   5. Applied rewrites 98.3%

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

   if 4.9999999999999996e-41 < (/.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. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. remove-double-neg N/A

         \[\leadsto \frac{1}{x} \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\sin x \cdot \sinh y\right)\right)\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{1}{x} \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\sin x \cdot \sinh y}\right)\right)\right)\right) \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto \frac{1}{x} \cdot \left(\mathsf{neg}\left(\color{blue}{\sin x \cdot \left(\mathsf{neg}\left(\sinh y\right)\right)}\right)\right) \]

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

         \[\leadsto \frac{1}{x} \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\sin x\right)\right) \cdot \left(\mathsf{neg}\left(\sinh y\right)\right)\right)} \]

      8. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{x} \cdot \left(\mathsf{neg}\left(\sin x\right)\right)\right) \cdot \left(\mathsf{neg}\left(\sinh y\right)\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{x} \cdot \left(\mathsf{neg}\left(\sin x\right)\right)\right) \cdot \left(\mathsf{neg}\left(\sinh y\right)\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\frac{1}{x} \cdot \left(\mathsf{neg}\left(\sin x\right)\right)\right)} \cdot \left(\mathsf{neg}\left(\sinh y\right)\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \left(\color{blue}{\frac{1}{x}} \cdot \left(\mathsf{neg}\left(\sin x\right)\right)\right) \cdot \left(\mathsf{neg}\left(\sinh y\right)\right) \]

      12. lower-neg.f64 N/A

          \[\leadsto \left(\frac{1}{x} \cdot \color{blue}{\left(\mathsf{neg}\left(\sin x\right)\right)}\right) \cdot \left(\mathsf{neg}\left(\sinh y\right)\right) \]

      13. lower-neg.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-1} \cdot \left(\mathsf{neg}\left(\sinh y\right)\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 79.1%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin x \cdot \sinh y}{x} \leq -\infty:\\ \;\;\;\;\left(-\sinh y\right) \cdot \left(0.16666666666666666 \cdot \left(x \cdot x\right)\right)\\ \mathbf{elif}\;\frac{\sin x \cdot \sinh y}{x} \leq 5 \cdot 10^{-41}:\\ \;\;\;\;y \cdot \frac{\sin x}{x}\\ \mathbf{else}:\\ \;\;\;\;\sinh y \cdot \left(--1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin x \cdot \sinh y}{x}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), x\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)}{x}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-41}:\\ \;\;\;\;y \cdot \frac{\sin x}{x}\\ \mathbf{else}:\\ \;\;\;\;\sinh y \cdot \left(--1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sin x) (sinh y)) x)))
   (if (<= t_0 (- INFINITY))
     (/
      (*
       (fma
        (* x x)
        (*
         x
         (fma
          (* x x)
          (fma (* x x) -0.0001984126984126984 0.008333333333333333)
          -0.16666666666666666))
        x)
       (fma
        (fma
         (* y y)
         (fma (* y y) 0.0001984126984126984 0.008333333333333333)
         0.16666666666666666)
        (* y (* y y))
        y))
      x)
     (if (<= t_0 5e-41) (* y (/ (sin x) x)) (* (sinh y) (- -1.0))))))

C

double code(double x, double y) {
	double t_0 = (sin(x) * sinh(y)) / 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 * y), fma((y * y), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666), (y * (y * y)), y)) / x;
	} else if (t_0 <= 5e-41) {
		tmp = y * (sin(x) / x);
	} else {
		tmp = sinh(y) * -(-1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(sin(x) * sinh(y)) / 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 * y), fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666), Float64(y * Float64(y * y)), y)) / x);
	elseif (t_0 <= 5e-41)
		tmp = Float64(y * Float64(sin(x) / x));
	else
		tmp = Float64(sinh(y) * Float64(-(-1.0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sin[x], $MachinePrecision] * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, 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 * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[t$95$0, 5e-41], N[(y * N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[Sinh[y], $MachinePrecision] * (--1.0)), $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 89.9

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

   5. Applied rewrites 89.9%

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

   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. associate-*l* N/A

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

      4. *-lft-identity N/A

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

      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. Applied rewrites 57.0%

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

   if -inf.0 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 4.9999999999999996e-41

   1. Initial program 78.9%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 98.3

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

   5. Applied rewrites 98.3%

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

   if 4.9999999999999996e-41 < (/.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. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. remove-double-neg N/A

         \[\leadsto \frac{1}{x} \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\sin x \cdot \sinh y\right)\right)\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{1}{x} \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\sin x \cdot \sinh y}\right)\right)\right)\right) \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto \frac{1}{x} \cdot \left(\mathsf{neg}\left(\color{blue}{\sin x \cdot \left(\mathsf{neg}\left(\sinh y\right)\right)}\right)\right) \]

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

         \[\leadsto \frac{1}{x} \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\sin x\right)\right) \cdot \left(\mathsf{neg}\left(\sinh y\right)\right)\right)} \]

      8. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{x} \cdot \left(\mathsf{neg}\left(\sin x\right)\right)\right) \cdot \left(\mathsf{neg}\left(\sinh y\right)\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{x} \cdot \left(\mathsf{neg}\left(\sin x\right)\right)\right) \cdot \left(\mathsf{neg}\left(\sinh y\right)\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\frac{1}{x} \cdot \left(\mathsf{neg}\left(\sin x\right)\right)\right)} \cdot \left(\mathsf{neg}\left(\sinh y\right)\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \left(\color{blue}{\frac{1}{x}} \cdot \left(\mathsf{neg}\left(\sin x\right)\right)\right) \cdot \left(\mathsf{neg}\left(\sinh y\right)\right) \]

      12. lower-neg.f64 N/A

          \[\leadsto \left(\frac{1}{x} \cdot \color{blue}{\left(\mathsf{neg}\left(\sin x\right)\right)}\right) \cdot \left(\mathsf{neg}\left(\sinh y\right)\right) \]

      13. lower-neg.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-1} \cdot \left(\mathsf{neg}\left(\sinh y\right)\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 79.1%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin x \cdot \sinh y}{x} \leq -\infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), x\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)}{x}\\ \mathbf{elif}\;\frac{\sin x \cdot \sinh y}{x} \leq 5 \cdot 10^{-41}:\\ \;\;\;\;y \cdot \frac{\sin x}{x}\\ \mathbf{else}:\\ \;\;\;\;\sinh y \cdot \left(--1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 62.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin x \cdot \sinh y}{x} \leq -5 \cdot 10^{-232}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right) \cdot \mathsf{fma}\left(x \cdot x, x \cdot -0.16666666666666666, x\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\sinh y \cdot \left(--1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (sin x) (sinh y)) x) -5e-232)
   (/
    (*
     (fma
      (fma
       (* y y)
       (fma (* y y) 0.0001984126984126984 0.008333333333333333)
       0.16666666666666666)
      (* y (* y y))
      y)
     (fma (* x x) (* x -0.16666666666666666) x))
    x)
   (* (sinh y) (- -1.0))))

C

double code(double x, double y) {
	double tmp;
	if (((sin(x) * sinh(y)) / x) <= -5e-232) {
		tmp = (fma(fma((y * y), fma((y * y), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666), (y * (y * y)), y) * fma((x * x), (x * -0.16666666666666666), x)) / x;
	} else {
		tmp = sinh(y) * -(-1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(sin(x) * sinh(y)) / x) <= -5e-232)
		tmp = Float64(Float64(fma(fma(Float64(y * y), fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666), Float64(y * Float64(y * y)), y) * fma(Float64(x * x), Float64(x * -0.16666666666666666), x)) / x);
	else
		tmp = Float64(sinh(y) * Float64(-(-1.0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[Sin[x], $MachinePrecision] * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], -5e-232], N[(N[(N[(N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * -0.16666666666666666), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[Sinh[y], $MachinePrecision] * (--1.0)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -4.9999999999999999e-232

   1. Initial program 98.7%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 90.7

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

   5. Applied rewrites 90.7%

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

   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. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-lft-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 57.3

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

   8. Applied rewrites 57.3%

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

   if -4.9999999999999999e-232 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 84.8%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. remove-double-neg N/A

         \[\leadsto \frac{1}{x} \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\sin x \cdot \sinh y\right)\right)\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{1}{x} \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\sin x \cdot \sinh y}\right)\right)\right)\right) \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto \frac{1}{x} \cdot \left(\mathsf{neg}\left(\color{blue}{\sin x \cdot \left(\mathsf{neg}\left(\sinh y\right)\right)}\right)\right) \]

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

         \[\leadsto \frac{1}{x} \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\sin x\right)\right) \cdot \left(\mathsf{neg}\left(\sinh y\right)\right)\right)} \]

      8. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{x} \cdot \left(\mathsf{neg}\left(\sin x\right)\right)\right) \cdot \left(\mathsf{neg}\left(\sinh y\right)\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{x} \cdot \left(\mathsf{neg}\left(\sin x\right)\right)\right) \cdot \left(\mathsf{neg}\left(\sinh y\right)\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\frac{1}{x} \cdot \left(\mathsf{neg}\left(\sin x\right)\right)\right)} \cdot \left(\mathsf{neg}\left(\sinh y\right)\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \left(\color{blue}{\frac{1}{x}} \cdot \left(\mathsf{neg}\left(\sin x\right)\right)\right) \cdot \left(\mathsf{neg}\left(\sinh y\right)\right) \]

      12. lower-neg.f64 N/A

          \[\leadsto \left(\frac{1}{x} \cdot \color{blue}{\left(\mathsf{neg}\left(\sin x\right)\right)}\right) \cdot \left(\mathsf{neg}\left(\sinh y\right)\right) \]

      13. lower-neg.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-1} \cdot \left(\mathsf{neg}\left(\sinh y\right)\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 59.7%

      \[\leadsto \color{blue}{-1} \cdot \left(-\sinh y\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin x \cdot \sinh y}{x} \leq -5 \cdot 10^{-232}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right) \cdot \mathsf{fma}\left(x \cdot x, x \cdot -0.16666666666666666, x\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\sinh y \cdot \left(--1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 58.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin x \cdot \sinh y}{x} \leq -5 \cdot 10^{-285}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right) \cdot \mathsf{fma}\left(x \cdot x, x \cdot -0.16666666666666666, x\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot 0.16666666666666666, y \cdot y, y\right)}{x} \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, x \cdot 0.008333333333333333, -0.16666666666666666\right), x \cdot \left(x \cdot x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (sin x) (sinh y)) x) -5e-285)
   (/
    (*
     (fma
      (fma
       (* y y)
       (fma (* y y) 0.0001984126984126984 0.008333333333333333)
       0.16666666666666666)
      (* y (* y y))
      y)
     (fma (* x x) (* x -0.16666666666666666) x))
    x)
   (*
    (/ (fma (* y 0.16666666666666666) (* y y) y) x)
    (fma
     (fma x (* x 0.008333333333333333) -0.16666666666666666)
     (* x (* x x))
     x))))

C

double code(double x, double y) {
	double tmp;
	if (((sin(x) * sinh(y)) / x) <= -5e-285) {
		tmp = (fma(fma((y * y), fma((y * y), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666), (y * (y * y)), y) * fma((x * x), (x * -0.16666666666666666), x)) / x;
	} else {
		tmp = (fma((y * 0.16666666666666666), (y * y), y) / x) * fma(fma(x, (x * 0.008333333333333333), -0.16666666666666666), (x * (x * x)), x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(sin(x) * sinh(y)) / x) <= -5e-285)
		tmp = Float64(Float64(fma(fma(Float64(y * y), fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666), Float64(y * Float64(y * y)), y) * fma(Float64(x * x), Float64(x * -0.16666666666666666), x)) / x);
	else
		tmp = Float64(Float64(fma(Float64(y * 0.16666666666666666), Float64(y * y), y) / x) * fma(fma(x, Float64(x * 0.008333333333333333), -0.16666666666666666), Float64(x * Float64(x * x)), x));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[Sin[x], $MachinePrecision] * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], -5e-285], N[(N[(N[(N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * -0.16666666666666666), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(N[(y * 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + y), $MachinePrecision] / x), $MachinePrecision] * N[(N[(x * N[(x * 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -5.00000000000000018e-285

   1. Initial program 98.7%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 91.3

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

   5. Applied rewrites 91.3%

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

   6. Taylor expanded in x around 0

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

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

      4. associate-*l* N/A

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

      5. *-lft-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 54.0

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

   8. Applied rewrites 54.0%

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

   if -5.00000000000000018e-285 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 84.0%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 44.8

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

   5. Applied rewrites 44.8%

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

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. associate-*r* N/A

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

      5. unpow2 N/A

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

      6. unpow3 N/A

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

      7. lower-fma.f64 N/A

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

      8. cube-mult N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 38.6

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

   8. Applied rewrites 38.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   10. Applied rewrites 58.5%

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

4. Final simplification 56.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin x \cdot \sinh y}{x} \leq -5 \cdot 10^{-285}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right) \cdot \mathsf{fma}\left(x \cdot x, x \cdot -0.16666666666666666, x\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot 0.16666666666666666, y \cdot y, y\right)}{x} \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, x \cdot 0.008333333333333333, -0.16666666666666666\right), x \cdot \left(x \cdot x\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 60.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot x\right)\\ \mathbf{if}\;\frac{\sin x \cdot \sinh y}{x} \leq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y\right)}{x} \cdot \mathsf{fma}\left(-0.16666666666666666, t\_0, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot 0.16666666666666666, y \cdot y, y\right)}{x} \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, x \cdot 0.008333333333333333, -0.16666666666666666\right), t\_0, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* x (* x x))))
   (if (<= (/ (* (sin x) (sinh y)) x) 0.0)
     (*
      (/
       (fma
        (* y y)
        (*
         y
         (fma
          (* y y)
          (fma y (* y 0.0001984126984126984) 0.008333333333333333)
          0.16666666666666666))
        y)
       x)
      (fma -0.16666666666666666 t_0 x))
     (*
      (/ (fma (* y 0.16666666666666666) (* y y) y) x)
      (fma (fma x (* x 0.008333333333333333) -0.16666666666666666) t_0 x)))))

C

double code(double x, double y) {
	double t_0 = x * (x * x);
	double tmp;
	if (((sin(x) * sinh(y)) / x) <= 0.0) {
		tmp = (fma((y * y), (y * fma((y * y), fma(y, (y * 0.0001984126984126984), 0.008333333333333333), 0.16666666666666666)), y) / x) * fma(-0.16666666666666666, t_0, x);
	} else {
		tmp = (fma((y * 0.16666666666666666), (y * y), y) / x) * fma(fma(x, (x * 0.008333333333333333), -0.16666666666666666), t_0, x);
	}
	return tmp;
}

Julia

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

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sin[x], $MachinePrecision] * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], 0.0], N[(N[(N[(N[(y * y), $MachinePrecision] * N[(y * N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * 0.0001984126984126984), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] / x), $MachinePrecision] * N[(-0.16666666666666666 * t$95$0 + x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(y * 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + y), $MachinePrecision] / x), $MachinePrecision] * N[(N[(x * N[(x * 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * t$95$0 + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 84.4%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 79.6

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

   5. Applied rewrites 79.6%

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

   6. Taylor expanded in x around 0

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

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

      4. associate-*l* N/A

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

      5. *-lft-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 36.6

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

   8. Applied rewrites 36.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   10. Applied rewrites 53.0%

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

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

   1. Initial program 98.8%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 67.7

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

   5. Applied rewrites 67.7%

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

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. associate-*r* N/A

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

      5. unpow2 N/A

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

      6. unpow3 N/A

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

      7. lower-fma.f64 N/A

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

      8. cube-mult N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 58.1

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

   8. Applied rewrites 58.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   10. Applied rewrites 65.8%

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

Alternative 9: 56.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin x \cdot \sinh y}{x} \leq -5 \cdot 10^{-285}:\\ \;\;\;\;\frac{x \cdot \left(y \cdot \left(\mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right)\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot 0.16666666666666666, y \cdot y, y\right)}{x} \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, x \cdot 0.008333333333333333, -0.16666666666666666\right), x \cdot \left(x \cdot x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (sin x) (sinh y)) x) -5e-285)
   (/
    (*
     x
     (*
      y
      (*
       (fma x (* x -0.16666666666666666) 1.0)
       (fma 0.16666666666666666 (* y y) 1.0))))
    x)
   (*
    (/ (fma (* y 0.16666666666666666) (* y y) y) x)
    (fma
     (fma x (* x 0.008333333333333333) -0.16666666666666666)
     (* x (* x x))
     x))))

C

double code(double x, double y) {
	double tmp;
	if (((sin(x) * sinh(y)) / x) <= -5e-285) {
		tmp = (x * (y * (fma(x, (x * -0.16666666666666666), 1.0) * fma(0.16666666666666666, (y * y), 1.0)))) / x;
	} else {
		tmp = (fma((y * 0.16666666666666666), (y * y), y) / x) * fma(fma(x, (x * 0.008333333333333333), -0.16666666666666666), (x * (x * x)), x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(sin(x) * sinh(y)) / x) <= -5e-285)
		tmp = Float64(Float64(x * Float64(y * Float64(fma(x, Float64(x * -0.16666666666666666), 1.0) * fma(0.16666666666666666, Float64(y * y), 1.0)))) / x);
	else
		tmp = Float64(Float64(fma(Float64(y * 0.16666666666666666), Float64(y * y), y) / x) * fma(fma(x, Float64(x * 0.008333333333333333), -0.16666666666666666), Float64(x * Float64(x * x)), x));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[Sin[x], $MachinePrecision] * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], -5e-285], N[(N[(x * N[(y * N[(N[(x * N[(x * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * N[(0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(N[(y * 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + y), $MachinePrecision] / x), $MachinePrecision] * N[(N[(x * N[(x * 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -5.00000000000000018e-285

   1. Initial program 98.7%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 86.6

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

   5. Applied rewrites 86.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 53.9%

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

   if -5.00000000000000018e-285 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 84.0%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 44.8

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

   5. Applied rewrites 44.8%

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

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. associate-*r* N/A

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

      5. unpow2 N/A

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

      6. unpow3 N/A

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

      7. lower-fma.f64 N/A

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

      8. cube-mult N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 38.6

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

   8. Applied rewrites 38.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   10. Applied rewrites 58.5%

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

Alternative 10: 56.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin x \cdot \sinh y}{x} \leq -5 \cdot 10^{-232}:\\ \;\;\;\;\frac{x \cdot \left(y \cdot \left(\mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right)\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y\right) \cdot \left(--1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (sin x) (sinh y)) x) -5e-232)
   (/
    (*
     x
     (*
      y
      (*
       (fma x (* x -0.16666666666666666) 1.0)
       (fma 0.16666666666666666 (* y y) 1.0))))
    x)
   (*
    (fma
     (* y y)
     (*
      y
      (fma
       (* y y)
       (fma (* y y) 0.0001984126984126984 0.008333333333333333)
       0.16666666666666666))
     y)
    (- -1.0))))

C

double code(double x, double y) {
	double tmp;
	if (((sin(x) * sinh(y)) / x) <= -5e-232) {
		tmp = (x * (y * (fma(x, (x * -0.16666666666666666), 1.0) * fma(0.16666666666666666, (y * y), 1.0)))) / x;
	} else {
		tmp = fma((y * y), (y * fma((y * y), fma((y * y), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666)), y) * -(-1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(sin(x) * sinh(y)) / x) <= -5e-232)
		tmp = Float64(Float64(x * Float64(y * Float64(fma(x, Float64(x * -0.16666666666666666), 1.0) * fma(0.16666666666666666, Float64(y * y), 1.0)))) / x);
	else
		tmp = Float64(fma(Float64(y * y), Float64(y * fma(Float64(y * y), fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666)), y) * Float64(-(-1.0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[Sin[x], $MachinePrecision] * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], -5e-232], N[(N[(x * N[(y * N[(N[(x * N[(x * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * N[(0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(y * y), $MachinePrecision] * N[(y * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] * (--1.0)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -4.9999999999999999e-232

   1. Initial program 98.7%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 85.5

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

   5. Applied rewrites 85.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 57.3%

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

   if -4.9999999999999999e-232 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 84.8%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. remove-double-neg N/A

         \[\leadsto \frac{1}{x} \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\sin x \cdot \sinh y\right)\right)\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{1}{x} \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\sin x \cdot \sinh y}\right)\right)\right)\right) \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto \frac{1}{x} \cdot \left(\mathsf{neg}\left(\color{blue}{\sin x \cdot \left(\mathsf{neg}\left(\sinh y\right)\right)}\right)\right) \]

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

         \[\leadsto \frac{1}{x} \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\sin x\right)\right) \cdot \left(\mathsf{neg}\left(\sinh y\right)\right)\right)} \]

      8. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{x} \cdot \left(\mathsf{neg}\left(\sin x\right)\right)\right) \cdot \left(\mathsf{neg}\left(\sinh y\right)\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{x} \cdot \left(\mathsf{neg}\left(\sin x\right)\right)\right) \cdot \left(\mathsf{neg}\left(\sinh y\right)\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\frac{1}{x} \cdot \left(\mathsf{neg}\left(\sin x\right)\right)\right)} \cdot \left(\mathsf{neg}\left(\sinh y\right)\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \left(\color{blue}{\frac{1}{x}} \cdot \left(\mathsf{neg}\left(\sin x\right)\right)\right) \cdot \left(\mathsf{neg}\left(\sinh y\right)\right) \]

      12. lower-neg.f64 N/A

          \[\leadsto \left(\frac{1}{x} \cdot \color{blue}{\left(\mathsf{neg}\left(\sin x\right)\right)}\right) \cdot \left(\mathsf{neg}\left(\sinh y\right)\right) \]

      13. lower-neg.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 57.4

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

   7. Applied rewrites 57.4%

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

   8. Taylor expanded in x around 0

      \[\leadsto -1 \cdot \left(\mathsf{neg}\left(\sinh y\right)\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 59.7%

       \[\leadsto -1 \cdot \left(-\sinh y\right) \]

   11. Taylor expanded in y around 0

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

       1. +-commutative N/A

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

       2. distribute-rgt-in N/A

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

       3. associate-*l* N/A

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

       4. *-lft-identity N/A

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

       5. lower-fma.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 N/A

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

       8. lower-*.f64 N/A

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

       9. +-commutative N/A

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

       10. lower-fma.f64 N/A

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

       11. unpow2 N/A

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

       12. lower-*.f64 N/A

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

       13. +-commutative N/A

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

       14. *-commutative N/A

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

       15. lower-fma.f64 N/A

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

       16. unpow2 N/A

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

       17. lower-*.f64 56.2

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

   13. Applied rewrites 56.2%

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

4. Final simplification 56.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin x \cdot \sinh y}{x} \leq -5 \cdot 10^{-232}:\\ \;\;\;\;\frac{x \cdot \left(y \cdot \left(\mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right)\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y\right) \cdot \left(--1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 58.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin x \cdot \sinh y}{x} \leq -5 \cdot 10^{-232}:\\ \;\;\;\;\mathsf{fma}\left(0.16666666666666666, x \cdot x, -1\right) \cdot \left(-\mathsf{fma}\left(y \cdot \left(y \cdot y\right), \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y\right) \cdot \left(--1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (sin x) (sinh y)) x) -5e-232)
   (*
    (fma 0.16666666666666666 (* x x) -1.0)
    (-
     (fma
      (* y (* y y))
      (fma y (* y 0.008333333333333333) 0.16666666666666666)
      y)))
   (*
    (fma
     (* y y)
     (*
      y
      (fma
       (* y y)
       (fma (* y y) 0.0001984126984126984 0.008333333333333333)
       0.16666666666666666))
     y)
    (- -1.0))))

C

double code(double x, double y) {
	double tmp;
	if (((sin(x) * sinh(y)) / x) <= -5e-232) {
		tmp = fma(0.16666666666666666, (x * x), -1.0) * -fma((y * (y * y)), fma(y, (y * 0.008333333333333333), 0.16666666666666666), y);
	} else {
		tmp = fma((y * y), (y * fma((y * y), fma((y * y), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666)), y) * -(-1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(sin(x) * sinh(y)) / x) <= -5e-232)
		tmp = Float64(fma(0.16666666666666666, Float64(x * x), -1.0) * Float64(-fma(Float64(y * Float64(y * y)), fma(y, Float64(y * 0.008333333333333333), 0.16666666666666666), y)));
	else
		tmp = Float64(fma(Float64(y * y), Float64(y * fma(Float64(y * y), fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666)), y) * Float64(-(-1.0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[Sin[x], $MachinePrecision] * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], -5e-232], N[(N[(0.16666666666666666 * N[(x * x), $MachinePrecision] + -1.0), $MachinePrecision] * (-N[(N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(y * N[(y * 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + y), $MachinePrecision])), $MachinePrecision], N[(N[(N[(y * y), $MachinePrecision] * N[(y * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] * (--1.0)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -4.9999999999999999e-232

   1. Initial program 98.7%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. remove-double-neg N/A

         \[\leadsto \frac{1}{x} \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\sin x \cdot \sinh y\right)\right)\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{1}{x} \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\sin x \cdot \sinh y}\right)\right)\right)\right) \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto \frac{1}{x} \cdot \left(\mathsf{neg}\left(\color{blue}{\sin x \cdot \left(\mathsf{neg}\left(\sinh y\right)\right)}\right)\right) \]

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

         \[\leadsto \frac{1}{x} \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\sin x\right)\right) \cdot \left(\mathsf{neg}\left(\sinh y\right)\right)\right)} \]

      8. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{x} \cdot \left(\mathsf{neg}\left(\sin x\right)\right)\right) \cdot \left(\mathsf{neg}\left(\sinh y\right)\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{x} \cdot \left(\mathsf{neg}\left(\sin x\right)\right)\right) \cdot \left(\mathsf{neg}\left(\sinh y\right)\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\frac{1}{x} \cdot \left(\mathsf{neg}\left(\sin x\right)\right)\right)} \cdot \left(\mathsf{neg}\left(\sinh y\right)\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \left(\color{blue}{\frac{1}{x}} \cdot \left(\mathsf{neg}\left(\sin x\right)\right)\right) \cdot \left(\mathsf{neg}\left(\sinh y\right)\right) \]

      12. lower-neg.f64 N/A

          \[\leadsto \left(\frac{1}{x} \cdot \color{blue}{\left(\mathsf{neg}\left(\sin x\right)\right)}\right) \cdot \left(\mathsf{neg}\left(\sinh y\right)\right) \]

      13. lower-neg.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 63.1

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

   7. Applied rewrites 63.1%

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

   8. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. cube-mult N/A

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

      6. *-rgt-identity N/A

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

      7. lower-fma.f64 N/A

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

      8. cube-mult N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. unpow2 N/A

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

      16. associate-*l* N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 57.6

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

   10. Applied rewrites 57.6%

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

   if -4.9999999999999999e-232 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 84.8%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. remove-double-neg N/A

         \[\leadsto \frac{1}{x} \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\sin x \cdot \sinh y\right)\right)\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{1}{x} \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\sin x \cdot \sinh y}\right)\right)\right)\right) \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto \frac{1}{x} \cdot \left(\mathsf{neg}\left(\color{blue}{\sin x \cdot \left(\mathsf{neg}\left(\sinh y\right)\right)}\right)\right) \]

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

         \[\leadsto \frac{1}{x} \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\sin x\right)\right) \cdot \left(\mathsf{neg}\left(\sinh y\right)\right)\right)} \]

      8. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{x} \cdot \left(\mathsf{neg}\left(\sin x\right)\right)\right) \cdot \left(\mathsf{neg}\left(\sinh y\right)\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{x} \cdot \left(\mathsf{neg}\left(\sin x\right)\right)\right) \cdot \left(\mathsf{neg}\left(\sinh y\right)\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\frac{1}{x} \cdot \left(\mathsf{neg}\left(\sin x\right)\right)\right)} \cdot \left(\mathsf{neg}\left(\sinh y\right)\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \left(\color{blue}{\frac{1}{x}} \cdot \left(\mathsf{neg}\left(\sin x\right)\right)\right) \cdot \left(\mathsf{neg}\left(\sinh y\right)\right) \]

      12. lower-neg.f64 N/A

          \[\leadsto \left(\frac{1}{x} \cdot \color{blue}{\left(\mathsf{neg}\left(\sin x\right)\right)}\right) \cdot \left(\mathsf{neg}\left(\sinh y\right)\right) \]

      13. lower-neg.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 57.4

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

   7. Applied rewrites 57.4%

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

   8. Taylor expanded in x around 0

      \[\leadsto -1 \cdot \left(\mathsf{neg}\left(\sinh y\right)\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 59.7%

       \[\leadsto -1 \cdot \left(-\sinh y\right) \]

   11. Taylor expanded in y around 0

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

       1. +-commutative N/A

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

       2. distribute-rgt-in N/A

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

       3. associate-*l* N/A

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

       4. *-lft-identity N/A

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

       5. lower-fma.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 N/A

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

       8. lower-*.f64 N/A

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

       9. +-commutative N/A

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

       10. lower-fma.f64 N/A

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

       11. unpow2 N/A

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

       12. lower-*.f64 N/A

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

       13. +-commutative N/A

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

       14. *-commutative N/A

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

       15. lower-fma.f64 N/A

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

       16. unpow2 N/A

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

       17. lower-*.f64 56.2

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

   13. Applied rewrites 56.2%

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

4. Final simplification 56.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin x \cdot \sinh y}{x} \leq -5 \cdot 10^{-232}:\\ \;\;\;\;\mathsf{fma}\left(0.16666666666666666, x \cdot x, -1\right) \cdot \left(-\mathsf{fma}\left(y \cdot \left(y \cdot y\right), \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y\right) \cdot \left(--1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 56.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin x \cdot \sinh y}{x} \leq -5 \cdot 10^{-232}:\\ \;\;\;\;\mathsf{fma}\left(y, 0.16666666666666666 \cdot \left(y \cdot y\right), y\right) \cdot \left(-\mathsf{fma}\left(0.16666666666666666, x \cdot x, -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y\right) \cdot \left(--1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (sin x) (sinh y)) x) -5e-232)
   (*
    (fma y (* 0.16666666666666666 (* y y)) y)
    (- (fma 0.16666666666666666 (* x x) -1.0)))
   (*
    (fma
     (* y y)
     (*
      y
      (fma
       (* y y)
       (fma (* y y) 0.0001984126984126984 0.008333333333333333)
       0.16666666666666666))
     y)
    (- -1.0))))

C

double code(double x, double y) {
	double tmp;
	if (((sin(x) * sinh(y)) / x) <= -5e-232) {
		tmp = fma(y, (0.16666666666666666 * (y * y)), y) * -fma(0.16666666666666666, (x * x), -1.0);
	} else {
		tmp = fma((y * y), (y * fma((y * y), fma((y * y), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666)), y) * -(-1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(sin(x) * sinh(y)) / x) <= -5e-232)
		tmp = Float64(fma(y, Float64(0.16666666666666666 * Float64(y * y)), y) * Float64(-fma(0.16666666666666666, Float64(x * x), -1.0)));
	else
		tmp = Float64(fma(Float64(y * y), Float64(y * fma(Float64(y * y), fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666)), y) * Float64(-(-1.0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[Sin[x], $MachinePrecision] * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], -5e-232], N[(N[(y * N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] * (-N[(0.16666666666666666 * N[(x * x), $MachinePrecision] + -1.0), $MachinePrecision])), $MachinePrecision], N[(N[(N[(y * y), $MachinePrecision] * N[(y * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] * (--1.0)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -4.9999999999999999e-232

   1. Initial program 98.7%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. remove-double-neg N/A

         \[\leadsto \frac{1}{x} \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\sin x \cdot \sinh y\right)\right)\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{1}{x} \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\sin x \cdot \sinh y}\right)\right)\right)\right) \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto \frac{1}{x} \cdot \left(\mathsf{neg}\left(\color{blue}{\sin x \cdot \left(\mathsf{neg}\left(\sinh y\right)\right)}\right)\right) \]

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

         \[\leadsto \frac{1}{x} \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\sin x\right)\right) \cdot \left(\mathsf{neg}\left(\sinh y\right)\right)\right)} \]

      8. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{x} \cdot \left(\mathsf{neg}\left(\sin x\right)\right)\right) \cdot \left(\mathsf{neg}\left(\sinh y\right)\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{x} \cdot \left(\mathsf{neg}\left(\sin x\right)\right)\right) \cdot \left(\mathsf{neg}\left(\sinh y\right)\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\frac{1}{x} \cdot \left(\mathsf{neg}\left(\sin x\right)\right)\right)} \cdot \left(\mathsf{neg}\left(\sinh y\right)\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \left(\color{blue}{\frac{1}{x}} \cdot \left(\mathsf{neg}\left(\sin x\right)\right)\right) \cdot \left(\mathsf{neg}\left(\sinh y\right)\right) \]

      12. lower-neg.f64 N/A

          \[\leadsto \left(\frac{1}{x} \cdot \color{blue}{\left(\mathsf{neg}\left(\sin x\right)\right)}\right) \cdot \left(\mathsf{neg}\left(\sinh y\right)\right) \]

      13. lower-neg.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 63.1

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

   7. Applied rewrites 63.1%

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

   8. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 57.5

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

   10. Applied rewrites 57.5%

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

   if -4.9999999999999999e-232 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 84.8%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. remove-double-neg N/A

         \[\leadsto \frac{1}{x} \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\sin x \cdot \sinh y\right)\right)\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{1}{x} \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\sin x \cdot \sinh y}\right)\right)\right)\right) \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto \frac{1}{x} \cdot \left(\mathsf{neg}\left(\color{blue}{\sin x \cdot \left(\mathsf{neg}\left(\sinh y\right)\right)}\right)\right) \]

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

         \[\leadsto \frac{1}{x} \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\sin x\right)\right) \cdot \left(\mathsf{neg}\left(\sinh y\right)\right)\right)} \]

      8. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{x} \cdot \left(\mathsf{neg}\left(\sin x\right)\right)\right) \cdot \left(\mathsf{neg}\left(\sinh y\right)\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{x} \cdot \left(\mathsf{neg}\left(\sin x\right)\right)\right) \cdot \left(\mathsf{neg}\left(\sinh y\right)\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\frac{1}{x} \cdot \left(\mathsf{neg}\left(\sin x\right)\right)\right)} \cdot \left(\mathsf{neg}\left(\sinh y\right)\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \left(\color{blue}{\frac{1}{x}} \cdot \left(\mathsf{neg}\left(\sin x\right)\right)\right) \cdot \left(\mathsf{neg}\left(\sinh y\right)\right) \]

      12. lower-neg.f64 N/A

          \[\leadsto \left(\frac{1}{x} \cdot \color{blue}{\left(\mathsf{neg}\left(\sin x\right)\right)}\right) \cdot \left(\mathsf{neg}\left(\sinh y\right)\right) \]

      13. lower-neg.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 57.4

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

   7. Applied rewrites 57.4%

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

   8. Taylor expanded in x around 0

      \[\leadsto -1 \cdot \left(\mathsf{neg}\left(\sinh y\right)\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 59.7%

       \[\leadsto -1 \cdot \left(-\sinh y\right) \]

   11. Taylor expanded in y around 0

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

       1. +-commutative N/A

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

       2. distribute-rgt-in N/A

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

       3. associate-*l* N/A

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

       4. *-lft-identity N/A

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

       5. lower-fma.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 N/A

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

       8. lower-*.f64 N/A

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

       9. +-commutative N/A

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

       10. lower-fma.f64 N/A

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

       11. unpow2 N/A

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

       12. lower-*.f64 N/A

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

       13. +-commutative N/A

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

       14. *-commutative N/A

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

       15. lower-fma.f64 N/A

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

       16. unpow2 N/A

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

       17. lower-*.f64 56.2

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

   13. Applied rewrites 56.2%

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

4. Final simplification 56.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin x \cdot \sinh y}{x} \leq -5 \cdot 10^{-232}:\\ \;\;\;\;\mathsf{fma}\left(y, 0.16666666666666666 \cdot \left(y \cdot y\right), y\right) \cdot \left(-\mathsf{fma}\left(0.16666666666666666, x \cdot x, -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y\right) \cdot \left(--1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 56.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin x \cdot \sinh y}{x} \leq -5 \cdot 10^{-232}:\\ \;\;\;\;\mathsf{fma}\left(y, 0.16666666666666666 \cdot \left(y \cdot y\right), y\right) \cdot \left(-\mathsf{fma}\left(0.16666666666666666, x \cdot x, -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot -0.0001984126984126984, -0.008333333333333333\right), -0.16666666666666666\right), -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (sin x) (sinh y)) x) -5e-232)
   (*
    (fma y (* 0.16666666666666666 (* y y)) y)
    (- (fma 0.16666666666666666 (* x x) -1.0)))
   (*
    -1.0
    (*
     y
     (fma
      (* y y)
      (fma
       (* y y)
       (fma y (* y -0.0001984126984126984) -0.008333333333333333)
       -0.16666666666666666)
      -1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (((sin(x) * sinh(y)) / x) <= -5e-232) {
		tmp = fma(y, (0.16666666666666666 * (y * y)), y) * -fma(0.16666666666666666, (x * x), -1.0);
	} else {
		tmp = -1.0 * (y * fma((y * y), fma((y * y), fma(y, (y * -0.0001984126984126984), -0.008333333333333333), -0.16666666666666666), -1.0));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(sin(x) * sinh(y)) / x) <= -5e-232)
		tmp = Float64(fma(y, Float64(0.16666666666666666 * Float64(y * y)), y) * Float64(-fma(0.16666666666666666, Float64(x * x), -1.0)));
	else
		tmp = Float64(-1.0 * Float64(y * fma(Float64(y * y), fma(Float64(y * y), fma(y, Float64(y * -0.0001984126984126984), -0.008333333333333333), -0.16666666666666666), -1.0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[Sin[x], $MachinePrecision] * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], -5e-232], N[(N[(y * N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] * (-N[(0.16666666666666666 * N[(x * x), $MachinePrecision] + -1.0), $MachinePrecision])), $MachinePrecision], N[(-1.0 * N[(y * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * -0.0001984126984126984), $MachinePrecision] + -0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -4.9999999999999999e-232

   1. Initial program 98.7%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. remove-double-neg N/A

         \[\leadsto \frac{1}{x} \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\sin x \cdot \sinh y\right)\right)\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{1}{x} \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\sin x \cdot \sinh y}\right)\right)\right)\right) \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto \frac{1}{x} \cdot \left(\mathsf{neg}\left(\color{blue}{\sin x \cdot \left(\mathsf{neg}\left(\sinh y\right)\right)}\right)\right) \]

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

         \[\leadsto \frac{1}{x} \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\sin x\right)\right) \cdot \left(\mathsf{neg}\left(\sinh y\right)\right)\right)} \]

      8. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{x} \cdot \left(\mathsf{neg}\left(\sin x\right)\right)\right) \cdot \left(\mathsf{neg}\left(\sinh y\right)\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{x} \cdot \left(\mathsf{neg}\left(\sin x\right)\right)\right) \cdot \left(\mathsf{neg}\left(\sinh y\right)\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\frac{1}{x} \cdot \left(\mathsf{neg}\left(\sin x\right)\right)\right)} \cdot \left(\mathsf{neg}\left(\sinh y\right)\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \left(\color{blue}{\frac{1}{x}} \cdot \left(\mathsf{neg}\left(\sin x\right)\right)\right) \cdot \left(\mathsf{neg}\left(\sinh y\right)\right) \]

      12. lower-neg.f64 N/A

          \[\leadsto \left(\frac{1}{x} \cdot \color{blue}{\left(\mathsf{neg}\left(\sin x\right)\right)}\right) \cdot \left(\mathsf{neg}\left(\sinh y\right)\right) \]

      13. lower-neg.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 63.1

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

   7. Applied rewrites 63.1%

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

   8. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 57.5

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

   10. Applied rewrites 57.5%

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

   if -4.9999999999999999e-232 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 84.8%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. remove-double-neg N/A

         \[\leadsto \frac{1}{x} \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\sin x \cdot \sinh y\right)\right)\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{1}{x} \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\sin x \cdot \sinh y}\right)\right)\right)\right) \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto \frac{1}{x} \cdot \left(\mathsf{neg}\left(\color{blue}{\sin x \cdot \left(\mathsf{neg}\left(\sinh y\right)\right)}\right)\right) \]

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

         \[\leadsto \frac{1}{x} \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\sin x\right)\right) \cdot \left(\mathsf{neg}\left(\sinh y\right)\right)\right)} \]

      8. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{x} \cdot \left(\mathsf{neg}\left(\sin x\right)\right)\right) \cdot \left(\mathsf{neg}\left(\sinh y\right)\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{x} \cdot \left(\mathsf{neg}\left(\sin x\right)\right)\right) \cdot \left(\mathsf{neg}\left(\sinh y\right)\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\frac{1}{x} \cdot \left(\mathsf{neg}\left(\sin x\right)\right)\right)} \cdot \left(\mathsf{neg}\left(\sinh y\right)\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \left(\color{blue}{\frac{1}{x}} \cdot \left(\mathsf{neg}\left(\sin x\right)\right)\right) \cdot \left(\mathsf{neg}\left(\sinh y\right)\right) \]

      12. lower-neg.f64 N/A

          \[\leadsto \left(\frac{1}{x} \cdot \color{blue}{\left(\mathsf{neg}\left(\sin x\right)\right)}\right) \cdot \left(\mathsf{neg}\left(\sinh y\right)\right) \]

      13. lower-neg.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 57.4

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

   7. Applied rewrites 57.4%

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

   8. Taylor expanded in x around 0

      \[\leadsto -1 \cdot \left(\mathsf{neg}\left(\sinh y\right)\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 59.7%

       \[\leadsto -1 \cdot \left(-\sinh y\right) \]

   11. Taylor expanded in y around 0

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

       1. lower-*.f64 N/A

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

       2. sub-neg N/A

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

       3. metadata-eval N/A

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

       4. lower-fma.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 N/A

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

       7. sub-neg N/A

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

       8. metadata-eval N/A

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

       9. lower-fma.f64 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f64 N/A

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

       12. sub-neg N/A

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

       13. *-commutative N/A

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

       14. unpow2 N/A

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

       15. associate-*l* N/A

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

       16. metadata-eval N/A

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

       17. lower-fma.f64 N/A

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

       18. lower-*.f64 56.2

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

   13. Applied rewrites 56.2%

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

4. Final simplification 56.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin x \cdot \sinh y}{x} \leq -5 \cdot 10^{-232}:\\ \;\;\;\;\mathsf{fma}\left(y, 0.16666666666666666 \cdot \left(y \cdot y\right), y\right) \cdot \left(-\mathsf{fma}\left(0.16666666666666666, x \cdot x, -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot -0.0001984126984126984, -0.008333333333333333\right), -0.16666666666666666\right), -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 55.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin x \cdot \sinh y}{x} \leq -5 \cdot 10^{-232}:\\ \;\;\;\;\mathsf{fma}\left(y, 0.16666666666666666 \cdot \left(y \cdot y\right), y\right) \cdot \left(-\mathsf{fma}\left(0.16666666666666666, x \cdot x, -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot -0.008333333333333333, -0.16666666666666666\right), -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (sin x) (sinh y)) x) -5e-232)
   (*
    (fma y (* 0.16666666666666666 (* y y)) y)
    (- (fma 0.16666666666666666 (* x x) -1.0)))
   (*
    -1.0
    (*
     y
     (fma
      y
      (* y (fma y (* y -0.008333333333333333) -0.16666666666666666))
      -1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (((sin(x) * sinh(y)) / x) <= -5e-232) {
		tmp = fma(y, (0.16666666666666666 * (y * y)), y) * -fma(0.16666666666666666, (x * x), -1.0);
	} else {
		tmp = -1.0 * (y * fma(y, (y * fma(y, (y * -0.008333333333333333), -0.16666666666666666)), -1.0));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(sin(x) * sinh(y)) / x) <= -5e-232)
		tmp = Float64(fma(y, Float64(0.16666666666666666 * Float64(y * y)), y) * Float64(-fma(0.16666666666666666, Float64(x * x), -1.0)));
	else
		tmp = Float64(-1.0 * Float64(y * fma(y, Float64(y * fma(y, Float64(y * -0.008333333333333333), -0.16666666666666666)), -1.0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[Sin[x], $MachinePrecision] * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], -5e-232], N[(N[(y * N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] * (-N[(0.16666666666666666 * N[(x * x), $MachinePrecision] + -1.0), $MachinePrecision])), $MachinePrecision], N[(-1.0 * N[(y * N[(y * N[(y * N[(y * N[(y * -0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -4.9999999999999999e-232

   1. Initial program 98.7%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. remove-double-neg N/A

         \[\leadsto \frac{1}{x} \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\sin x \cdot \sinh y\right)\right)\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{1}{x} \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\sin x \cdot \sinh y}\right)\right)\right)\right) \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto \frac{1}{x} \cdot \left(\mathsf{neg}\left(\color{blue}{\sin x \cdot \left(\mathsf{neg}\left(\sinh y\right)\right)}\right)\right) \]

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

         \[\leadsto \frac{1}{x} \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\sin x\right)\right) \cdot \left(\mathsf{neg}\left(\sinh y\right)\right)\right)} \]

      8. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{x} \cdot \left(\mathsf{neg}\left(\sin x\right)\right)\right) \cdot \left(\mathsf{neg}\left(\sinh y\right)\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{x} \cdot \left(\mathsf{neg}\left(\sin x\right)\right)\right) \cdot \left(\mathsf{neg}\left(\sinh y\right)\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\frac{1}{x} \cdot \left(\mathsf{neg}\left(\sin x\right)\right)\right)} \cdot \left(\mathsf{neg}\left(\sinh y\right)\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \left(\color{blue}{\frac{1}{x}} \cdot \left(\mathsf{neg}\left(\sin x\right)\right)\right) \cdot \left(\mathsf{neg}\left(\sinh y\right)\right) \]

      12. lower-neg.f64 N/A

          \[\leadsto \left(\frac{1}{x} \cdot \color{blue}{\left(\mathsf{neg}\left(\sin x\right)\right)}\right) \cdot \left(\mathsf{neg}\left(\sinh y\right)\right) \]

      13. lower-neg.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 63.1

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

   7. Applied rewrites 63.1%

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

   8. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 57.5

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

   10. Applied rewrites 57.5%

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

   if -4.9999999999999999e-232 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 84.8%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. remove-double-neg N/A

         \[\leadsto \frac{1}{x} \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\sin x \cdot \sinh y\right)\right)\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{1}{x} \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\sin x \cdot \sinh y}\right)\right)\right)\right) \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto \frac{1}{x} \cdot \left(\mathsf{neg}\left(\color{blue}{\sin x \cdot \left(\mathsf{neg}\left(\sinh y\right)\right)}\right)\right) \]

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

         \[\leadsto \frac{1}{x} \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\sin x\right)\right) \cdot \left(\mathsf{neg}\left(\sinh y\right)\right)\right)} \]

      8. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{x} \cdot \left(\mathsf{neg}\left(\sin x\right)\right)\right) \cdot \left(\mathsf{neg}\left(\sinh y\right)\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{x} \cdot \left(\mathsf{neg}\left(\sin x\right)\right)\right) \cdot \left(\mathsf{neg}\left(\sinh y\right)\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\frac{1}{x} \cdot \left(\mathsf{neg}\left(\sin x\right)\right)\right)} \cdot \left(\mathsf{neg}\left(\sinh y\right)\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \left(\color{blue}{\frac{1}{x}} \cdot \left(\mathsf{neg}\left(\sin x\right)\right)\right) \cdot \left(\mathsf{neg}\left(\sinh y\right)\right) \]

      12. lower-neg.f64 N/A

          \[\leadsto \left(\frac{1}{x} \cdot \color{blue}{\left(\mathsf{neg}\left(\sin x\right)\right)}\right) \cdot \left(\mathsf{neg}\left(\sinh y\right)\right) \]

      13. lower-neg.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 57.4

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

   7. Applied rewrites 57.4%

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

   8. Taylor expanded in x around 0

      \[\leadsto -1 \cdot \left(\mathsf{neg}\left(\sinh y\right)\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 59.7%

       \[\leadsto -1 \cdot \left(-\sinh y\right) \]

   11. Taylor expanded in y around 0

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

       1. lower-*.f64 N/A

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

       2. sub-neg N/A

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

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

       5. metadata-eval N/A

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

       6. lower-fma.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. sub-neg N/A

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

       9. *-commutative N/A

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

       10. unpow2 N/A

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

       11. associate-*l* N/A

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

       12. metadata-eval N/A

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

       13. lower-fma.f64 N/A

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

       14. lower-*.f64 54.4

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

   13. Applied rewrites 54.4%

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

4. Final simplification 55.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin x \cdot \sinh y}{x} \leq -5 \cdot 10^{-232}:\\ \;\;\;\;\mathsf{fma}\left(y, 0.16666666666666666 \cdot \left(y \cdot y\right), y\right) \cdot \left(-\mathsf{fma}\left(0.16666666666666666, x \cdot x, -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot -0.008333333333333333, -0.16666666666666666\right), -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 44.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin x \cdot \sinh y}{x} \leq -5 \cdot 10^{-232}:\\ \;\;\;\;\mathsf{fma}\left(x, y \cdot \left(x \cdot -0.16666666666666666\right), y\right)\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(y \cdot \mathsf{fma}\left(y, y \cdot -0.16666666666666666, -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (sin x) (sinh y)) x) -5e-232)
   (fma x (* y (* x -0.16666666666666666)) y)
   (* -1.0 (* y (fma y (* y -0.16666666666666666) -1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (((sin(x) * sinh(y)) / x) <= -5e-232) {
		tmp = fma(x, (y * (x * -0.16666666666666666)), y);
	} else {
		tmp = -1.0 * (y * fma(y, (y * -0.16666666666666666), -1.0));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(sin(x) * sinh(y)) / x) <= -5e-232)
		tmp = fma(x, Float64(y * Float64(x * -0.16666666666666666)), y);
	else
		tmp = Float64(-1.0 * Float64(y * fma(y, Float64(y * -0.16666666666666666), -1.0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[Sin[x], $MachinePrecision] * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], -5e-232], N[(x * N[(y * N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision], N[(-1.0 * N[(y * N[(y * N[(y * -0.16666666666666666), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -4.9999999999999999e-232

   1. Initial program 98.7%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 31.3

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

   5. Applied rewrites 31.3%

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

   6. Taylor expanded in x around 0

      \[\leadsto y \cdot 1 \]
   7. Step-by-step derivation

   8. Applied rewrites 21.3%

      \[\leadsto y \cdot 1 \]

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 34.7%

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

   13. Applied rewrites 34.7%

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

   if -4.9999999999999999e-232 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 84.8%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. remove-double-neg N/A

         \[\leadsto \frac{1}{x} \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\sin x \cdot \sinh y\right)\right)\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{1}{x} \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\sin x \cdot \sinh y}\right)\right)\right)\right) \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto \frac{1}{x} \cdot \left(\mathsf{neg}\left(\color{blue}{\sin x \cdot \left(\mathsf{neg}\left(\sinh y\right)\right)}\right)\right) \]

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

         \[\leadsto \frac{1}{x} \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\sin x\right)\right) \cdot \left(\mathsf{neg}\left(\sinh y\right)\right)\right)} \]

      8. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{x} \cdot \left(\mathsf{neg}\left(\sin x\right)\right)\right) \cdot \left(\mathsf{neg}\left(\sinh y\right)\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{x} \cdot \left(\mathsf{neg}\left(\sin x\right)\right)\right) \cdot \left(\mathsf{neg}\left(\sinh y\right)\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\frac{1}{x} \cdot \left(\mathsf{neg}\left(\sin x\right)\right)\right)} \cdot \left(\mathsf{neg}\left(\sinh y\right)\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \left(\color{blue}{\frac{1}{x}} \cdot \left(\mathsf{neg}\left(\sin x\right)\right)\right) \cdot \left(\mathsf{neg}\left(\sinh y\right)\right) \]

      12. lower-neg.f64 N/A

          \[\leadsto \left(\frac{1}{x} \cdot \color{blue}{\left(\mathsf{neg}\left(\sin x\right)\right)}\right) \cdot \left(\mathsf{neg}\left(\sinh y\right)\right) \]

      13. lower-neg.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 57.4

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

   7. Applied rewrites 57.4%

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

   8. Taylor expanded in x around 0

      \[\leadsto -1 \cdot \left(\mathsf{neg}\left(\sinh y\right)\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 59.7%

       \[\leadsto -1 \cdot \left(-\sinh y\right) \]

   11. Taylor expanded in y around 0

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

       1. lower-*.f64 N/A

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

       2. sub-neg N/A

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

       3. *-commutative N/A

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

       4. unpow2 N/A

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

       5. associate-*l* N/A

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

       6. *-commutative N/A

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

       7. metadata-eval N/A

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

       8. lower-fma.f64 N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 51.5

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

   13. Applied rewrites 51.5%

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

4. Final simplification 45.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin x \cdot \sinh y}{x} \leq -5 \cdot 10^{-232}:\\ \;\;\;\;\mathsf{fma}\left(x, y \cdot \left(x \cdot -0.16666666666666666\right), y\right)\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(y \cdot \mathsf{fma}\left(y, y \cdot -0.16666666666666666, -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 32.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin x \cdot \sinh y}{x} \leq -5 \cdot 10^{-232}:\\ \;\;\;\;\mathsf{fma}\left(x, y \cdot \left(x \cdot -0.16666666666666666\right), y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (sin x) (sinh y)) x) -5e-232)
   (fma x (* y (* x -0.16666666666666666)) y)
   (* y 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (((sin(x) * sinh(y)) / x) <= -5e-232) {
		tmp = fma(x, (y * (x * -0.16666666666666666)), y);
	} else {
		tmp = y * 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(sin(x) * sinh(y)) / x) <= -5e-232)
		tmp = fma(x, Float64(y * Float64(x * -0.16666666666666666)), y);
	else
		tmp = Float64(y * 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[Sin[x], $MachinePrecision] * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], -5e-232], N[(x * N[(y * N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision], N[(y * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -4.9999999999999999e-232

   1. Initial program 98.7%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 31.3

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

   5. Applied rewrites 31.3%

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

   6. Taylor expanded in x around 0

      \[\leadsto y \cdot 1 \]
   7. Step-by-step derivation

   8. Applied rewrites 21.3%

      \[\leadsto y \cdot 1 \]

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 34.7%

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

   13. Applied rewrites 34.7%

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

   if -4.9999999999999999e-232 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 84.8%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 62.3

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

   5. Applied rewrites 62.3%

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

   6. Taylor expanded in x around 0

      \[\leadsto y \cdot 1 \]
   7. Step-by-step derivation

   8. Applied rewrites 31.2%

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

4. Final simplification 32.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin x \cdot \sinh y}{x} \leq -5 \cdot 10^{-232}:\\ \;\;\;\;\mathsf{fma}\left(x, y \cdot \left(x \cdot -0.16666666666666666\right), y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 25.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin x \cdot \sinh y}{x} \leq -5 \cdot 10^{-232}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(y \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (sin x) (sinh y)) x) -5e-232)
   (* (* x x) (* y -0.16666666666666666))
   (* y 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (((sin(x) * sinh(y)) / x) <= -5e-232) {
		tmp = (x * x) * (y * -0.16666666666666666);
	} else {
		tmp = y * 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (((sin(x) * sinh(y)) / x) <= (-5d-232)) then
        tmp = (x * x) * (y * (-0.16666666666666666d0))
    else
        tmp = y * 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (((Math.sin(x) * Math.sinh(y)) / x) <= -5e-232) {
		tmp = (x * x) * (y * -0.16666666666666666);
	} else {
		tmp = y * 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((math.sin(x) * math.sinh(y)) / x) <= -5e-232:
		tmp = (x * x) * (y * -0.16666666666666666)
	else:
		tmp = y * 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(sin(x) * sinh(y)) / x) <= -5e-232)
		tmp = Float64(Float64(x * x) * Float64(y * -0.16666666666666666));
	else
		tmp = Float64(y * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((sin(x) * sinh(y)) / x) <= -5e-232)
		tmp = (x * x) * (y * -0.16666666666666666);
	else
		tmp = y * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[Sin[x], $MachinePrecision] * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], -5e-232], N[(N[(x * x), $MachinePrecision] * N[(y * -0.16666666666666666), $MachinePrecision]), $MachinePrecision], N[(y * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -4.9999999999999999e-232

   1. Initial program 98.7%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 31.3

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

   5. Applied rewrites 31.3%

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

   6. Taylor expanded in x around 0

      \[\leadsto y \cdot 1 \]
   7. Step-by-step derivation

   8. Applied rewrites 21.3%

      \[\leadsto y \cdot 1 \]

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 34.7%

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

   12. Taylor expanded in x around inf

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

   14. Applied rewrites 16.3%

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

   if -4.9999999999999999e-232 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 84.8%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 62.3

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

   5. Applied rewrites 62.3%

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

   6. Taylor expanded in x around 0

      \[\leadsto y \cdot 1 \]
   7. Step-by-step derivation

   8. Applied rewrites 31.2%

      \[\leadsto y \cdot 1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 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]

Derivation

1. Initial program 89.9%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 99.8

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

4. Applied rewrites 99.8%

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

5. Final simplification 99.8%

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

Alternative 19: 61.7% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.8 \cdot 10^{+31}:\\ \;\;\;\;-1 \cdot \left(y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot -0.008333333333333333, -0.16666666666666666\right), -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(x \cdot \left(0.16666666666666666 \cdot \left(y \cdot y\right)\right)\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 4.8e+31)
   (*
    -1.0
    (*
     y
     (fma
      y
      (* y (fma y (* y -0.008333333333333333) -0.16666666666666666))
      -1.0)))
   (/ (* y (* x (* 0.16666666666666666 (* y y)))) x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 4.8e+31) {
		tmp = -1.0 * (y * fma(y, (y * fma(y, (y * -0.008333333333333333), -0.16666666666666666)), -1.0));
	} else {
		tmp = (y * (x * (0.16666666666666666 * (y * y)))) / x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 4.8e+31)
		tmp = Float64(-1.0 * Float64(y * fma(y, Float64(y * fma(y, Float64(y * -0.008333333333333333), -0.16666666666666666)), -1.0)));
	else
		tmp = Float64(Float64(y * Float64(x * Float64(0.16666666666666666 * Float64(y * y)))) / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 4.8e+31], N[(-1.0 * N[(y * N[(y * N[(y * N[(y * N[(y * -0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * N[(x * N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.79999999999999965e31

   1. Initial program 87.2%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. remove-double-neg N/A

         \[\leadsto \frac{1}{x} \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\sin x \cdot \sinh y\right)\right)\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{1}{x} \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\sin x \cdot \sinh y}\right)\right)\right)\right) \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto \frac{1}{x} \cdot \left(\mathsf{neg}\left(\color{blue}{\sin x \cdot \left(\mathsf{neg}\left(\sinh y\right)\right)}\right)\right) \]

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

         \[\leadsto \frac{1}{x} \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\sin x\right)\right) \cdot \left(\mathsf{neg}\left(\sinh y\right)\right)\right)} \]

      8. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{x} \cdot \left(\mathsf{neg}\left(\sin x\right)\right)\right) \cdot \left(\mathsf{neg}\left(\sinh y\right)\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{x} \cdot \left(\mathsf{neg}\left(\sin x\right)\right)\right) \cdot \left(\mathsf{neg}\left(\sinh y\right)\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\frac{1}{x} \cdot \left(\mathsf{neg}\left(\sin x\right)\right)\right)} \cdot \left(\mathsf{neg}\left(\sinh y\right)\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \left(\color{blue}{\frac{1}{x}} \cdot \left(\mathsf{neg}\left(\sin x\right)\right)\right) \cdot \left(\mathsf{neg}\left(\sinh y\right)\right) \]

      12. lower-neg.f64 N/A

          \[\leadsto \left(\frac{1}{x} \cdot \color{blue}{\left(\mathsf{neg}\left(\sin x\right)\right)}\right) \cdot \left(\mathsf{neg}\left(\sinh y\right)\right) \]

      13. lower-neg.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 68.9

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

   7. Applied rewrites 68.9%

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

   8. Taylor expanded in x around 0

      \[\leadsto -1 \cdot \left(\mathsf{neg}\left(\sinh y\right)\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 71.3%

       \[\leadsto -1 \cdot \left(-\sinh y\right) \]

   11. Taylor expanded in y around 0

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

       1. lower-*.f64 N/A

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

       2. sub-neg N/A

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

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

       5. metadata-eval N/A

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

       6. lower-fma.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. sub-neg N/A

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

       9. *-commutative N/A

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

       10. unpow2 N/A

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

       11. associate-*l* N/A

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

       12. metadata-eval N/A

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

       13. lower-fma.f64 N/A

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

       14. lower-*.f64 66.9

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

   13. Applied rewrites 66.9%

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

   if 4.79999999999999965e31 < 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 77.5

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

   5. Applied rewrites 77.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 27.5%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 46.1%

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

Alternative 20: 57.3% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.8 \cdot 10^{+31}:\\ \;\;\;\;-1 \cdot \left(y \cdot \mathsf{fma}\left(y, y \cdot -0.16666666666666666, -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(x \cdot \left(0.16666666666666666 \cdot \left(y \cdot y\right)\right)\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 4.8e+31)
   (* -1.0 (* y (fma y (* y -0.16666666666666666) -1.0)))
   (/ (* y (* x (* 0.16666666666666666 (* y y)))) x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 4.8e+31) {
		tmp = -1.0 * (y * fma(y, (y * -0.16666666666666666), -1.0));
	} else {
		tmp = (y * (x * (0.16666666666666666 * (y * y)))) / x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 4.8e+31)
		tmp = Float64(-1.0 * Float64(y * fma(y, Float64(y * -0.16666666666666666), -1.0)));
	else
		tmp = Float64(Float64(y * Float64(x * Float64(0.16666666666666666 * Float64(y * y)))) / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 4.8e+31], N[(-1.0 * N[(y * N[(y * N[(y * -0.16666666666666666), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * N[(x * N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.79999999999999965e31

   1. Initial program 87.2%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. remove-double-neg N/A

         \[\leadsto \frac{1}{x} \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\sin x \cdot \sinh y\right)\right)\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{1}{x} \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\sin x \cdot \sinh y}\right)\right)\right)\right) \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto \frac{1}{x} \cdot \left(\mathsf{neg}\left(\color{blue}{\sin x \cdot \left(\mathsf{neg}\left(\sinh y\right)\right)}\right)\right) \]

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

         \[\leadsto \frac{1}{x} \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\sin x\right)\right) \cdot \left(\mathsf{neg}\left(\sinh y\right)\right)\right)} \]

      8. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{x} \cdot \left(\mathsf{neg}\left(\sin x\right)\right)\right) \cdot \left(\mathsf{neg}\left(\sinh y\right)\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{x} \cdot \left(\mathsf{neg}\left(\sin x\right)\right)\right) \cdot \left(\mathsf{neg}\left(\sinh y\right)\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\frac{1}{x} \cdot \left(\mathsf{neg}\left(\sin x\right)\right)\right)} \cdot \left(\mathsf{neg}\left(\sinh y\right)\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \left(\color{blue}{\frac{1}{x}} \cdot \left(\mathsf{neg}\left(\sin x\right)\right)\right) \cdot \left(\mathsf{neg}\left(\sinh y\right)\right) \]

      12. lower-neg.f64 N/A

          \[\leadsto \left(\frac{1}{x} \cdot \color{blue}{\left(\mathsf{neg}\left(\sin x\right)\right)}\right) \cdot \left(\mathsf{neg}\left(\sinh y\right)\right) \]

      13. lower-neg.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 68.9

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

   7. Applied rewrites 68.9%

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

   8. Taylor expanded in x around 0

      \[\leadsto -1 \cdot \left(\mathsf{neg}\left(\sinh y\right)\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 71.3%

       \[\leadsto -1 \cdot \left(-\sinh y\right) \]

   11. Taylor expanded in y around 0

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

       1. lower-*.f64 N/A

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

       2. sub-neg N/A

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

       3. *-commutative N/A

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

       4. unpow2 N/A

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

       5. associate-*l* N/A

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

       6. *-commutative N/A

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

       7. metadata-eval N/A

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

       8. lower-fma.f64 N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 65.0

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

   13. Applied rewrites 65.0%

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

   if 4.79999999999999965e31 < 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 77.5

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

   5. Applied rewrites 77.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 27.5%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 46.1%

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

Alternative 21: 27.1% accurate, 36.2× speedup

Math

\[\begin{array}{l} \\ y \cdot 1 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* y 1.0))

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = y * 1.0d0
end function

Java

public static double code(double x, double y) {
	return y * 1.0;
}

Python

def code(x, y):
	return y * 1.0

Julia

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

MATLAB

function tmp = code(x, y)
	tmp = y * 1.0;
end

Wolfram

code[x_, y_] := N[(y * 1.0), $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 50.8

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

5. Applied rewrites 50.8%

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

6. Taylor expanded in x around 0

   \[\leadsto y \cdot 1 \]
7. Step-by-step derivation

8. Applied rewrites 27.5%

   \[\leadsto y \cdot 1 \]
9. Add Preprocessing

Developer Target 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024232 
(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))