Linear.Quaternion:$ccosh from linear-1.19.1.3

Percentage Accurate: 89.1% → 99.8%

Time: 13.0s

Alternatives: 23

Speedup: 1.4×

• 50.6% 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: 89.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

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(Float64(sinh(y) / x) * sin(x))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.2%

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

   1. lift-/.f64 N/A

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

   2. 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. Add Preprocessing

Alternative 2: 82.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y \cdot \sin x}{x}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right) \cdot \left(\mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right) \cdot \left(y \cdot x\right)\right)}{x}\\ \mathbf{elif}\;t\_0 \leq 0.001:\\ \;\;\;\;\sin x \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{y} - e^{-y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sinh y) (sin x)) x)))
   (if (<= t_0 (- INFINITY))
     (/
      (*
       (fma (* x x) -0.16666666666666666 1.0)
       (* (fma 0.16666666666666666 (* y y) 1.0) (* y x)))
      x)
     (if (<= t_0 0.001)
       (*
        (sin x)
        (/
         (fma
          (fma y (* y 0.008333333333333333) 0.16666666666666666)
          (* y (* y y))
          y)
         x))
       (* 0.5 (- (exp y) (exp (- y))))))))

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), -0.16666666666666666, 1.0) * (fma(0.16666666666666666, (y * y), 1.0) * (y * x))) / x;
	} else if (t_0 <= 0.001) {
		tmp = sin(x) * (fma(fma(y, (y * 0.008333333333333333), 0.16666666666666666), (y * (y * y)), y) / x);
	} else {
		tmp = 0.5 * (exp(y) - exp(-y));
	}
	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), -0.16666666666666666, 1.0) * Float64(fma(0.16666666666666666, Float64(y * y), 1.0) * Float64(y * x))) / x);
	elseif (t_0 <= 0.001)
		tmp = Float64(sin(x) * Float64(fma(fma(y, Float64(y * 0.008333333333333333), 0.16666666666666666), Float64(y * Float64(y * y)), y) / x));
	else
		tmp = Float64(0.5 * Float64(exp(y) - exp(Float64(-y))));
	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] * -0.16666666666666666 + 1.0), $MachinePrecision] * N[(N[(0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[t$95$0, 0.001], N[(N[Sin[x], $MachinePrecision] * N[(N[(N[(y * N[(y * 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Exp[y], $MachinePrecision] - N[Exp[(-y)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-lft-identity N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt-in N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 77.5

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

   5. Applied rewrites 77.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 67.2%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 60.4%

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

   if -inf.0 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 1e-3

   1. Initial program 75.0%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. unpow2 N/A

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

      6. unpow3 N/A

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

      7. *-lft-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. cube-mult N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 N/A

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

      20. unpow2 N/A

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

      21. lower-*.f64 98.4

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

   7. Applied rewrites 98.4%

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

   if 1e-3 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. rec-exp N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-neg.f64 75.4

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

   5. Applied rewrites 75.4%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -\infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right) \cdot \left(\mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right) \cdot \left(y \cdot x\right)\right)}{x}\\ \mathbf{elif}\;\frac{\sinh y \cdot \sin x}{x} \leq 0.001:\\ \;\;\;\;\sin x \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{y} - e^{-y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y \cdot \sin x}{x}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right) \cdot \left(\mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right) \cdot \left(y \cdot x\right)\right)}{x}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\sin x \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sinh y \cdot \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.008333333333333333, -0.16666666666666666\right), x \cdot \left(x \cdot x\right), x\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sinh y) (sin x)) x)))
   (if (<= t_0 (- INFINITY))
     (/
      (*
       (fma (* x x) -0.16666666666666666 1.0)
       (* (fma 0.16666666666666666 (* y y) 1.0) (* y x)))
      x)
     (if (<= t_0 5e-8)
       (*
        (sin x)
        (/
         (fma
          (fma y (* y 0.008333333333333333) 0.16666666666666666)
          (* y (* y y))
          y)
         x))
       (/
        (*
         (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 = (fma((x * x), -0.16666666666666666, 1.0) * (fma(0.16666666666666666, (y * y), 1.0) * (y * x))) / x;
	} else if (t_0 <= 5e-8) {
		tmp = sin(x) * (fma(fma(y, (y * 0.008333333333333333), 0.16666666666666666), (y * (y * y)), y) / x);
	} 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(fma(Float64(x * x), -0.16666666666666666, 1.0) * Float64(fma(0.16666666666666666, Float64(y * y), 1.0) * Float64(y * x))) / x);
	elseif (t_0 <= 5e-8)
		tmp = Float64(sin(x) * Float64(fma(fma(y, Float64(y * 0.008333333333333333), 0.16666666666666666), Float64(y * Float64(y * y)), y) / x));
	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[(N[(x * x), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * N[(N[(0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[t$95$0, 5e-8], N[(N[Sin[x], $MachinePrecision] * N[(N[(N[(y * N[(y * 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] / x), $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 y around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-lft-identity N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt-in N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 77.5

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

   5. Applied rewrites 77.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 67.2%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 60.4%

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

   if -inf.0 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 4.9999999999999998e-8

   1. Initial program 75.0%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. unpow2 N/A

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

      6. unpow3 N/A

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

      7. *-lft-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. cube-mult N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 N/A

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

      20. unpow2 N/A

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

      21. lower-*.f64 98.4

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

   7. Applied rewrites 98.4%

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

   if 4.9999999999999998e-8 < (/.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 75.4

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

   5. Applied rewrites 75.4%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -\infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right) \cdot \left(\mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right) \cdot \left(y \cdot x\right)\right)}{x}\\ \mathbf{elif}\;\frac{\sinh y \cdot \sin x}{x} \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\sin x \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sinh y \cdot \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.008333333333333333, -0.16666666666666666\right), x \cdot \left(x \cdot x\right), x\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y \cdot \sin x}{x}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right) \cdot \left(\mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right) \cdot \left(y \cdot x\right)\right)}{x}\\ \mathbf{elif}\;t\_0 \leq 10^{-21}:\\ \;\;\;\;\frac{\sin x}{x} \cdot \mathsf{fma}\left(y \cdot y, y \cdot 0.16666666666666666, y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.008333333333333333, -0.16666666666666666\right), x \cdot \left(x \cdot x\right), x\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sinh y) (sin x)) x)))
   (if (<= t_0 (- INFINITY))
     (/
      (*
       (fma (* x x) -0.16666666666666666 1.0)
       (* (fma 0.16666666666666666 (* y y) 1.0) (* y x)))
      x)
     (if (<= t_0 1e-21)
       (* (/ (sin x) x) (fma (* y y) (* y 0.16666666666666666) y))
       (/
        (*
         (fma
          (fma (* x x) 0.008333333333333333 -0.16666666666666666)
          (* x (* x x))
          x)
         (fma
          (fma
           y
           (* y (fma (* y y) 0.0001984126984126984 0.008333333333333333))
           0.16666666666666666)
          (* y (* y y))
          y))
        x)))))

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), -0.16666666666666666, 1.0) * (fma(0.16666666666666666, (y * y), 1.0) * (y * x))) / x;
	} else if (t_0 <= 1e-21) {
		tmp = (sin(x) / x) * fma((y * y), (y * 0.16666666666666666), y);
	} else {
		tmp = (fma(fma((x * x), 0.008333333333333333, -0.16666666666666666), (x * (x * x)), x) * fma(fma(y, (y * fma((y * y), 0.0001984126984126984, 0.008333333333333333)), 0.16666666666666666), (y * (y * y)), y)) / 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(fma(Float64(x * x), -0.16666666666666666, 1.0) * Float64(fma(0.16666666666666666, Float64(y * y), 1.0) * Float64(y * x))) / x);
	elseif (t_0 <= 1e-21)
		tmp = Float64(Float64(sin(x) / x) * fma(Float64(y * y), Float64(y * 0.16666666666666666), y));
	else
		tmp = Float64(Float64(fma(fma(Float64(x * x), 0.008333333333333333, -0.16666666666666666), Float64(x * Float64(x * x)), x) * fma(fma(y, Float64(y * fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333)), 0.16666666666666666), Float64(y * Float64(y * y)), y)) / 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[(N[(x * x), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * N[(N[(0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[t$95$0, 1e-21], N[(N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(y * 0.16666666666666666), $MachinePrecision] + y), $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[(y * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision] + y), $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 y around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-lft-identity N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt-in N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 77.5

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

   5. Applied rewrites 77.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 67.2%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 60.4%

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

   if -inf.0 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 9.99999999999999908e-22

   1. Initial program 74.4%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 98.2%

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

   if 9.99999999999999908e-22 < (/.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 76.4

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

   5. Applied rewrites 76.4%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 58.5

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

   8. Applied rewrites 58.5%

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

   9. Taylor expanded in y around 0

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

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

       3. *-commutative N/A

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

       4. associate-*l* N/A

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

       5. unpow2 N/A

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

       6. unpow3 N/A

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

       7. *-lft-identity N/A

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

       8. lower-fma.f64 N/A

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

   11. Applied rewrites 70.8%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -\infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right) \cdot \left(\mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right) \cdot \left(y \cdot x\right)\right)}{x}\\ \mathbf{elif}\;\frac{\sinh y \cdot \sin x}{x} \leq 10^{-21}:\\ \;\;\;\;\frac{\sin x}{x} \cdot \mathsf{fma}\left(y \cdot y, y \cdot 0.16666666666666666, y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.008333333333333333, -0.16666666666666666\right), x \cdot \left(x \cdot x\right), x\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y \cdot \sin x}{x}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right) \cdot \left(\mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right) \cdot \left(y \cdot x\right)\right)}{x}\\ \mathbf{elif}\;t\_0 \leq 10^{-21}:\\ \;\;\;\;y \cdot \frac{\sin x}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.008333333333333333, -0.16666666666666666\right), x \cdot \left(x \cdot x\right), x\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sinh y) (sin x)) x)))
   (if (<= t_0 (- INFINITY))
     (/
      (*
       (fma (* x x) -0.16666666666666666 1.0)
       (* (fma 0.16666666666666666 (* y y) 1.0) (* y x)))
      x)
     (if (<= t_0 1e-21)
       (* y (/ (sin x) x))
       (/
        (*
         (fma
          (fma (* x x) 0.008333333333333333 -0.16666666666666666)
          (* x (* x x))
          x)
         (fma
          (fma
           y
           (* y (fma (* y y) 0.0001984126984126984 0.008333333333333333))
           0.16666666666666666)
          (* y (* y y))
          y))
        x)))))

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), -0.16666666666666666, 1.0) * (fma(0.16666666666666666, (y * y), 1.0) * (y * x))) / x;
	} else if (t_0 <= 1e-21) {
		tmp = y * (sin(x) / x);
	} else {
		tmp = (fma(fma((x * x), 0.008333333333333333, -0.16666666666666666), (x * (x * x)), x) * fma(fma(y, (y * fma((y * y), 0.0001984126984126984, 0.008333333333333333)), 0.16666666666666666), (y * (y * y)), y)) / 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(fma(Float64(x * x), -0.16666666666666666, 1.0) * Float64(fma(0.16666666666666666, Float64(y * y), 1.0) * Float64(y * x))) / x);
	elseif (t_0 <= 1e-21)
		tmp = Float64(y * Float64(sin(x) / x));
	else
		tmp = Float64(Float64(fma(fma(Float64(x * x), 0.008333333333333333, -0.16666666666666666), Float64(x * Float64(x * x)), x) * fma(fma(y, Float64(y * fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333)), 0.16666666666666666), Float64(y * Float64(y * y)), y)) / 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[(N[(x * x), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * N[(N[(0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[t$95$0, 1e-21], N[(y * N[(N[Sin[x], $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[(y * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision] + y), $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 y around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-lft-identity N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt-in N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 77.5

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

   5. Applied rewrites 77.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 67.2%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 60.4%

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

   if -inf.0 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 9.99999999999999908e-22

   1. Initial program 74.4%

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

   3. Taylor expanded in y around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sin.f64 96.8

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

   5. Applied rewrites 96.8%

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

   if 9.99999999999999908e-22 < (/.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 76.4

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

   5. Applied rewrites 76.4%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 58.5

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

   8. Applied rewrites 58.5%

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

   9. Taylor expanded in y around 0

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

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

       3. *-commutative N/A

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

       4. associate-*l* N/A

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

       5. unpow2 N/A

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

       6. unpow3 N/A

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

       7. *-lft-identity N/A

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

       8. lower-fma.f64 N/A

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

   11. Applied rewrites 70.8%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -\infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right) \cdot \left(\mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right) \cdot \left(y \cdot x\right)\right)}{x}\\ \mathbf{elif}\;\frac{\sinh y \cdot \sin x}{x} \leq 10^{-21}:\\ \;\;\;\;y \cdot \frac{\sin x}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.008333333333333333, -0.16666666666666666\right), x \cdot \left(x \cdot x\right), x\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 60.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot y\right)\\ \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq 10^{-21}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), t\_0, y\right)}{x} \cdot \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot -0.16666666666666666, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.008333333333333333, -0.16666666666666666\right), x \cdot \left(x \cdot x\right), x\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), t\_0, y\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y y))))
   (if (<= (/ (* (sinh y) (sin x)) x) 1e-21)
     (*
      (/ (fma (fma y (* y 0.008333333333333333) 0.16666666666666666) t_0 y) x)
      (fma x (* (* x x) -0.16666666666666666) x))
     (/
      (*
       (fma
        (fma (* x x) 0.008333333333333333 -0.16666666666666666)
        (* x (* x x))
        x)
       (fma
        (fma
         y
         (* y (fma (* y y) 0.0001984126984126984 0.008333333333333333))
         0.16666666666666666)
        t_0
        y))
      x))))

C

double code(double x, double y) {
	double t_0 = y * (y * y);
	double tmp;
	if (((sinh(y) * sin(x)) / x) <= 1e-21) {
		tmp = (fma(fma(y, (y * 0.008333333333333333), 0.16666666666666666), t_0, y) / x) * fma(x, ((x * x) * -0.16666666666666666), x);
	} else {
		tmp = (fma(fma((x * x), 0.008333333333333333, -0.16666666666666666), (x * (x * x)), x) * fma(fma(y, (y * fma((y * y), 0.0001984126984126984, 0.008333333333333333)), 0.16666666666666666), t_0, y)) / x;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(y * Float64(y * y))
	tmp = 0.0
	if (Float64(Float64(sinh(y) * sin(x)) / x) <= 1e-21)
		tmp = Float64(Float64(fma(fma(y, Float64(y * 0.008333333333333333), 0.16666666666666666), t_0, y) / x) * fma(x, Float64(Float64(x * x) * -0.16666666666666666), x));
	else
		tmp = Float64(Float64(fma(fma(Float64(x * x), 0.008333333333333333, -0.16666666666666666), Float64(x * Float64(x * x)), x) * fma(fma(y, Float64(y * fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333)), 0.16666666666666666), t_0, y)) / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sinh[y], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], 1e-21], N[(N[(N[(N[(y * N[(y * 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * t$95$0 + y), $MachinePrecision] / x), $MachinePrecision] * N[(x * N[(N[(x * x), $MachinePrecision] * -0.16666666666666666), $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[(y * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * t$95$0 + y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 9.99999999999999908e-22

   1. Initial program 82.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. unpow2 N/A

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

      6. unpow3 N/A

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

      7. *-lft-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. cube-mult N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 N/A

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

      20. unpow2 N/A

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

      21. lower-*.f64 95.3

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

   7. Applied rewrites 95.3%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 56.9

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

   10. Applied rewrites 56.9%

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

   if 9.99999999999999908e-22 < (/.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 76.4

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

   5. Applied rewrites 76.4%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 58.5

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

   8. Applied rewrites 58.5%

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

   9. Taylor expanded in y around 0

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

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

       3. *-commutative N/A

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

       4. associate-*l* N/A

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

       5. unpow2 N/A

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

       6. unpow3 N/A

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

       7. *-lft-identity N/A

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

       8. lower-fma.f64 N/A

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

   11. Applied rewrites 70.8%

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

4. Final simplification 60.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq 10^{-21}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)}{x} \cdot \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot -0.16666666666666666, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.008333333333333333, -0.16666666666666666\right), x \cdot \left(x \cdot x\right), x\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 59.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)\\ \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq 10^{-21}:\\ \;\;\;\;\frac{t\_0}{x} \cdot \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot -0.16666666666666666, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 \cdot \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.008333333333333333, -0.16666666666666666\right), x \cdot \left(x \cdot x\right), x\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (fma
          (fma y (* y 0.008333333333333333) 0.16666666666666666)
          (* y (* y y))
          y)))
   (if (<= (/ (* (sinh y) (sin x)) x) 1e-21)
     (* (/ t_0 x) (fma x (* (* x x) -0.16666666666666666) x))
     (/
      (*
       t_0
       (fma
        (fma (* x x) 0.008333333333333333 -0.16666666666666666)
        (* x (* x x))
        x))
      x))))

C

double code(double x, double y) {
	double t_0 = fma(fma(y, (y * 0.008333333333333333), 0.16666666666666666), (y * (y * y)), y);
	double tmp;
	if (((sinh(y) * sin(x)) / x) <= 1e-21) {
		tmp = (t_0 / x) * fma(x, ((x * x) * -0.16666666666666666), x);
	} else {
		tmp = (t_0 * fma(fma((x * x), 0.008333333333333333, -0.16666666666666666), (x * (x * x)), x)) / x;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(fma(y, Float64(y * 0.008333333333333333), 0.16666666666666666), Float64(y * Float64(y * y)), y)
	tmp = 0.0
	if (Float64(Float64(sinh(y) * sin(x)) / x) <= 1e-21)
		tmp = Float64(Float64(t_0 / x) * fma(x, Float64(Float64(x * x) * -0.16666666666666666), x));
	else
		tmp = Float64(Float64(t_0 * 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[(y * N[(y * 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sinh[y], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], 1e-21], N[(N[(t$95$0 / x), $MachinePrecision] * N[(x * N[(N[(x * x), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * 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 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 9.99999999999999908e-22

   1. Initial program 82.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. unpow2 N/A

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

      6. unpow3 N/A

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

      7. *-lft-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. cube-mult N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 N/A

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

      20. unpow2 N/A

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

      21. lower-*.f64 95.3

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

   7. Applied rewrites 95.3%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 56.9

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

   10. Applied rewrites 56.9%

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

   if 9.99999999999999908e-22 < (/.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 76.4

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

   5. Applied rewrites 76.4%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. unpow2 N/A

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

      6. unpow3 N/A

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

      7. *-lft-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. cube-mult N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 N/A

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

      20. unpow2 N/A

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

      21. lower-*.f64 68.1

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

   8. Applied rewrites 68.1%

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

4. Final simplification 59.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq 10^{-21}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)}{x} \cdot \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot -0.16666666666666666, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.008333333333333333, -0.16666666666666666\right), x \cdot \left(x \cdot x\right), x\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 45.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq 3 \cdot 10^{-19}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot \left(y \cdot \left(y \cdot 0.16666666666666666\right)\right), y, y \cdot x\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (sinh y) (sin x)) x) 3e-19)
   (*
    y
    (fma
     x
     (*
      x
      (fma
       (* x x)
       (fma (* x x) -0.0001984126984126984 0.008333333333333333)
       -0.16666666666666666))
     1.0))
   (/ (fma (* x (* y (* y 0.16666666666666666))) y (* y x)) x)))

C

double code(double x, double y) {
	double tmp;
	if (((sinh(y) * sin(x)) / x) <= 3e-19) {
		tmp = y * fma(x, (x * fma((x * x), fma((x * x), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666)), 1.0);
	} else {
		tmp = fma((x * (y * (y * 0.16666666666666666))), y, (y * x)) / x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(sinh(y) * sin(x)) / x) <= 3e-19)
		tmp = Float64(y * fma(x, Float64(x * fma(Float64(x * x), fma(Float64(x * x), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666)), 1.0));
	else
		tmp = Float64(fma(Float64(x * Float64(y * Float64(y * 0.16666666666666666))), y, Float64(y * 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], 3e-19], N[(y * N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * N[(y * N[(y * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y + N[(y * x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 2.99999999999999993e-19

   1. Initial program 82.8%

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

   3. Taylor expanded in y around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sin.f64 67.7

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

   5. Applied rewrites 67.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 38.6%

      \[\leadsto y \cdot 1 \]

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 43.0%

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

   if 2.99999999999999993e-19 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-lft-identity N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt-in N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 68.2

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

   5. Applied rewrites 68.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 54.4%

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

   10. Applied rewrites 54.4%

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

4. Final simplification 45.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq 3 \cdot 10^{-19}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot \left(y \cdot \left(y \cdot 0.16666666666666666\right)\right), y, y \cdot x\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 51.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq 3 \cdot 10^{-19}:\\ \;\;\;\;\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot -0.16666666666666666, x\right) \cdot \left(y \cdot \frac{1}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot \left(y \cdot \left(y \cdot 0.16666666666666666\right)\right), y, y \cdot x\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (sinh y) (sin x)) x) 3e-19)
   (* (fma x (* (* x x) -0.16666666666666666) x) (* y (/ 1.0 x)))
   (/ (fma (* x (* y (* y 0.16666666666666666))) y (* y x)) x)))

C

double code(double x, double y) {
	double tmp;
	if (((sinh(y) * sin(x)) / x) <= 3e-19) {
		tmp = fma(x, ((x * x) * -0.16666666666666666), x) * (y * (1.0 / x));
	} else {
		tmp = fma((x * (y * (y * 0.16666666666666666))), y, (y * x)) / x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(sinh(y) * sin(x)) / x) <= 3e-19)
		tmp = Float64(fma(x, Float64(Float64(x * x) * -0.16666666666666666), x) * Float64(y * Float64(1.0 / x)));
	else
		tmp = Float64(fma(Float64(x * Float64(y * Float64(y * 0.16666666666666666))), y, Float64(y * 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], 3e-19], N[(N[(x * N[(N[(x * x), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + x), $MachinePrecision] * N[(y * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * N[(y * N[(y * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y + N[(y * x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 2.99999999999999993e-19

   1. Initial program 82.8%

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

   3. Taylor expanded in y around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sin.f64 67.7

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

   5. Applied rewrites 67.7%

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

   7. Applied rewrites 74.9%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 50.4%

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

   if 2.99999999999999993e-19 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-lft-identity N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt-in N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 68.2

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

   5. Applied rewrites 68.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 54.4%

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

   10. Applied rewrites 54.4%

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

4. Final simplification 51.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq 3 \cdot 10^{-19}:\\ \;\;\;\;\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot -0.16666666666666666, x\right) \cdot \left(y \cdot \frac{1}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot \left(y \cdot \left(y \cdot 0.16666666666666666\right)\right), y, y \cdot x\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 51.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq 3 \cdot 10^{-19}:\\ \;\;\;\;\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot -0.16666666666666666, x\right) \cdot \left(y \cdot \frac{1}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \mathsf{fma}\left(y, 0.16666666666666666 \cdot \left(y \cdot y\right), y\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (sinh y) (sin x)) x) 3e-19)
   (* (fma x (* (* x x) -0.16666666666666666) x) (* y (/ 1.0 x)))
   (/ (* x (fma y (* 0.16666666666666666 (* y y)) y)) x)))

C

double code(double x, double y) {
	double tmp;
	if (((sinh(y) * sin(x)) / x) <= 3e-19) {
		tmp = fma(x, ((x * x) * -0.16666666666666666), x) * (y * (1.0 / x));
	} else {
		tmp = (x * fma(y, (0.16666666666666666 * (y * y)), y)) / x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(sinh(y) * sin(x)) / x) <= 3e-19)
		tmp = Float64(fma(x, Float64(Float64(x * x) * -0.16666666666666666), x) * Float64(y * Float64(1.0 / x)));
	else
		tmp = Float64(Float64(x * fma(y, Float64(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], 3e-19], N[(N[(x * N[(N[(x * x), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + x), $MachinePrecision] * N[(y * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(y * N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 2.99999999999999993e-19

   1. Initial program 82.8%

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

   3. Taylor expanded in y around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sin.f64 67.7

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

   5. Applied rewrites 67.7%

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

   7. Applied rewrites 74.9%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 50.4%

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

   if 2.99999999999999993e-19 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-lft-identity N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt-in N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 68.2

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

   5. Applied rewrites 68.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 54.4%

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

4. Final simplification 51.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq 3 \cdot 10^{-19}:\\ \;\;\;\;\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot -0.16666666666666666, x\right) \cdot \left(y \cdot \frac{1}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \mathsf{fma}\left(y, 0.16666666666666666 \cdot \left(y \cdot y\right), y\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 44.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq 3 \cdot 10^{-19}:\\ \;\;\;\;y \cdot \frac{\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot -0.16666666666666666, x\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \mathsf{fma}\left(y, 0.16666666666666666 \cdot \left(y \cdot y\right), y\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (sinh y) (sin x)) x) 3e-19)
   (* y (/ (fma x (* (* x x) -0.16666666666666666) x) x))
   (/ (* x (fma y (* 0.16666666666666666 (* y y)) y)) x)))

C

double code(double x, double y) {
	double tmp;
	if (((sinh(y) * sin(x)) / x) <= 3e-19) {
		tmp = y * (fma(x, ((x * x) * -0.16666666666666666), x) / x);
	} else {
		tmp = (x * fma(y, (0.16666666666666666 * (y * y)), y)) / x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(sinh(y) * sin(x)) / x) <= 3e-19)
		tmp = Float64(y * Float64(fma(x, Float64(Float64(x * x) * -0.16666666666666666), x) / x));
	else
		tmp = Float64(Float64(x * fma(y, Float64(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], 3e-19], N[(y * N[(N[(x * N[(N[(x * x), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(y * N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 2.99999999999999993e-19

   1. Initial program 82.8%

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

   3. Taylor expanded in y around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sin.f64 67.7

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

   5. Applied rewrites 67.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 42.4%

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

   if 2.99999999999999993e-19 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-lft-identity N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt-in N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 68.2

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

   5. Applied rewrites 68.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 54.4%

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

4. Final simplification 45.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq 3 \cdot 10^{-19}:\\ \;\;\;\;y \cdot \frac{\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot -0.16666666666666666, x\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \mathsf{fma}\left(y, 0.16666666666666666 \cdot \left(y \cdot y\right), y\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 44.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq 0.001:\\ \;\;\;\;y \cdot \frac{\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot -0.16666666666666666, x\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(y \cdot \left(y \cdot y\right)\right) \cdot \left(x \cdot 0.16666666666666666\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (sinh y) (sin x)) x) 0.001)
   (* y (/ (fma x (* (* x x) -0.16666666666666666) x) x))
   (/ (* (* y (* y y)) (* x 0.16666666666666666)) x)))

C

double code(double x, double y) {
	double tmp;
	if (((sinh(y) * sin(x)) / x) <= 0.001) {
		tmp = y * (fma(x, ((x * x) * -0.16666666666666666), x) / x);
	} else {
		tmp = ((y * (y * y)) * (x * 0.16666666666666666)) / x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(sinh(y) * sin(x)) / x) <= 0.001)
		tmp = Float64(y * Float64(fma(x, Float64(Float64(x * x) * -0.16666666666666666), x) / x));
	else
		tmp = Float64(Float64(Float64(y * Float64(y * y)) * Float64(x * 0.16666666666666666)) / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[Sinh[y], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], 0.001], N[(y * N[(N[(x * N[(N[(x * x), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 1e-3

   1. Initial program 82.9%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 67.9

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

   5. Applied rewrites 67.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 42.7%

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

   if 1e-3 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-lft-identity N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt-in N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 67.7

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

   5. Applied rewrites 67.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 53.7%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 53.7%

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

4. Final simplification 45.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq 0.001:\\ \;\;\;\;y \cdot \frac{\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot -0.16666666666666666, x\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(y \cdot \left(y \cdot y\right)\right) \cdot \left(x \cdot 0.16666666666666666\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 44.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq 0.001:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, x \cdot -0.16666666666666666, y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(y \cdot \left(y \cdot y\right)\right) \cdot \left(x \cdot 0.16666666666666666\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (sinh y) (sin x)) x) 0.001)
   (fma (* y x) (* x -0.16666666666666666) y)
   (/ (* (* y (* y y)) (* x 0.16666666666666666)) x)))

C

double code(double x, double y) {
	double tmp;
	if (((sinh(y) * sin(x)) / x) <= 0.001) {
		tmp = fma((y * x), (x * -0.16666666666666666), y);
	} else {
		tmp = ((y * (y * y)) * (x * 0.16666666666666666)) / x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(sinh(y) * sin(x)) / x) <= 0.001)
		tmp = fma(Float64(y * x), Float64(x * -0.16666666666666666), y);
	else
		tmp = Float64(Float64(Float64(y * Float64(y * y)) * Float64(x * 0.16666666666666666)) / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[Sinh[y], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], 0.001], N[(N[(y * x), $MachinePrecision] * N[(x * -0.16666666666666666), $MachinePrecision] + y), $MachinePrecision], N[(N[(N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 1e-3

   1. Initial program 82.9%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 67.9

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

   5. Applied rewrites 67.9%

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

   7. Applied rewrites 75.0%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 42.8%

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

   12. Applied rewrites 42.8%

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

   if 1e-3 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-lft-identity N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt-in N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 67.7

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

   5. Applied rewrites 67.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 53.7%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 53.7%

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

4. Final simplification 45.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq 0.001:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, x \cdot -0.16666666666666666, y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(y \cdot \left(y \cdot y\right)\right) \cdot \left(x \cdot 0.16666666666666666\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 37.1% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(sinh(y) * sin(x)) / x) <= 0.0)
		tmp = fma(Float64(y * x), Float64(x * -0.16666666666666666), y);
	else
		tmp = Float64(y * fma(Float64(x * x), fma(0.008333333333333333, Float64(x * x), -0.16666666666666666), 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[(y * x), $MachinePrecision] * N[(x * -0.16666666666666666), $MachinePrecision] + y), $MachinePrecision], N[(y * N[(N[(x * x), $MachinePrecision] * N[(0.008333333333333333 * N[(x * x), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 79.3%

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

   3. Taylor expanded in y around 0

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

      1. associate-/l* N/A

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

      2. 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 60.3

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

   5. Applied rewrites 60.3%

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

   7. Applied rewrites 69.2%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 38.1%

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

   12. Applied rewrites 38.2%

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

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

   1. Initial program 98.9%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 39.4

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

   5. Applied rewrites 39.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 34.5%

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

4. Final simplification 36.7%

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

Alternative 15: 35.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq 3 \cdot 10^{-19}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, x \cdot -0.16666666666666666, y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (sinh y) (sin x)) x) 3e-19)
   (fma (* y x) (* x -0.16666666666666666) y)
   (/ (* y x) x)))

C

double code(double x, double y) {
	double tmp;
	if (((sinh(y) * sin(x)) / x) <= 3e-19) {
		tmp = fma((y * x), (x * -0.16666666666666666), y);
	} else {
		tmp = (y * x) / x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(sinh(y) * sin(x)) / x) <= 3e-19)
		tmp = fma(Float64(y * x), Float64(x * -0.16666666666666666), y);
	else
		tmp = Float64(Float64(y * 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], 3e-19], N[(N[(y * x), $MachinePrecision] * N[(x * -0.16666666666666666), $MachinePrecision] + y), $MachinePrecision], N[(N[(y * x), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 2.99999999999999993e-19

   1. Initial program 82.8%

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

   3. Taylor expanded in y around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sin.f64 67.7

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

   5. Applied rewrites 67.7%

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

   7. Applied rewrites 74.9%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 42.5%

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

   12. Applied rewrites 42.5%

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

   if 2.99999999999999993e-19 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-lft-identity N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt-in N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 68.2

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

   5. Applied rewrites 68.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 54.4%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 14.3%

       \[\leadsto \frac{y \cdot x}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 35.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq 3 \cdot 10^{-19}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, x \cdot -0.16666666666666666, y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 26.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -4 \cdot 10^{-199}:\\ \;\;\;\;y \cdot \left(\left(x \cdot x\right) \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (sinh y) (sin x)) x) -4e-199)
   (* y (* (* x x) -0.16666666666666666))
   (* y 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (((sinh(y) * sin(x)) / x) <= -4e-199) {
		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) <= (-4d-199)) 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) <= -4e-199) {
		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) <= -4e-199:
		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) <= -4e-199)
		tmp = Float64(y * Float64(Float64(x * 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) <= -4e-199)
		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], -4e-199], N[(y * N[(N[(x * x), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision], N[(y * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -3.99999999999999993e-199

   1. Initial program 98.9%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 25.5

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

   5. Applied rewrites 25.5%

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

   7. Applied rewrites 42.7%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 25.3%

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

   11. Taylor expanded in x around inf

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

   13. Applied rewrites 13.9%

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

   if -3.99999999999999993e-199 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 81.8%

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

   3. Taylor expanded in y around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sin.f64 64.1

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

   5. Applied rewrites 64.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 37.8%

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

4. Final simplification 30.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -4 \cdot 10^{-199}:\\ \;\;\;\;y \cdot \left(\left(x \cdot x\right) \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 94.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right)\\ \mathbf{if}\;y \leq 0.38:\\ \;\;\;\;\sin x \cdot \frac{\mathsf{fma}\left(t\_0, y \cdot \left(y \cdot y\right), y\right)}{x}\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{+53}:\\ \;\;\;\;\frac{\sinh y \cdot \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot -0.16666666666666666, x\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(\sin x \cdot \mathsf{fma}\left(y \cdot y, t\_0, 1\right)\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma y (* y 0.008333333333333333) 0.16666666666666666)))
   (if (<= y 0.38)
     (* (sin x) (/ (fma t_0 (* y (* y y)) y) x))
     (if (<= y 4.8e+53)
       (/ (* (sinh y) (fma x (* (* x x) -0.16666666666666666) x)) x)
       (/ (* y (* (sin x) (fma (* y y) t_0 1.0))) x)))))

C

double code(double x, double y) {
	double t_0 = fma(y, (y * 0.008333333333333333), 0.16666666666666666);
	double tmp;
	if (y <= 0.38) {
		tmp = sin(x) * (fma(t_0, (y * (y * y)), y) / x);
	} else if (y <= 4.8e+53) {
		tmp = (sinh(y) * fma(x, ((x * x) * -0.16666666666666666), x)) / x;
	} else {
		tmp = (y * (sin(x) * fma((y * y), t_0, 1.0))) / x;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(y, Float64(y * 0.008333333333333333), 0.16666666666666666)
	tmp = 0.0
	if (y <= 0.38)
		tmp = Float64(sin(x) * Float64(fma(t_0, Float64(y * Float64(y * y)), y) / x));
	elseif (y <= 4.8e+53)
		tmp = Float64(Float64(sinh(y) * fma(x, Float64(Float64(x * x) * -0.16666666666666666), x)) / x);
	else
		tmp = Float64(Float64(y * Float64(sin(x) * fma(Float64(y * y), t_0, 1.0))) / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(y * 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]}, If[LessEqual[y, 0.38], N[(N[Sin[x], $MachinePrecision] * N[(N[(t$95$0 * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.8e+53], N[(N[(N[Sinh[y], $MachinePrecision] * N[(x * N[(N[(x * x), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(y * N[(N[Sin[x], $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 0.38

   1. Initial program 83.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. unpow2 N/A

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

      6. unpow3 N/A

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

      7. *-lft-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. cube-mult N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 N/A

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

      20. unpow2 N/A

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

      21. lower-*.f64 94.9

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

   7. Applied rewrites 94.9%

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

   if 0.38 < y < 4.8e53

   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 70.0

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

   5. Applied rewrites 70.0%

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

   if 4.8e53 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. *-rgt-identity N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. distribute-lft-in N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. distribute-rgt-out N/A

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

   5. Applied rewrites 98.1%

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

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.38:\\ \;\;\;\;\sin x \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)}{x}\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{+53}:\\ \;\;\;\;\frac{\sinh y \cdot \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot -0.16666666666666666, x\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(\sin x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), 1\right)\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 94.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin x \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)}{x}\\ \mathbf{if}\;y \leq 0.38:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{+53}:\\ \;\;\;\;\frac{\sinh y \cdot \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot -0.16666666666666666, x\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (*
          (sin x)
          (/
           (fma
            (fma y (* y 0.008333333333333333) 0.16666666666666666)
            (* y (* y y))
            y)
           x))))
   (if (<= y 0.38)
     t_0
     (if (<= y 4.8e+53)
       (/ (* (sinh y) (fma x (* (* x x) -0.16666666666666666) x)) x)
       t_0))))

C

double code(double x, double y) {
	double t_0 = sin(x) * (fma(fma(y, (y * 0.008333333333333333), 0.16666666666666666), (y * (y * y)), y) / x);
	double tmp;
	if (y <= 0.38) {
		tmp = t_0;
	} else if (y <= 4.8e+53) {
		tmp = (sinh(y) * fma(x, ((x * x) * -0.16666666666666666), x)) / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(x) * Float64(fma(fma(y, Float64(y * 0.008333333333333333), 0.16666666666666666), Float64(y * Float64(y * y)), y) / x))
	tmp = 0.0
	if (y <= 0.38)
		tmp = t_0;
	elseif (y <= 4.8e+53)
		tmp = Float64(Float64(sinh(y) * fma(x, Float64(Float64(x * x) * -0.16666666666666666), x)) / x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[x], $MachinePrecision] * N[(N[(N[(y * N[(y * 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 0.38], t$95$0, If[LessEqual[y, 4.8e+53], N[(N[(N[Sinh[y], $MachinePrecision] * N[(x * N[(N[(x * x), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < 0.38 or 4.8e53 < y

   1. Initial program 86.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. unpow2 N/A

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

      6. unpow3 N/A

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

      7. *-lft-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. cube-mult N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 N/A

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

      20. unpow2 N/A

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

      21. lower-*.f64 95.6

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

   7. Applied rewrites 95.6%

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

   if 0.38 < y < 4.8e53

   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 70.0

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

   5. Applied rewrites 70.0%

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

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.38:\\ \;\;\;\;\sin x \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)}{x}\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{+53}:\\ \;\;\;\;\frac{\sinh y \cdot \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot -0.16666666666666666, x\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 65.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot y\right)\\ \mathbf{if}\;x \leq 6 \cdot 10^{+14}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), t\_0, y\right)}{x} \cdot \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot -0.16666666666666666, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\frac{\frac{6}{x} + \frac{-36}{y \cdot \left(y \cdot x\right)}}{t\_0}}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y y))))
   (if (<= x 6e+14)
     (*
      (/ (fma (fma y (* y 0.008333333333333333) 0.16666666666666666) t_0 y) x)
      (fma x (* (* x x) -0.16666666666666666) x))
     (/ (/ 1.0 (/ (+ (/ 6.0 x) (/ -36.0 (* y (* y x)))) t_0)) x))))

C

double code(double x, double y) {
	double t_0 = y * (y * y);
	double tmp;
	if (x <= 6e+14) {
		tmp = (fma(fma(y, (y * 0.008333333333333333), 0.16666666666666666), t_0, y) / x) * fma(x, ((x * x) * -0.16666666666666666), x);
	} else {
		tmp = (1.0 / (((6.0 / x) + (-36.0 / (y * (y * x)))) / t_0)) / x;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(y * Float64(y * y))
	tmp = 0.0
	if (x <= 6e+14)
		tmp = Float64(Float64(fma(fma(y, Float64(y * 0.008333333333333333), 0.16666666666666666), t_0, y) / x) * fma(x, Float64(Float64(x * x) * -0.16666666666666666), x));
	else
		tmp = Float64(Float64(1.0 / Float64(Float64(Float64(6.0 / x) + Float64(-36.0 / Float64(y * Float64(y * x)))) / t_0)) / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 6e+14], N[(N[(N[(N[(y * N[(y * 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * t$95$0 + y), $MachinePrecision] / x), $MachinePrecision] * N[(x * N[(N[(x * x), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[(N[(6.0 / x), $MachinePrecision] + N[(-36.0 / N[(y * N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 6e14

   1. Initial program 83.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. unpow2 N/A

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

      6. unpow3 N/A

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

      7. *-lft-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. cube-mult N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 N/A

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

      20. unpow2 N/A

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

      21. lower-*.f64 93.5

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

   7. Applied rewrites 93.5%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 73.6

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

   10. Applied rewrites 73.6%

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

   if 6e14 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-lft-identity N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt-in N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 86.8

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

   5. Applied rewrites 86.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 24.4%

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

   10. Applied rewrites 19.6%

       \[\leadsto \frac{\frac{1}{\frac{\mathsf{fma}\left(y, y \cdot \left(y \cdot 0.16666666666666666\right), -y\right)}{\color{blue}{\left(\left(y \cdot y\right) \cdot \left(0.027777777777777776 \cdot \left(\left(y \cdot \left(y \cdot y\right)\right) \cdot y\right)\right) - y \cdot y\right) \cdot x}}}}{x} \]

   11. Taylor expanded in y around inf

       \[\leadsto \frac{\frac{1}{\frac{6 \cdot \frac{1}{x} - 36 \cdot \frac{1}{x \cdot {y}^{2}}}{{y}^{\color{blue}{3}}}}}{x} \]
   12. Step-by-step derivation

   13. Applied rewrites 47.3%

       \[\leadsto \frac{\frac{1}{\frac{\frac{6}{x} + \frac{-36}{y \cdot \left(y \cdot x\right)}}{y \cdot \color{blue}{\left(y \cdot y\right)}}}}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 65.1% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot y\right)\\ \mathbf{if}\;x \leq 1.9 \cdot 10^{+62}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot 0.008333333333333333, 0.16666666666666666\right), t\_0, y\right)}{x} \cdot \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot -0.16666666666666666, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 \cdot \left(x \cdot 0.16666666666666666\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y y))))
   (if (<= x 1.9e+62)
     (*
      (/ (fma (fma y (* y 0.008333333333333333) 0.16666666666666666) t_0 y) x)
      (fma x (* (* x x) -0.16666666666666666) x))
     (/ (* t_0 (* x 0.16666666666666666)) x))))

C

double code(double x, double y) {
	double t_0 = y * (y * y);
	double tmp;
	if (x <= 1.9e+62) {
		tmp = (fma(fma(y, (y * 0.008333333333333333), 0.16666666666666666), t_0, y) / x) * fma(x, ((x * x) * -0.16666666666666666), x);
	} else {
		tmp = (t_0 * (x * 0.16666666666666666)) / x;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(y * Float64(y * y))
	tmp = 0.0
	if (x <= 1.9e+62)
		tmp = Float64(Float64(fma(fma(y, Float64(y * 0.008333333333333333), 0.16666666666666666), t_0, y) / x) * fma(x, Float64(Float64(x * x) * -0.16666666666666666), x));
	else
		tmp = Float64(Float64(t_0 * Float64(x * 0.16666666666666666)) / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 1.9e+62], N[(N[(N[(N[(y * N[(y * 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * t$95$0 + y), $MachinePrecision] / x), $MachinePrecision] * N[(x * N[(N[(x * x), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.89999999999999992e62

   1. Initial program 84.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. unpow2 N/A

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

      6. unpow3 N/A

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

      7. *-lft-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. cube-mult N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 N/A

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

      20. unpow2 N/A

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

      21. lower-*.f64 93.9

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

   7. Applied rewrites 93.9%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 70.9

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

   10. Applied rewrites 70.9%

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

   if 1.89999999999999992e62 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-lft-identity N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt-in N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. associate-*r* N/A

          \[\leadsto \frac{y \cdot \left(\sin x \cdot \left(\color{blue}{\left(\frac{1}{6} \cdot y\right) \cdot y} + 1\right)\right)}{x} \]

      14. *-commutative N/A

          \[\leadsto \frac{y \cdot \left(\sin x \cdot \left(\color{blue}{y \cdot \left(\frac{1}{6} \cdot y\right)} + 1\right)\right)}{x} \]

      15. lower-fma.f64 N/A

          \[\leadsto \frac{y \cdot \left(\sin x \cdot \color{blue}{\mathsf{fma}\left(y, \frac{1}{6} \cdot y, 1\right)}\right)}{x} \]

      16. *-commutative N/A

          \[\leadsto \frac{y \cdot \left(\sin x \cdot \mathsf{fma}\left(y, \color{blue}{y \cdot \frac{1}{6}}, 1\right)\right)}{x} \]

      17. lower-*.f64 85.5

          \[\leadsto \frac{y \cdot \left(\sin x \cdot \mathsf{fma}\left(y, \color{blue}{y \cdot 0.16666666666666666}, 1\right)\right)}{x} \]

   5. Applied rewrites 85.5%

      \[\leadsto \frac{\color{blue}{y \cdot \left(\sin x \cdot \mathsf{fma}\left(y, y \cdot 0.16666666666666666, 1\right)\right)}}{x} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{x \cdot \color{blue}{\left(y \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)}}{x} \]
   7. Step-by-step derivation

   8. Applied rewrites 27.5%

      \[\leadsto \frac{x \cdot \color{blue}{\mathsf{fma}\left(y, 0.16666666666666666 \cdot \left(y \cdot y\right), y\right)}}{x} \]

   9. Taylor expanded in y around inf

      \[\leadsto \frac{\frac{1}{6} \cdot \left(x \cdot \color{blue}{{y}^{3}}\right)}{x} \]
   10. Step-by-step derivation

   11. Applied rewrites 48.6%

       \[\leadsto \frac{\left(y \cdot \left(y \cdot y\right)\right) \cdot \left(x \cdot \color{blue}{0.16666666666666666}\right)}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 36.6% 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 87.2%

   \[\frac{\sin x \cdot \sinh y}{x} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\frac{y \cdot \sin x}{x}} \]
4. Step-by-step derivation

   1. associate-/l* N/A

      \[\leadsto \color{blue}{y \cdot \frac{\sin x}{x}} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{y \cdot \frac{\sin x}{x}} \]

   3. lower-/.f64 N/A

      \[\leadsto y \cdot \color{blue}{\frac{\sin x}{x}} \]

   4. lower-sin.f64 51.9

      \[\leadsto y \cdot \frac{\color{blue}{\sin x}}{x} \]

5. Applied rewrites 51.9%

   \[\leadsto \color{blue}{y \cdot \frac{\sin x}{x}} \]
6. Step-by-step derivation

7. Applied rewrites 64.4%

   \[\leadsto \left(y \cdot \frac{1}{x}\right) \cdot \color{blue}{\sin x} \]

8. Taylor expanded in x around 0

   \[\leadsto y + \color{blue}{\frac{-1}{6} \cdot \left({x}^{2} \cdot y\right)} \]
9. Step-by-step derivation

10. Applied rewrites 37.6%

    \[\leadsto \mathsf{fma}\left(y, \color{blue}{\left(x \cdot x\right) \cdot -0.16666666666666666}, y\right) \]
11. Step-by-step derivation

12. Applied rewrites 37.6%

    \[\leadsto \color{blue}{\mathsf{fma}\left(y \cdot x, x \cdot -0.16666666666666666, y\right)} \]
13. Add Preprocessing

Alternative 22: 36.6% 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 87.2%

   \[\frac{\sin x \cdot \sinh y}{x} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\frac{y \cdot \sin x}{x}} \]
4. Step-by-step derivation

   1. associate-/l* N/A

      \[\leadsto \color{blue}{y \cdot \frac{\sin x}{x}} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{y \cdot \frac{\sin x}{x}} \]

   3. lower-/.f64 N/A

      \[\leadsto y \cdot \color{blue}{\frac{\sin x}{x}} \]

   4. lower-sin.f64 51.9

      \[\leadsto y \cdot \frac{\color{blue}{\sin x}}{x} \]

5. Applied rewrites 51.9%

   \[\leadsto \color{blue}{y \cdot \frac{\sin x}{x}} \]

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 37.6%

   \[\leadsto \mathsf{fma}\left(y, \color{blue}{\left(x \cdot x\right) \cdot -0.16666666666666666}, y\right) \]
9. Add Preprocessing

Alternative 23: 27.9% 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 87.2%

   \[\frac{\sin x \cdot \sinh y}{x} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\frac{y \cdot \sin x}{x}} \]
4. Step-by-step derivation

   1. associate-/l* N/A

      \[\leadsto \color{blue}{y \cdot \frac{\sin x}{x}} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{y \cdot \frac{\sin x}{x}} \]

   3. lower-/.f64 N/A

      \[\leadsto y \cdot \color{blue}{\frac{\sin x}{x}} \]

   4. lower-sin.f64 51.9

      \[\leadsto y \cdot \frac{\color{blue}{\sin x}}{x} \]

5. Applied rewrites 51.9%

   \[\leadsto \color{blue}{y \cdot \frac{\sin x}{x}} \]

6. Taylor expanded in x around 0

   \[\leadsto y \cdot 1 \]
7. Step-by-step derivation

8. Applied rewrites 30.2%

   \[\leadsto y \cdot 1 \]
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 2024221 
(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))