Linear.Quaternion:$ccosh from linear-1.19.1.3

Percentage Accurate: 89.0% → 99.9%

Time: 11.4s

Alternatives: 22

Speedup: 1.0×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 89.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.0%

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

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

   4. associate-/r* N/A

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

   5. clear-num N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 100.0

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

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\frac{\sinh y}{\frac{x}{\sin x}}} \]
5. Add Preprocessing

Alternative 2: 87.2% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 79.4

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

   5. Applied rewrites 79.4%

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

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

   1. Initial program 79.3%

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

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

   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. 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 72.9

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

   5. Applied rewrites 72.9%

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

4. Final simplification 86.7%

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

Alternative 3: 87.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y \cdot \sin x}{x}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\frac{\sinh y \cdot \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot -0.16666666666666666, x\right)}{x}\\ \mathbf{elif}\;t\_0 \leq 0.004:\\ \;\;\;\;\frac{\sin x}{x} \cdot \mathsf{fma}\left(y \cdot y, y \cdot 0.16666666666666666, y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sinh y}{1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sinh y) (sin x)) x)))
   (if (<= t_0 (- INFINITY))
     (/ (* (sinh y) (fma x (* (* x x) -0.16666666666666666) x)) x)
     (if (<= t_0 0.004)
       (* (/ (sin x) x) (fma (* y y) (* y 0.16666666666666666) y))
       (/ (sinh y) 1.0)))))

C

double code(double x, double y) {
	double t_0 = (sinh(y) * sin(x)) / x;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (sinh(y) * fma(x, ((x * x) * -0.16666666666666666), x)) / x;
	} else if (t_0 <= 0.004) {
		tmp = (sin(x) / x) * fma((y * y), (y * 0.16666666666666666), y);
	} else {
		tmp = sinh(y) / 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(sinh(y) * sin(x)) / x)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(sinh(y) * fma(x, Float64(Float64(x * x) * -0.16666666666666666), x)) / x);
	elseif (t_0 <= 0.004)
		tmp = Float64(Float64(sin(x) / x) * fma(Float64(y * y), Float64(y * 0.16666666666666666), y));
	else
		tmp = Float64(sinh(y) / 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sinh[y], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[Sinh[y], $MachinePrecision] * N[(x * N[(N[(x * x), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[t$95$0, 0.004], N[(N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(y * 0.16666666666666666), $MachinePrecision] + y), $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 x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 79.4

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

   5. Applied rewrites 79.4%

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

   if -inf.0 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 0.0040000000000000001

   1. Initial program 79.3%

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

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

   if 0.0040000000000000001 < (/.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. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. clear-num N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\sinh y}{\color{blue}{1}} \]
   6. Step-by-step derivation

   7. Applied rewrites 72.9%

      \[\leadsto \frac{\sinh y}{\color{blue}{1}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -\infty:\\ \;\;\;\;\frac{\sinh y \cdot \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot -0.16666666666666666, x\right)}{x}\\ \mathbf{elif}\;\frac{\sinh y \cdot \sin x}{x} \leq 0.004:\\ \;\;\;\;\frac{\sin x}{x} \cdot \mathsf{fma}\left(y \cdot y, y \cdot 0.16666666666666666, y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sinh y}{1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y \cdot \sin x}{x}\\ t_1 := \mathsf{fma}\left(y \cdot y, y \cdot 0.16666666666666666, y\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), x\right) \cdot t\_1}{x}\\ \mathbf{elif}\;t\_0 \leq 0.004:\\ \;\;\;\;\frac{\sin x}{x} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\sinh y}{1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sinh y) (sin x)) x))
        (t_1 (fma (* y y) (* y 0.16666666666666666) y)))
   (if (<= t_0 (- INFINITY))
     (/
      (*
       (fma
        (* x x)
        (*
         x
         (fma
          (* x x)
          (fma (* x x) -0.0001984126984126984 0.008333333333333333)
          -0.16666666666666666))
        x)
       t_1)
      x)
     (if (<= t_0 0.004) (* (/ (sin x) x) t_1) (/ (sinh y) 1.0)))))

C

double code(double x, double y) {
	double t_0 = (sinh(y) * sin(x)) / x;
	double t_1 = fma((y * y), (y * 0.16666666666666666), y);
	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) * t_1) / x;
	} else if (t_0 <= 0.004) {
		tmp = (sin(x) / x) * t_1;
	} else {
		tmp = sinh(y) / 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(sinh(y) * sin(x)) / x)
	t_1 = fma(Float64(y * y), Float64(y * 0.16666666666666666), y)
	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) * t_1) / x);
	elseif (t_0 <= 0.004)
		tmp = Float64(Float64(sin(x) / x) * t_1);
	else
		tmp = Float64(sinh(y) / 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sinh[y], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, Block[{t$95$1 = N[(N[(y * y), $MachinePrecision] * N[(y * 0.16666666666666666), $MachinePrecision] + y), $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] * t$95$1), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[t$95$0, 0.004], N[(N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision] * t$95$1), $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

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-lft-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 71.3

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

   5. Applied rewrites 71.3%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-*l* N/A

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

      4. *-lft-identity N/A

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

      5. lower-fma.f64 N/A

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

   8. Applied rewrites 62.9%

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

   if -inf.0 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 0.0040000000000000001

   1. Initial program 79.3%

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

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

   if 0.0040000000000000001 < (/.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. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. clear-num N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\sinh y}{\color{blue}{1}} \]
   6. Step-by-step derivation

   7. Applied rewrites 72.9%

      \[\leadsto \frac{\sinh y}{\color{blue}{1}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -\infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), x\right) \cdot \mathsf{fma}\left(y \cdot y, y \cdot 0.16666666666666666, y\right)}{x}\\ \mathbf{elif}\;\frac{\sinh y \cdot \sin x}{x} \leq 0.004:\\ \;\;\;\;\frac{\sin x}{x} \cdot \mathsf{fma}\left(y \cdot y, y \cdot 0.16666666666666666, y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sinh y}{1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y \cdot \sin x}{x}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), x\right) \cdot \mathsf{fma}\left(y \cdot y, y \cdot 0.16666666666666666, y\right)}{x}\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{-14}:\\ \;\;\;\;y \cdot \frac{\sin x}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sinh y}{1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sinh y) (sin x)) x)))
   (if (<= t_0 (- INFINITY))
     (/
      (*
       (fma
        (* x x)
        (*
         x
         (fma
          (* x x)
          (fma (* x x) -0.0001984126984126984 0.008333333333333333)
          -0.16666666666666666))
        x)
       (fma (* y y) (* y 0.16666666666666666) y))
      x)
     (if (<= t_0 4e-14) (* y (/ (sin x) x)) (/ (sinh y) 1.0)))))

C

double code(double x, double y) {
	double t_0 = (sinh(y) * sin(x)) / x;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (fma((x * x), (x * fma((x * x), fma((x * x), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666)), x) * fma((y * y), (y * 0.16666666666666666), y)) / x;
	} else if (t_0 <= 4e-14) {
		tmp = y * (sin(x) / x);
	} else {
		tmp = sinh(y) / 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(sinh(y) * sin(x)) / x)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(fma(Float64(x * x), Float64(x * fma(Float64(x * x), fma(Float64(x * x), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666)), x) * fma(Float64(y * y), Float64(y * 0.16666666666666666), y)) / x);
	elseif (t_0 <= 4e-14)
		tmp = Float64(y * Float64(sin(x) / x));
	else
		tmp = Float64(sinh(y) / 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sinh[y], $MachinePrecision] * N[Sin[x], $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[(y * y), $MachinePrecision] * N[(y * 0.16666666666666666), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[t$95$0, 4e-14], 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

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-lft-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 71.3

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

   5. Applied rewrites 71.3%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-*l* N/A

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

      4. *-lft-identity N/A

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

      5. lower-fma.f64 N/A

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

   8. Applied rewrites 62.9%

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

   if -inf.0 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 4e-14

   1. Initial program 79.1%

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

   3. Taylor expanded in y around 0

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

      1. associate-/l* N/A

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

      2. 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 98.0

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

   5. Applied rewrites 98.0%

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

   if 4e-14 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 100.0%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing
   3. 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. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. clear-num N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\sinh y}{\color{blue}{1}} \]
   6. Step-by-step derivation

   7. Applied rewrites 73.2%

      \[\leadsto \frac{\sinh y}{\color{blue}{1}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -\infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), x\right) \cdot \mathsf{fma}\left(y \cdot y, y \cdot 0.16666666666666666, y\right)}{x}\\ \mathbf{elif}\;\frac{\sinh y \cdot \sin x}{x} \leq 4 \cdot 10^{-14}:\\ \;\;\;\;y \cdot \frac{\sin x}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sinh y}{1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 62.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq 0.004:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, y \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right)\right), y\right)}{x} \cdot \mathsf{fma}\left(x, x \cdot \left(x \cdot -0.16666666666666666\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sinh y}{1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (sinh y) (sin x)) x) 0.004)
   (*
    (/
     (fma
      y
      (*
       y
       (*
        y
        (fma
         (* y y)
         (fma (* y y) 0.0001984126984126984 0.008333333333333333)
         0.16666666666666666)))
      y)
     x)
    (fma x (* x (* x -0.16666666666666666)) x))
   (/ (sinh y) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (((sinh(y) * sin(x)) / x) <= 0.004) {
		tmp = (fma(y, (y * (y * fma((y * y), fma((y * y), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666))), y) / x) * fma(x, (x * (x * -0.16666666666666666)), x);
	} else {
		tmp = sinh(y) / 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(sinh(y) * sin(x)) / x) <= 0.004)
		tmp = Float64(Float64(fma(y, Float64(y * Float64(y * fma(Float64(y * y), fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666))), y) / x) * fma(x, Float64(x * Float64(x * -0.16666666666666666)), x));
	else
		tmp = Float64(sinh(y) / 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[Sinh[y], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], 0.004], N[(N[(N[(y * N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] / x), $MachinePrecision] * N[(x * N[(x * N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]), $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) < 0.0040000000000000001

   1. Initial program 86.3%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-lft-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 75.6

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

   5. Applied rewrites 75.6%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 41.0

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

   8. Applied rewrites 41.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   10. Applied rewrites 58.9%

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

   11. Taylor expanded in y around 0

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

       1. +-commutative N/A

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

       2. distribute-lft-in N/A

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

       3. *-rgt-identity N/A

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

       4. lower-fma.f64 N/A

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

   13. Applied rewrites 60.5%

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

   if 0.0040000000000000001 < (/.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. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. clear-num N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\sinh y}{\color{blue}{1}} \]
   6. Step-by-step derivation

   7. Applied rewrites 72.9%

      \[\leadsto \frac{\sinh y}{\color{blue}{1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq 0.004:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, y \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right)\right), y\right)}{x} \cdot \mathsf{fma}\left(x, x \cdot \left(x \cdot -0.16666666666666666\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sinh y}{1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 61.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq 4 \cdot 10^{-308}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, y \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right)\right), y\right)}{x} \cdot \mathsf{fma}\left(x, x \cdot \left(x \cdot -0.16666666666666666\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, y \cdot \left(y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right)\right), y\right)}{x} \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, x \cdot 0.008333333333333333, -0.16666666666666666\right), x \cdot \left(x \cdot x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (sinh y) (sin x)) x) 4e-308)
   (*
    (/
     (fma
      y
      (*
       y
       (*
        y
        (fma
         (* y y)
         (fma (* y y) 0.0001984126984126984 0.008333333333333333)
         0.16666666666666666)))
      y)
     x)
    (fma x (* x (* x -0.16666666666666666)) x))
   (*
    (/
     (fma
      y
      (*
       y
       (*
        y
        (fma
         y
         (* y (fma y (* y 0.0001984126984126984) 0.008333333333333333))
         0.16666666666666666)))
      y)
     x)
    (fma
     (fma x (* x 0.008333333333333333) -0.16666666666666666)
     (* x (* x x))
     x))))

C

double code(double x, double y) {
	double tmp;
	if (((sinh(y) * sin(x)) / x) <= 4e-308) {
		tmp = (fma(y, (y * (y * fma((y * y), fma((y * y), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666))), y) / x) * fma(x, (x * (x * -0.16666666666666666)), x);
	} else {
		tmp = (fma(y, (y * (y * fma(y, (y * fma(y, (y * 0.0001984126984126984), 0.008333333333333333)), 0.16666666666666666))), 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(sinh(y) * sin(x)) / x) <= 4e-308)
		tmp = Float64(Float64(fma(y, Float64(y * Float64(y * fma(Float64(y * y), fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666))), y) / x) * fma(x, Float64(x * Float64(x * -0.16666666666666666)), x));
	else
		tmp = Float64(Float64(fma(y, Float64(y * Float64(y * fma(y, Float64(y * fma(y, Float64(y * 0.0001984126984126984), 0.008333333333333333)), 0.16666666666666666))), 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[Sinh[y], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], 4e-308], N[(N[(N[(y * N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] / x), $MachinePrecision] * N[(x * N[(x * N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y * N[(y * N[(y * N[(y * N[(y * N[(y * N[(y * 0.0001984126984126984), $MachinePrecision] + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $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) < 4.00000000000000013e-308

   1. Initial program 84.2%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-lft-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 71.7

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

   5. Applied rewrites 71.7%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 38.0

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

   8. Applied rewrites 38.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   10. Applied rewrites 58.8%

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

   11. Taylor expanded in y around 0

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

       1. +-commutative N/A

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

       2. distribute-lft-in N/A

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

       3. *-rgt-identity N/A

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

       4. lower-fma.f64 N/A

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

   13. Applied rewrites 60.7%

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

   if 4.00000000000000013e-308 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 99.9%

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

   3. Taylor expanded in 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 68.9

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

   5. Applied rewrites 68.9%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

   8. Applied rewrites 63.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   10. Applied rewrites 66.0%

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

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq 4 \cdot 10^{-308}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, y \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right)\right), y\right)}{x} \cdot \mathsf{fma}\left(x, x \cdot \left(x \cdot -0.16666666666666666\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, y \cdot \left(y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right)\right), y\right)}{x} \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, x \cdot 0.008333333333333333, -0.16666666666666666\right), x \cdot \left(x \cdot x\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 60.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq 2 \cdot 10^{-183}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, y \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right)\right), y\right)}{x} \cdot \mathsf{fma}\left(x, x \cdot \left(x \cdot -0.16666666666666666\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.008333333333333333, -0.16666666666666666\right), x \cdot \left(x \cdot x\right), x\right) \cdot \mathsf{fma}\left(y \cdot \mathsf{fma}\left(y \cdot y, y \cdot \left(y \cdot 0.0001984126984126984\right), 0.16666666666666666\right), y \cdot y, y\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (sinh y) (sin x)) x) 2e-183)
   (*
    (/
     (fma
      y
      (*
       y
       (*
        y
        (fma
         (* y y)
         (fma (* y y) 0.0001984126984126984 0.008333333333333333)
         0.16666666666666666)))
      y)
     x)
    (fma x (* x (* x -0.16666666666666666)) x))
   (/
    (*
     (fma
      (fma (* x x) 0.008333333333333333 -0.16666666666666666)
      (* x (* x x))
      x)
     (fma
      (* y (fma (* y y) (* y (* y 0.0001984126984126984)) 0.16666666666666666))
      (* y y)
      y))
    x)))

C

double code(double x, double y) {
	double tmp;
	if (((sinh(y) * sin(x)) / x) <= 2e-183) {
		tmp = (fma(y, (y * (y * fma((y * y), fma((y * y), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666))), y) / x) * fma(x, (x * (x * -0.16666666666666666)), x);
	} else {
		tmp = (fma(fma((x * x), 0.008333333333333333, -0.16666666666666666), (x * (x * x)), x) * fma((y * fma((y * y), (y * (y * 0.0001984126984126984)), 0.16666666666666666)), (y * y), y)) / x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(sinh(y) * sin(x)) / x) <= 2e-183)
		tmp = Float64(Float64(fma(y, Float64(y * Float64(y * fma(Float64(y * y), fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666))), y) / x) * fma(x, Float64(x * Float64(x * -0.16666666666666666)), x));
	else
		tmp = Float64(Float64(fma(fma(Float64(x * x), 0.008333333333333333, -0.16666666666666666), Float64(x * Float64(x * x)), x) * fma(Float64(y * fma(Float64(y * y), Float64(y * Float64(y * 0.0001984126984126984)), 0.16666666666666666)), Float64(y * y), y)) / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[Sinh[y], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], 2e-183], N[(N[(N[(y * N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] / x), $MachinePrecision] * N[(x * N[(x * N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(x * x), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision] * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] * N[(N[(y * N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * 0.0001984126984126984), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] * N[(y * y), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 2.00000000000000001e-183

   1. Initial program 85.2%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-lft-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 73.6

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

   5. Applied rewrites 73.6%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 39.5

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

   8. Applied rewrites 39.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   10. Applied rewrites 58.9%

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

   11. Taylor expanded in y around 0

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

       1. +-commutative N/A

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

       2. distribute-lft-in N/A

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

       3. *-rgt-identity N/A

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

       4. lower-fma.f64 N/A

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

   13. Applied rewrites 60.6%

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

   if 2.00000000000000001e-183 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 99.9%

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

   3. Taylor expanded in 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 70.3

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

   5. Applied rewrites 70.3%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

   8. Applied rewrites 63.5%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 63.5%

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

4. Final simplification 61.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq 2 \cdot 10^{-183}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, y \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right)\right), y\right)}{x} \cdot \mathsf{fma}\left(x, x \cdot \left(x \cdot -0.16666666666666666\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.008333333333333333, -0.16666666666666666\right), x \cdot \left(x \cdot x\right), x\right) \cdot \mathsf{fma}\left(y \cdot \mathsf{fma}\left(y \cdot y, y \cdot \left(y \cdot 0.0001984126984126984\right), 0.16666666666666666\right), y \cdot y, y\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 60.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y, y \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right)\right), y\right)\\ \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq 0.004:\\ \;\;\;\;\frac{t\_0}{x} \cdot \mathsf{fma}\left(x, x \cdot \left(x \cdot -0.16666666666666666\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (fma
          y
          (*
           y
           (*
            y
            (fma
             (* y y)
             (fma (* y y) 0.0001984126984126984 0.008333333333333333)
             0.16666666666666666)))
          y)))
   (if (<= (/ (* (sinh y) (sin x)) x) 0.004)
     (* (/ t_0 x) (fma x (* x (* x -0.16666666666666666)) x))
     (/ t_0 1.0))))

C

double code(double x, double y) {
	double t_0 = fma(y, (y * (y * fma((y * y), fma((y * y), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666))), y);
	double tmp;
	if (((sinh(y) * sin(x)) / x) <= 0.004) {
		tmp = (t_0 / x) * fma(x, (x * (x * -0.16666666666666666)), x);
	} else {
		tmp = t_0 / 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(y, Float64(y * Float64(y * fma(Float64(y * y), fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666))), y)
	tmp = 0.0
	if (Float64(Float64(sinh(y) * sin(x)) / x) <= 0.004)
		tmp = Float64(Float64(t_0 / x) * fma(x, Float64(x * Float64(x * -0.16666666666666666)), x));
	else
		tmp = Float64(t_0 / 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sinh[y], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], 0.004], N[(N[(t$95$0 / x), $MachinePrecision] * N[(x * N[(x * N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 0.0040000000000000001

   1. Initial program 86.3%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-lft-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 75.6

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

   5. Applied rewrites 75.6%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 41.0

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

   8. Applied rewrites 41.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   10. Applied rewrites 58.9%

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

   11. Taylor expanded in y around 0

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

       1. +-commutative N/A

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

       2. distribute-lft-in N/A

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

       3. *-rgt-identity N/A

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

       4. lower-fma.f64 N/A

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

   13. Applied rewrites 60.5%

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

   if 0.0040000000000000001 < (/.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. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. clear-num N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\sinh y}{\color{blue}{1}} \]
   6. Step-by-step derivation

   7. Applied rewrites 72.9%

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

   8. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

   10. Applied rewrites 61.9%

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

4. Final simplification 60.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq 0.004:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, y \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right)\right), y\right)}{x} \cdot \mathsf{fma}\left(x, x \cdot \left(x \cdot -0.16666666666666666\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, y \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right)\right), y\right)}{1}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 59.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq 0.004:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot \left(x \cdot -0.16666666666666666\right), x\right) \cdot \frac{\mathsf{fma}\left(y, y \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right)\right), y\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, y \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right)\right), y\right)}{1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (sinh y) (sin x)) x) 0.004)
   (*
    (fma x (* x (* x -0.16666666666666666)) x)
    (/
     (fma
      y
      (* y (* y (fma (* y y) 0.008333333333333333 0.16666666666666666)))
      y)
     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 (((sinh(y) * sin(x)) / x) <= 0.004) {
		tmp = fma(x, (x * (x * -0.16666666666666666)), x) * (fma(y, (y * (y * fma((y * y), 0.008333333333333333, 0.16666666666666666))), y) / 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(sinh(y) * sin(x)) / x) <= 0.004)
		tmp = Float64(fma(x, Float64(x * Float64(x * -0.16666666666666666)), x) * Float64(fma(y, Float64(y * Float64(y * fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666))), y) / x));
	else
		tmp = Float64(fma(y, Float64(y * Float64(y * fma(Float64(y * y), fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666))), y) / 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[Sinh[y], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], 0.004], N[(N[(x * N[(x * N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] * N[(N[(y * N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[(y * N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $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) < 0.0040000000000000001

   1. Initial program 86.3%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-lft-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 75.6

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

   5. Applied rewrites 75.6%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 41.0

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

   8. Applied rewrites 41.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   10. Applied rewrites 58.9%

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

   11. Taylor expanded in y around 0

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

       1. +-commutative N/A

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

       2. distribute-lft-in N/A

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

       3. *-rgt-identity N/A

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

       4. lower-fma.f64 N/A

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

       5. unpow2 N/A

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

       6. associate-*l* N/A

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

       7. lower-*.f64 N/A

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

       8. lower-*.f64 N/A

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

       9. +-commutative N/A

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

       10. *-commutative N/A

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

       11. lower-fma.f64 N/A

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

       12. unpow2 N/A

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

       13. lower-*.f64 60.0

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

   13. Applied rewrites 60.0%

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

   if 0.0040000000000000001 < (/.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. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. clear-num N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\sinh y}{\color{blue}{1}} \]
   6. Step-by-step derivation

   7. Applied rewrites 72.9%

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

   8. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

   10. Applied rewrites 61.9%

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

4. Final simplification 60.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq 0.004:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot \left(x \cdot -0.16666666666666666\right), x\right) \cdot \frac{\mathsf{fma}\left(y, y \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right)\right), y\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, y \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right)\right), y\right)}{1}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 57.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -5 \cdot 10^{-159}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, y \cdot 0.16666666666666666, y\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, y \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right)\right), y\right)}{1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (sinh y) (sin x)) x) -5e-159)
   (*
    (fma (* y y) (* y 0.16666666666666666) y)
    (fma
     (* x x)
     (fma
      (* x x)
      (fma (* x x) -0.0001984126984126984 0.008333333333333333)
      -0.16666666666666666)
     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 (((sinh(y) * sin(x)) / x) <= -5e-159) {
		tmp = fma((y * y), (y * 0.16666666666666666), y) * fma((x * x), fma((x * x), fma((x * x), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666), 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(sinh(y) * sin(x)) / x) <= -5e-159)
		tmp = Float64(fma(Float64(y * y), Float64(y * 0.16666666666666666), y) * fma(Float64(x * x), fma(Float64(x * x), fma(Float64(x * x), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666), 1.0));
	else
		tmp = Float64(fma(y, Float64(y * Float64(y * fma(Float64(y * y), fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666))), y) / 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[Sinh[y], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], -5e-159], N[(N[(N[(y * y), $MachinePrecision] * N[(y * 0.16666666666666666), $MachinePrecision] + y), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(y * N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $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) < -5.00000000000000032e-159

   1. Initial program 99.9%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 77.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 64.0%

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

   if -5.00000000000000032e-159 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 84.9%

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

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

      4. associate-/r* N/A

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

      5. clear-num N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\sinh y}{\color{blue}{1}} \]
   6. Step-by-step derivation

   7. Applied rewrites 58.3%

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

   8. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

   10. Applied rewrites 53.7%

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

4. Final simplification 57.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -5 \cdot 10^{-159}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, y \cdot 0.16666666666666666, y\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, y \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right)\right), y\right)}{1}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 57.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -5 \cdot 10^{-159}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot -0.16666666666666666, x\right) \cdot \mathsf{fma}\left(y \cdot y, y \cdot 0.16666666666666666, y\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, y \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right)\right), y\right)}{1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (sinh y) (sin x)) x) -5e-159)
   (/
    (*
     (fma x (* (* x x) -0.16666666666666666) x)
     (fma (* y y) (* y 0.16666666666666666) y))
    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 (((sinh(y) * sin(x)) / x) <= -5e-159) {
		tmp = (fma(x, ((x * x) * -0.16666666666666666), x) * fma((y * y), (y * 0.16666666666666666), y)) / 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(sinh(y) * sin(x)) / x) <= -5e-159)
		tmp = Float64(Float64(fma(x, Float64(Float64(x * x) * -0.16666666666666666), x) * fma(Float64(y * y), Float64(y * 0.16666666666666666), y)) / x);
	else
		tmp = Float64(fma(y, Float64(y * Float64(y * fma(Float64(y * y), fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666))), y) / 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[Sinh[y], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], -5e-159], N[(N[(N[(x * N[(N[(x * x), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + x), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(y * 0.16666666666666666), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(y * N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $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) < -5.00000000000000032e-159

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-lft-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 77.2

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

   5. Applied rewrites 77.2%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 63.8

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

   8. Applied rewrites 63.8%

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

   if -5.00000000000000032e-159 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 84.9%

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

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

      4. associate-/r* N/A

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

      5. clear-num N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\sinh y}{\color{blue}{1}} \]
   6. Step-by-step derivation

   7. Applied rewrites 58.3%

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

   8. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

   10. Applied rewrites 53.7%

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

4. Final simplification 57.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -5 \cdot 10^{-159}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot -0.16666666666666666, x\right) \cdot \mathsf{fma}\left(y \cdot y, y \cdot 0.16666666666666666, y\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, y \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right)\right), y\right)}{1}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 55.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -5 \cdot 10^{-159}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot -0.16666666666666666, x\right) \cdot \mathsf{fma}\left(y \cdot y, y \cdot 0.16666666666666666, y\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, y \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right)\right), y\right)}{1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (sinh y) (sin x)) x) -5e-159)
   (/
    (*
     (fma x (* (* x x) -0.16666666666666666) x)
     (fma (* y y) (* y 0.16666666666666666) y))
    x)
   (/
    (fma
     y
     (* y (* y (fma (* y y) 0.008333333333333333 0.16666666666666666)))
     y)
    1.0)))

C

double code(double x, double y) {
	double tmp;
	if (((sinh(y) * sin(x)) / x) <= -5e-159) {
		tmp = (fma(x, ((x * x) * -0.16666666666666666), x) * fma((y * y), (y * 0.16666666666666666), y)) / x;
	} else {
		tmp = fma(y, (y * (y * fma((y * y), 0.008333333333333333, 0.16666666666666666))), y) / 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(sinh(y) * sin(x)) / x) <= -5e-159)
		tmp = Float64(Float64(fma(x, Float64(Float64(x * x) * -0.16666666666666666), x) * fma(Float64(y * y), Float64(y * 0.16666666666666666), y)) / x);
	else
		tmp = Float64(fma(y, Float64(y * Float64(y * fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666))), y) / 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[Sinh[y], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], -5e-159], N[(N[(N[(x * N[(N[(x * x), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + x), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(y * 0.16666666666666666), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(y * N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision]), $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) < -5.00000000000000032e-159

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-lft-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 77.2

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

   5. Applied rewrites 77.2%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 63.8

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

   8. Applied rewrites 63.8%

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

   if -5.00000000000000032e-159 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 84.9%

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

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

      4. associate-/r* N/A

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

      5. clear-num N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\sinh y}{\color{blue}{1}} \]
   6. Step-by-step derivation

   7. Applied rewrites 58.3%

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

   8. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 53.7

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

   10. Applied rewrites 53.7%

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

4. Final simplification 57.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -5 \cdot 10^{-159}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot -0.16666666666666666, x\right) \cdot \mathsf{fma}\left(y \cdot y, y \cdot 0.16666666666666666, y\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, y \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right)\right), y\right)}{1}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 57.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq 0:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot \left(x \cdot -0.16666666666666666\right), x\right) \cdot \frac{\mathsf{fma}\left(y, y \cdot \left(y \cdot 0.16666666666666666\right), y\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, y \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right)\right), y\right)}{1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (sinh y) (sin x)) x) 0.0)
   (*
    (fma x (* x (* x -0.16666666666666666)) x)
    (/ (fma y (* y (* y 0.16666666666666666)) y) x))
   (/
    (fma
     y
     (* y (* y (fma (* y y) 0.008333333333333333 0.16666666666666666)))
     y)
    1.0)))

C

double code(double x, double y) {
	double tmp;
	if (((sinh(y) * sin(x)) / x) <= 0.0) {
		tmp = fma(x, (x * (x * -0.16666666666666666)), x) * (fma(y, (y * (y * 0.16666666666666666)), y) / x);
	} else {
		tmp = fma(y, (y * (y * fma((y * y), 0.008333333333333333, 0.16666666666666666))), y) / 1.0;
	}
	return tmp;
}

Julia

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

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[Sinh[y], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], 0.0], N[(N[(x * N[(x * N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] * N[(N[(y * N[(y * N[(y * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[(y * N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision]), $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) < 0.0

   1. Initial program 84.2%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-lft-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 71.7

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

   5. Applied rewrites 71.7%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 38.0

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

   8. Applied rewrites 38.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   10. Applied rewrites 58.8%

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

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

   1. Initial program 99.9%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing
   3. 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. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. clear-num N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\sinh y}{\color{blue}{1}} \]
   6. Step-by-step derivation

   7. Applied rewrites 69.2%

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

   8. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 61.3

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

   10. Applied rewrites 61.3%

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

4. Final simplification 59.7%

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

Alternative 15: 55.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -5 \cdot 10^{-159}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, y \cdot 0.16666666666666666, y\right) \cdot \mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, y \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right)\right), y\right)}{1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (sinh y) (sin x)) x) -5e-159)
   (*
    (fma (* y y) (* y 0.16666666666666666) y)
    (fma (* x x) -0.16666666666666666 1.0))
   (/
    (fma
     y
     (* y (* y (fma (* y y) 0.008333333333333333 0.16666666666666666)))
     y)
    1.0)))

C

double code(double x, double y) {
	double tmp;
	if (((sinh(y) * sin(x)) / x) <= -5e-159) {
		tmp = fma((y * y), (y * 0.16666666666666666), y) * fma((x * x), -0.16666666666666666, 1.0);
	} else {
		tmp = fma(y, (y * (y * fma((y * y), 0.008333333333333333, 0.16666666666666666))), y) / 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(sinh(y) * sin(x)) / x) <= -5e-159)
		tmp = Float64(fma(Float64(y * y), Float64(y * 0.16666666666666666), y) * fma(Float64(x * x), -0.16666666666666666, 1.0));
	else
		tmp = Float64(fma(y, Float64(y * Float64(y * fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666))), y) / 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[Sinh[y], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], -5e-159], N[(N[(N[(y * y), $MachinePrecision] * N[(y * 0.16666666666666666), $MachinePrecision] + y), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(y * N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision]), $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) < -5.00000000000000032e-159

   1. Initial program 99.9%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 77.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 63.8%

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

   if -5.00000000000000032e-159 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 84.9%

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

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

      4. associate-/r* N/A

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

      5. clear-num N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\sinh y}{\color{blue}{1}} \]
   6. Step-by-step derivation

   7. Applied rewrites 58.3%

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

   8. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 53.7

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

   10. Applied rewrites 53.7%

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

4. Final simplification 57.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -5 \cdot 10^{-159}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, y \cdot 0.16666666666666666, y\right) \cdot \mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, y \cdot \left(y \cdot \mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right)\right), y\right)}{1}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 53.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -5 \cdot 10^{-159}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, y \cdot 0.16666666666666666, y\right) \cdot \mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \left(y \cdot y\right) \cdot 0.16666666666666666, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (sinh y) (sin x)) x) -5e-159)
   (*
    (fma (* y y) (* y 0.16666666666666666) y)
    (fma (* x x) -0.16666666666666666 1.0))
   (fma y (* (* y y) 0.16666666666666666) y)))

C

double code(double x, double y) {
	double tmp;
	if (((sinh(y) * sin(x)) / x) <= -5e-159) {
		tmp = fma((y * y), (y * 0.16666666666666666), y) * fma((x * x), -0.16666666666666666, 1.0);
	} else {
		tmp = fma(y, ((y * y) * 0.16666666666666666), y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(sinh(y) * sin(x)) / x) <= -5e-159)
		tmp = Float64(fma(Float64(y * y), Float64(y * 0.16666666666666666), y) * fma(Float64(x * x), -0.16666666666666666, 1.0));
	else
		tmp = fma(y, Float64(Float64(y * y) * 0.16666666666666666), y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[Sinh[y], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], -5e-159], N[(N[(N[(y * y), $MachinePrecision] * N[(y * 0.16666666666666666), $MachinePrecision] + y), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -5.00000000000000032e-159

   1. Initial program 99.9%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 77.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 63.8%

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

   if -5.00000000000000032e-159 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 84.9%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 88.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 50.4%

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

4. Final simplification 55.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -5 \cdot 10^{-159}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, y \cdot 0.16666666666666666, y\right) \cdot \mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \left(y \cdot y\right) \cdot 0.16666666666666666, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 39.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -5 \cdot 10^{-159}:\\ \;\;\;\;y \cdot \left(x \cdot \left(x \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \left(y \cdot y\right) \cdot 0.16666666666666666, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (sinh y) (sin x)) x) -5e-159)
   (* y (* x (* x -0.16666666666666666)))
   (fma y (* (* y y) 0.16666666666666666) y)))

C

double code(double x, double y) {
	double tmp;
	if (((sinh(y) * sin(x)) / x) <= -5e-159) {
		tmp = y * (x * (x * -0.16666666666666666));
	} else {
		tmp = fma(y, ((y * y) * 0.16666666666666666), y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(sinh(y) * sin(x)) / x) <= -5e-159)
		tmp = Float64(y * Float64(x * Float64(x * -0.16666666666666666)));
	else
		tmp = fma(y, Float64(Float64(y * y) * 0.16666666666666666), y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[Sinh[y], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], -5e-159], N[(y * N[(x * N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -5.00000000000000032e-159

   1. Initial program 99.9%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 29.1

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

   5. Applied rewrites 29.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 37.8%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 18.7%

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

   if -5.00000000000000032e-159 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 84.9%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 88.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 50.4%

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

4. Final simplification 39.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -5 \cdot 10^{-159}:\\ \;\;\;\;y \cdot \left(x \cdot \left(x \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \left(y \cdot y\right) \cdot 0.16666666666666666, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 27.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -5 \cdot 10^{-159}:\\ \;\;\;\;y \cdot \left(x \cdot \left(x \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (sinh y) (sin x)) x) -5e-159)
   (* y (* x (* x -0.16666666666666666)))
   (* y 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (((sinh(y) * sin(x)) / x) <= -5e-159) {
		tmp = y * (x * (x * -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 (((sinh(y) * sin(x)) / x) <= (-5d-159)) then
        tmp = y * (x * (x * (-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.sinh(y) * Math.sin(x)) / x) <= -5e-159) {
		tmp = y * (x * (x * -0.16666666666666666));
	} else {
		tmp = y * 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((math.sinh(y) * math.sin(x)) / x) <= -5e-159:
		tmp = y * (x * (x * -0.16666666666666666))
	else:
		tmp = y * 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(sinh(y) * sin(x)) / x) <= -5e-159)
		tmp = Float64(y * Float64(x * Float64(x * -0.16666666666666666)));
	else
		tmp = Float64(y * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((sinh(y) * sin(x)) / x) <= -5e-159)
		tmp = y * (x * (x * -0.16666666666666666));
	else
		tmp = y * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[Sinh[y], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], -5e-159], N[(y * N[(x * N[(x * -0.16666666666666666), $MachinePrecision]), $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) < -5.00000000000000032e-159

   1. Initial program 99.9%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 29.1

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

   5. Applied rewrites 29.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 37.8%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 18.7%

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

   if -5.00000000000000032e-159 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 84.9%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 60.2

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

   5. Applied rewrites 60.2%

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

   6. Taylor expanded in x around 0

      \[\leadsto y \cdot 1 \]
   7. Step-by-step derivation

   8. Applied rewrites 29.6%

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

4. Final simplification 25.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -5 \cdot 10^{-159}:\\ \;\;\;\;y \cdot \left(x \cdot \left(x \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 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 90.0%

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

   1. lift-/.f64 N/A

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

   2. 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 99.9

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

4. Applied rewrites 99.9%

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

5. Final simplification 99.9%

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

Alternative 20: 36.0% accurate, 12.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 90.0%

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

3. Taylor expanded in y around 0

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

   1. associate-/l* N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-sin.f64 49.5

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

5. Applied rewrites 49.5%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 37.1%

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

10. Applied rewrites 37.1%

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

Alternative 21: 36.0% accurate, 12.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 90.0%

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

3. Taylor expanded in y around 0

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

   1. associate-/l* N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-sin.f64 49.5

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

5. Applied rewrites 49.5%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 37.1%

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

Alternative 22: 28.0% 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 90.0%

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

3. Taylor expanded in y around 0

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

   1. associate-/l* N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-sin.f64 49.5

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

5. Applied rewrites 49.5%

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

6. Taylor expanded in x around 0

   \[\leadsto y \cdot 1 \]
7. Step-by-step derivation

8. Applied rewrites 26.7%

   \[\leadsto y \cdot 1 \]
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 2024220 
(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))