Linear.Quaternion:$ccosh from linear-1.19.1.3

Percentage Accurate: 89.2% → 99.8%

Time: 9.9s

Alternatives: 16

Speedup: 1.0×

• 50.4% 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: 89.2% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.9%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. lift-*.f64 N/A

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

   4. associate-/r* N/A

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

   5. clear-num N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 100.0

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

4. Applied rewrites 100.0%

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

Alternative 2: 84.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin x \cdot \sinh y}{x}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(y \cdot y\right) \cdot 0.008333333333333333, y \cdot y, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, x \cdot x, 0.008333333333333333\right), x \cdot x, -0.16666666666666666\right), x \cdot x, 1\right)\right) \cdot y\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-54}:\\ \;\;\;\;\frac{y}{\frac{x}{\sin x}}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sin x) (sinh y)) x)))
   (if (<= t_0 (- INFINITY))
     (*
      (*
       (fma (* (* y y) 0.008333333333333333) (* y y) 1.0)
       (fma
        (fma
         (fma -0.0001984126984126984 (* x x) 0.008333333333333333)
         (* x x)
         -0.16666666666666666)
        (* x x)
        1.0))
      y)
     (if (<= t_0 2e-54) (/ y (/ x (sin x))) (sinh y)))))

C

double code(double x, double y) {
	double t_0 = (sin(x) * sinh(y)) / x;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (fma(((y * y) * 0.008333333333333333), (y * y), 1.0) * fma(fma(fma(-0.0001984126984126984, (x * x), 0.008333333333333333), (x * x), -0.16666666666666666), (x * x), 1.0)) * y;
	} else if (t_0 <= 2e-54) {
		tmp = y / (x / sin(x));
	} else {
		tmp = sinh(y);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(sin(x) * sinh(y)) / x)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(fma(Float64(Float64(y * y) * 0.008333333333333333), Float64(y * y), 1.0) * fma(fma(fma(-0.0001984126984126984, Float64(x * x), 0.008333333333333333), Float64(x * x), -0.16666666666666666), Float64(x * x), 1.0)) * y);
	elseif (t_0 <= 2e-54)
		tmp = Float64(y / Float64(x / sin(x)));
	else
		tmp = sinh(y);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sin[x], $MachinePrecision] * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[(N[(y * y), $MachinePrecision] * 0.008333333333333333), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(N[(-0.0001984126984126984 * N[(x * x), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t$95$0, 2e-54], N[(y / N[(x / N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sinh[y], $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

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 74.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 55.4%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 55.4%

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

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 67.6%

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

   if -inf.0 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 2.0000000000000001e-54

   1. Initial program 83.3%

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

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sin.f64 99.3

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

   5. Applied rewrites 99.3%

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

   7. Applied rewrites 99.3%

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

   if 2.0000000000000001e-54 < (/.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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. rec-exp N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-neg.f64 63.7

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

   5. Applied rewrites 63.7%

      \[\leadsto \color{blue}{\left(e^{y} - e^{-y}\right) \cdot 0.5} \]
   6. Step-by-step derivation

   7. Applied rewrites 66.7%

      \[\leadsto \sinh y \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin x \cdot \sinh y}{x} \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(y \cdot y\right) \cdot 0.008333333333333333, y \cdot y, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, x \cdot x, 0.008333333333333333\right), x \cdot x, -0.16666666666666666\right), x \cdot x, 1\right)\right) \cdot y\\ \mathbf{elif}\;\frac{\sin x \cdot \sinh y}{x} \leq 2 \cdot 10^{-54}:\\ \;\;\;\;\frac{y}{\frac{x}{\sin x}}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin x \cdot \sinh y}{x}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(y \cdot y\right) \cdot 0.008333333333333333, y \cdot y, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, x \cdot x, 0.008333333333333333\right), x \cdot x, -0.16666666666666666\right), x \cdot x, 1\right)\right) \cdot y\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-54}:\\ \;\;\;\;\frac{\sin x}{x} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sin x) (sinh y)) x)))
   (if (<= t_0 (- INFINITY))
     (*
      (*
       (fma (* (* y y) 0.008333333333333333) (* y y) 1.0)
       (fma
        (fma
         (fma -0.0001984126984126984 (* x x) 0.008333333333333333)
         (* x x)
         -0.16666666666666666)
        (* x x)
        1.0))
      y)
     (if (<= t_0 2e-54) (* (/ (sin x) x) y) (sinh y)))))

C

double code(double x, double y) {
	double t_0 = (sin(x) * sinh(y)) / x;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (fma(((y * y) * 0.008333333333333333), (y * y), 1.0) * fma(fma(fma(-0.0001984126984126984, (x * x), 0.008333333333333333), (x * x), -0.16666666666666666), (x * x), 1.0)) * y;
	} else if (t_0 <= 2e-54) {
		tmp = (sin(x) / x) * y;
	} else {
		tmp = sinh(y);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(sin(x) * sinh(y)) / x)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(fma(Float64(Float64(y * y) * 0.008333333333333333), Float64(y * y), 1.0) * fma(fma(fma(-0.0001984126984126984, Float64(x * x), 0.008333333333333333), Float64(x * x), -0.16666666666666666), Float64(x * x), 1.0)) * y);
	elseif (t_0 <= 2e-54)
		tmp = Float64(Float64(sin(x) / x) * y);
	else
		tmp = sinh(y);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sin[x], $MachinePrecision] * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[(N[(y * y), $MachinePrecision] * 0.008333333333333333), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(N[(-0.0001984126984126984 * N[(x * x), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t$95$0, 2e-54], N[(N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision] * y), $MachinePrecision], N[Sinh[y], $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

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 74.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 55.4%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 55.4%

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

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 67.6%

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

   if -inf.0 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 2.0000000000000001e-54

   1. Initial program 83.3%

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

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sin.f64 99.3

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

   5. Applied rewrites 99.3%

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

   if 2.0000000000000001e-54 < (/.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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. rec-exp N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-neg.f64 63.7

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

   5. Applied rewrites 63.7%

      \[\leadsto \color{blue}{\left(e^{y} - e^{-y}\right) \cdot 0.5} \]
   6. Step-by-step derivation

   7. Applied rewrites 66.7%

      \[\leadsto \sinh y \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin x \cdot \sinh y}{x} \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(y \cdot y\right) \cdot 0.008333333333333333, y \cdot y, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, x \cdot x, 0.008333333333333333\right), x \cdot x, -0.16666666666666666\right), x \cdot x, 1\right)\right) \cdot y\\ \mathbf{elif}\;\frac{\sin x \cdot \sinh y}{x} \leq 2 \cdot 10^{-54}:\\ \;\;\;\;\frac{\sin x}{x} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin x \cdot \sinh y}{x}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(y \cdot y\right) \cdot 0.008333333333333333, y \cdot y, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, x \cdot x, 0.008333333333333333\right), x \cdot x, -0.16666666666666666\right), x \cdot x, 1\right)\right) \cdot y\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-54}:\\ \;\;\;\;\frac{y}{x} \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sin x) (sinh y)) x)))
   (if (<= t_0 (- INFINITY))
     (*
      (*
       (fma (* (* y y) 0.008333333333333333) (* y y) 1.0)
       (fma
        (fma
         (fma -0.0001984126984126984 (* x x) 0.008333333333333333)
         (* x x)
         -0.16666666666666666)
        (* x x)
        1.0))
      y)
     (if (<= t_0 2e-54) (* (/ y x) (sin x)) (sinh y)))))

C

double code(double x, double y) {
	double t_0 = (sin(x) * sinh(y)) / x;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (fma(((y * y) * 0.008333333333333333), (y * y), 1.0) * fma(fma(fma(-0.0001984126984126984, (x * x), 0.008333333333333333), (x * x), -0.16666666666666666), (x * x), 1.0)) * y;
	} else if (t_0 <= 2e-54) {
		tmp = (y / x) * sin(x);
	} else {
		tmp = sinh(y);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(sin(x) * sinh(y)) / x)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(fma(Float64(Float64(y * y) * 0.008333333333333333), Float64(y * y), 1.0) * fma(fma(fma(-0.0001984126984126984, Float64(x * x), 0.008333333333333333), Float64(x * x), -0.16666666666666666), Float64(x * x), 1.0)) * y);
	elseif (t_0 <= 2e-54)
		tmp = Float64(Float64(y / x) * sin(x));
	else
		tmp = sinh(y);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sin[x], $MachinePrecision] * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[(N[(y * y), $MachinePrecision] * 0.008333333333333333), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(N[(-0.0001984126984126984 * N[(x * x), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t$95$0, 2e-54], N[(N[(y / x), $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision], N[Sinh[y], $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

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 74.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 55.4%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 55.4%

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

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 67.6%

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

   if -inf.0 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 2.0000000000000001e-54

   1. Initial program 83.3%

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

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sin.f64 99.3

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

   5. Applied rewrites 99.3%

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

   7. Applied rewrites 99.2%

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

   if 2.0000000000000001e-54 < (/.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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. rec-exp N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-neg.f64 63.7

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

   5. Applied rewrites 63.7%

      \[\leadsto \color{blue}{\left(e^{y} - e^{-y}\right) \cdot 0.5} \]
   6. Step-by-step derivation

   7. Applied rewrites 66.7%

      \[\leadsto \sinh y \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin x \cdot \sinh y}{x} \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(y \cdot y\right) \cdot 0.008333333333333333, y \cdot y, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, x \cdot x, 0.008333333333333333\right), x \cdot x, -0.16666666666666666\right), x \cdot x, 1\right)\right) \cdot y\\ \mathbf{elif}\;\frac{\sin x \cdot \sinh y}{x} \leq 2 \cdot 10^{-54}:\\ \;\;\;\;\frac{y}{x} \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 59.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin x \cdot \sinh y}{x}\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{-84}:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(y \cdot y\right) \cdot 0.008333333333333333, y \cdot y, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, x \cdot x, 0.008333333333333333\right), x \cdot x, -0.16666666666666666\right), x \cdot x, 1\right)\right) \cdot y\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-276}:\\ \;\;\;\;0.5 \cdot \left(1 - 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sin x) (sinh y)) x)))
   (if (<= t_0 -4e-84)
     (*
      (*
       (fma (* (* y y) 0.008333333333333333) (* y y) 1.0)
       (fma
        (fma
         (fma -0.0001984126984126984 (* x x) 0.008333333333333333)
         (* x x)
         -0.16666666666666666)
        (* x x)
        1.0))
      y)
     (if (<= t_0 5e-276) (* 0.5 (- 1.0 1.0)) (sinh y)))))

C

double code(double x, double y) {
	double t_0 = (sin(x) * sinh(y)) / x;
	double tmp;
	if (t_0 <= -4e-84) {
		tmp = (fma(((y * y) * 0.008333333333333333), (y * y), 1.0) * fma(fma(fma(-0.0001984126984126984, (x * x), 0.008333333333333333), (x * x), -0.16666666666666666), (x * x), 1.0)) * y;
	} else if (t_0 <= 5e-276) {
		tmp = 0.5 * (1.0 - 1.0);
	} else {
		tmp = sinh(y);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(sin(x) * sinh(y)) / x)
	tmp = 0.0
	if (t_0 <= -4e-84)
		tmp = Float64(Float64(fma(Float64(Float64(y * y) * 0.008333333333333333), Float64(y * y), 1.0) * fma(fma(fma(-0.0001984126984126984, Float64(x * x), 0.008333333333333333), Float64(x * x), -0.16666666666666666), Float64(x * x), 1.0)) * y);
	elseif (t_0 <= 5e-276)
		tmp = Float64(0.5 * Float64(1.0 - 1.0));
	else
		tmp = sinh(y);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sin[x], $MachinePrecision] * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[t$95$0, -4e-84], N[(N[(N[(N[(N[(y * y), $MachinePrecision] * 0.008333333333333333), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(N[(-0.0001984126984126984 * N[(x * x), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t$95$0, 5e-276], N[(0.5 * N[(1.0 - 1.0), $MachinePrecision]), $MachinePrecision], N[Sinh[y], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -4.0000000000000001e-84

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 77.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 60.7%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 59.8%

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

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 70.5%

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

   if -4.0000000000000001e-84 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 4.99999999999999967e-276

   1. Initial program 76.4%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. rec-exp N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-neg.f64 44.0

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

   5. Applied rewrites 44.0%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 44.0%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 44.1%

       \[\leadsto \left(1 - 1\right) \cdot 0.5 \]

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

   1. Initial program 99.4%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. rec-exp N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-neg.f64 45.4

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

   5. Applied rewrites 45.4%

      \[\leadsto \color{blue}{\left(e^{y} - e^{-y}\right) \cdot 0.5} \]
   6. Step-by-step derivation

   7. Applied rewrites 58.3%

      \[\leadsto \sinh y \]
3. Recombined 3 regimes into one program.

4. Final simplification 57.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin x \cdot \sinh y}{x} \leq -4 \cdot 10^{-84}:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(y \cdot y\right) \cdot 0.008333333333333333, y \cdot y, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, x \cdot x, 0.008333333333333333\right), x \cdot x, -0.16666666666666666\right), x \cdot x, 1\right)\right) \cdot y\\ \mathbf{elif}\;\frac{\sin x \cdot \sinh y}{x} \leq 5 \cdot 10^{-276}:\\ \;\;\;\;0.5 \cdot \left(1 - 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 56.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin x \cdot \sinh y}{x}\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{-84}:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(y \cdot y\right) \cdot 0.008333333333333333, y \cdot y, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, x \cdot x, 0.008333333333333333\right), x \cdot x, -0.16666666666666666\right), x \cdot x, 1\right)\right) \cdot y\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-276}:\\ \;\;\;\;0.5 \cdot \left(1 - 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0003968253968253968, y \cdot y, 0.016666666666666666\right), y \cdot y, 0.3333333333333333\right), y \cdot y, 2\right) \cdot y\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sin x) (sinh y)) x)))
   (if (<= t_0 -4e-84)
     (*
      (*
       (fma (* (* y y) 0.008333333333333333) (* y y) 1.0)
       (fma
        (fma
         (fma -0.0001984126984126984 (* x x) 0.008333333333333333)
         (* x x)
         -0.16666666666666666)
        (* x x)
        1.0))
      y)
     (if (<= t_0 5e-276)
       (* 0.5 (- 1.0 1.0))
       (*
        (*
         (fma
          (fma
           (fma 0.0003968253968253968 (* y y) 0.016666666666666666)
           (* y y)
           0.3333333333333333)
          (* y y)
          2.0)
         y)
        0.5)))))

C

double code(double x, double y) {
	double t_0 = (sin(x) * sinh(y)) / x;
	double tmp;
	if (t_0 <= -4e-84) {
		tmp = (fma(((y * y) * 0.008333333333333333), (y * y), 1.0) * fma(fma(fma(-0.0001984126984126984, (x * x), 0.008333333333333333), (x * x), -0.16666666666666666), (x * x), 1.0)) * y;
	} else if (t_0 <= 5e-276) {
		tmp = 0.5 * (1.0 - 1.0);
	} else {
		tmp = (fma(fma(fma(0.0003968253968253968, (y * y), 0.016666666666666666), (y * y), 0.3333333333333333), (y * y), 2.0) * y) * 0.5;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(sin(x) * sinh(y)) / x)
	tmp = 0.0
	if (t_0 <= -4e-84)
		tmp = Float64(Float64(fma(Float64(Float64(y * y) * 0.008333333333333333), Float64(y * y), 1.0) * fma(fma(fma(-0.0001984126984126984, Float64(x * x), 0.008333333333333333), Float64(x * x), -0.16666666666666666), Float64(x * x), 1.0)) * y);
	elseif (t_0 <= 5e-276)
		tmp = Float64(0.5 * Float64(1.0 - 1.0));
	else
		tmp = Float64(Float64(fma(fma(fma(0.0003968253968253968, Float64(y * y), 0.016666666666666666), Float64(y * y), 0.3333333333333333), Float64(y * y), 2.0) * y) * 0.5);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sin[x], $MachinePrecision] * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[t$95$0, -4e-84], N[(N[(N[(N[(N[(y * y), $MachinePrecision] * 0.008333333333333333), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(N[(-0.0001984126984126984 * N[(x * x), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t$95$0, 5e-276], N[(0.5 * N[(1.0 - 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(0.0003968253968253968 * N[(y * y), $MachinePrecision] + 0.016666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(y * y), $MachinePrecision] + 2.0), $MachinePrecision] * y), $MachinePrecision] * 0.5), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -4.0000000000000001e-84

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 77.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 60.7%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 59.8%

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

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 70.5%

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

   if -4.0000000000000001e-84 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 4.99999999999999967e-276

   1. Initial program 76.4%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. rec-exp N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-neg.f64 44.0

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

   5. Applied rewrites 44.0%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 44.0%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 44.1%

       \[\leadsto \left(1 - 1\right) \cdot 0.5 \]

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

   1. Initial program 99.4%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. rec-exp N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-neg.f64 45.4

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

   5. Applied rewrites 45.4%

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

   6. Taylor expanded in y around 0

      \[\leadsto \left(y \cdot \left(2 + {y}^{2} \cdot \left(\frac{1}{3} + {y}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {y}^{2}\right)\right)\right)\right) \cdot \frac{1}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 50.4%

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0003968253968253968, y \cdot y, 0.016666666666666666\right), y \cdot y, 0.3333333333333333\right), y \cdot y, 2\right) \cdot y\right) \cdot 0.5 \]
3. Recombined 3 regimes into one program.

4. Final simplification 54.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin x \cdot \sinh y}{x} \leq -4 \cdot 10^{-84}:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(y \cdot y\right) \cdot 0.008333333333333333, y \cdot y, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, x \cdot x, 0.008333333333333333\right), x \cdot x, -0.16666666666666666\right), x \cdot x, 1\right)\right) \cdot y\\ \mathbf{elif}\;\frac{\sin x \cdot \sinh y}{x} \leq 5 \cdot 10^{-276}:\\ \;\;\;\;0.5 \cdot \left(1 - 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0003968253968253968, y \cdot y, 0.016666666666666666\right), y \cdot y, 0.3333333333333333\right), y \cdot y, 2\right) \cdot y\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 57.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin x \cdot \sinh y}{x}\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{-149}:\\ \;\;\;\;\left(\mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(\left(y \cdot y\right) \cdot 0.008333333333333333, y \cdot y, 1\right)\right) \cdot y\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-276}:\\ \;\;\;\;0.5 \cdot \left(1 - 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0003968253968253968, y \cdot y, 0.016666666666666666\right), y \cdot y, 0.3333333333333333\right), y \cdot y, 2\right) \cdot y\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sin x) (sinh y)) x)))
   (if (<= t_0 -4e-149)
     (*
      (*
       (fma (* x x) -0.16666666666666666 1.0)
       (fma (* (* y y) 0.008333333333333333) (* y y) 1.0))
      y)
     (if (<= t_0 5e-276)
       (* 0.5 (- 1.0 1.0))
       (*
        (*
         (fma
          (fma
           (fma 0.0003968253968253968 (* y y) 0.016666666666666666)
           (* y y)
           0.3333333333333333)
          (* y y)
          2.0)
         y)
        0.5)))))

C

double code(double x, double y) {
	double t_0 = (sin(x) * sinh(y)) / x;
	double tmp;
	if (t_0 <= -4e-149) {
		tmp = (fma((x * x), -0.16666666666666666, 1.0) * fma(((y * y) * 0.008333333333333333), (y * y), 1.0)) * y;
	} else if (t_0 <= 5e-276) {
		tmp = 0.5 * (1.0 - 1.0);
	} else {
		tmp = (fma(fma(fma(0.0003968253968253968, (y * y), 0.016666666666666666), (y * y), 0.3333333333333333), (y * y), 2.0) * y) * 0.5;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(sin(x) * sinh(y)) / x)
	tmp = 0.0
	if (t_0 <= -4e-149)
		tmp = Float64(Float64(fma(Float64(x * x), -0.16666666666666666, 1.0) * fma(Float64(Float64(y * y) * 0.008333333333333333), Float64(y * y), 1.0)) * y);
	elseif (t_0 <= 5e-276)
		tmp = Float64(0.5 * Float64(1.0 - 1.0));
	else
		tmp = Float64(Float64(fma(fma(fma(0.0003968253968253968, Float64(y * y), 0.016666666666666666), Float64(y * y), 0.3333333333333333), Float64(y * y), 2.0) * y) * 0.5);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sin[x], $MachinePrecision] * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[t$95$0, -4e-149], N[(N[(N[(N[(x * x), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * N[(N[(N[(y * y), $MachinePrecision] * 0.008333333333333333), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t$95$0, 5e-276], N[(0.5 * N[(1.0 - 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(0.0003968253968253968 * N[(y * y), $MachinePrecision] + 0.016666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(y * y), $MachinePrecision] + 2.0), $MachinePrecision] * y), $MachinePrecision] * 0.5), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -3.99999999999999992e-149

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 79.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 60.7%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 59.9%

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

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 68.5%

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

   if -3.99999999999999992e-149 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 4.99999999999999967e-276

   1. Initial program 73.3%

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

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. rec-exp N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-neg.f64 49.3

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

   5. Applied rewrites 49.3%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 49.3%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 49.4%

       \[\leadsto \left(1 - 1\right) \cdot 0.5 \]

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

   1. Initial program 99.4%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. rec-exp N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-neg.f64 45.4

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

   5. Applied rewrites 45.4%

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

   6. Taylor expanded in y around 0

      \[\leadsto \left(y \cdot \left(2 + {y}^{2} \cdot \left(\frac{1}{3} + {y}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {y}^{2}\right)\right)\right)\right) \cdot \frac{1}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 50.4%

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0003968253968253968, y \cdot y, 0.016666666666666666\right), y \cdot y, 0.3333333333333333\right), y \cdot y, 2\right) \cdot y\right) \cdot 0.5 \]
3. Recombined 3 regimes into one program.

4. Final simplification 56.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin x \cdot \sinh y}{x} \leq -4 \cdot 10^{-149}:\\ \;\;\;\;\left(\mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(\left(y \cdot y\right) \cdot 0.008333333333333333, y \cdot y, 1\right)\right) \cdot y\\ \mathbf{elif}\;\frac{\sin x \cdot \sinh y}{x} \leq 5 \cdot 10^{-276}:\\ \;\;\;\;0.5 \cdot \left(1 - 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0003968253968253968, y \cdot y, 0.016666666666666666\right), y \cdot y, 0.3333333333333333\right), y \cdot y, 2\right) \cdot y\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 56.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin x \cdot \sinh y}{x}\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{-149}:\\ \;\;\;\;\left(\mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(\left(y \cdot y\right) \cdot 0.008333333333333333, y \cdot y, 1\right)\right) \cdot y\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-276}:\\ \;\;\;\;0.5 \cdot \left(1 - 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right) \cdot y, y, 1\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sin x) (sinh y)) x)))
   (if (<= t_0 -4e-149)
     (*
      (*
       (fma (* x x) -0.16666666666666666 1.0)
       (fma (* (* y y) 0.008333333333333333) (* y y) 1.0))
      y)
     (if (<= t_0 5e-276)
       (* 0.5 (- 1.0 1.0))
       (*
        (fma
         (* (fma (* y y) 0.008333333333333333 0.16666666666666666) y)
         y
         1.0)
        y)))))

C

double code(double x, double y) {
	double t_0 = (sin(x) * sinh(y)) / x;
	double tmp;
	if (t_0 <= -4e-149) {
		tmp = (fma((x * x), -0.16666666666666666, 1.0) * fma(((y * y) * 0.008333333333333333), (y * y), 1.0)) * y;
	} else if (t_0 <= 5e-276) {
		tmp = 0.5 * (1.0 - 1.0);
	} else {
		tmp = fma((fma((y * y), 0.008333333333333333, 0.16666666666666666) * y), y, 1.0) * y;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(sin(x) * sinh(y)) / x)
	tmp = 0.0
	if (t_0 <= -4e-149)
		tmp = Float64(Float64(fma(Float64(x * x), -0.16666666666666666, 1.0) * fma(Float64(Float64(y * y) * 0.008333333333333333), Float64(y * y), 1.0)) * y);
	elseif (t_0 <= 5e-276)
		tmp = Float64(0.5 * Float64(1.0 - 1.0));
	else
		tmp = Float64(fma(Float64(fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666) * y), y, 1.0) * y);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sin[x], $MachinePrecision] * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[t$95$0, -4e-149], N[(N[(N[(N[(x * x), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * N[(N[(N[(y * y), $MachinePrecision] * 0.008333333333333333), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t$95$0, 5e-276], N[(0.5 * N[(1.0 - 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision] * y), $MachinePrecision] * y + 1.0), $MachinePrecision] * y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -3.99999999999999992e-149

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 79.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 60.7%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 59.9%

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

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 68.5%

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

   if -3.99999999999999992e-149 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 4.99999999999999967e-276

   1. Initial program 73.3%

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

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. rec-exp N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-neg.f64 49.3

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

   5. Applied rewrites 49.3%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 49.3%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 49.4%

       \[\leadsto \left(1 - 1\right) \cdot 0.5 \]

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

   1. Initial program 99.4%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 84.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 49.4%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot y \]
   9. Step-by-step derivation

   10. Applied rewrites 49.4%

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

4. Final simplification 55.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin x \cdot \sinh y}{x} \leq -4 \cdot 10^{-149}:\\ \;\;\;\;\left(\mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(\left(y \cdot y\right) \cdot 0.008333333333333333, y \cdot y, 1\right)\right) \cdot y\\ \mathbf{elif}\;\frac{\sin x \cdot \sinh y}{x} \leq 5 \cdot 10^{-276}:\\ \;\;\;\;0.5 \cdot \left(1 - 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right) \cdot y, y, 1\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 45.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin x \cdot \sinh y}{x}\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{-84}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, x \cdot x, 0.008333333333333333\right), x \cdot x, -0.16666666666666666\right), x \cdot x, 1\right) \cdot y\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-276}:\\ \;\;\;\;0.5 \cdot \left(1 - 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right) \cdot y, y, 1\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sin x) (sinh y)) x)))
   (if (<= t_0 -4e-84)
     (*
      (fma
       (fma
        (fma -0.0001984126984126984 (* x x) 0.008333333333333333)
        (* x x)
        -0.16666666666666666)
       (* x x)
       1.0)
      y)
     (if (<= t_0 5e-276)
       (* 0.5 (- 1.0 1.0))
       (*
        (fma
         (* (fma (* y y) 0.008333333333333333 0.16666666666666666) y)
         y
         1.0)
        y)))))

C

double code(double x, double y) {
	double t_0 = (sin(x) * sinh(y)) / x;
	double tmp;
	if (t_0 <= -4e-84) {
		tmp = fma(fma(fma(-0.0001984126984126984, (x * x), 0.008333333333333333), (x * x), -0.16666666666666666), (x * x), 1.0) * y;
	} else if (t_0 <= 5e-276) {
		tmp = 0.5 * (1.0 - 1.0);
	} else {
		tmp = fma((fma((y * y), 0.008333333333333333, 0.16666666666666666) * y), y, 1.0) * y;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(sin(x) * sinh(y)) / x)
	tmp = 0.0
	if (t_0 <= -4e-84)
		tmp = Float64(fma(fma(fma(-0.0001984126984126984, Float64(x * x), 0.008333333333333333), Float64(x * x), -0.16666666666666666), Float64(x * x), 1.0) * y);
	elseif (t_0 <= 5e-276)
		tmp = Float64(0.5 * Float64(1.0 - 1.0));
	else
		tmp = Float64(fma(Float64(fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666) * y), y, 1.0) * y);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sin[x], $MachinePrecision] * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[t$95$0, -4e-84], N[(N[(N[(N[(-0.0001984126984126984 * N[(x * x), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t$95$0, 5e-276], N[(0.5 * N[(1.0 - 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision] * y), $MachinePrecision] * y + 1.0), $MachinePrecision] * y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -4.0000000000000001e-84

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sin.f64 14.7

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

   5. Applied rewrites 14.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 37.4%

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

   if -4.0000000000000001e-84 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 4.99999999999999967e-276

   1. Initial program 76.4%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. rec-exp N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-neg.f64 44.0

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

   5. Applied rewrites 44.0%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 44.0%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 44.1%

       \[\leadsto \left(1 - 1\right) \cdot 0.5 \]

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

   1. Initial program 99.4%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 84.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 49.4%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot y \]
   9. Step-by-step derivation

   10. Applied rewrites 49.4%

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

4. Final simplification 44.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin x \cdot \sinh y}{x} \leq -4 \cdot 10^{-84}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, x \cdot x, 0.008333333333333333\right), x \cdot x, -0.16666666666666666\right), x \cdot x, 1\right) \cdot y\\ \mathbf{elif}\;\frac{\sin x \cdot \sinh y}{x} \leq 5 \cdot 10^{-276}:\\ \;\;\;\;0.5 \cdot \left(1 - 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right) \cdot y, y, 1\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 46.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin x \cdot \sinh y}{x}\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{-149}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right) \cdot y\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-276}:\\ \;\;\;\;0.5 \cdot \left(1 - 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right) \cdot y, y, 1\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sin x) (sinh y)) x)))
   (if (<= t_0 -4e-149)
     (* (fma (* x x) -0.16666666666666666 1.0) y)
     (if (<= t_0 5e-276)
       (* 0.5 (- 1.0 1.0))
       (*
        (fma
         (* (fma (* y y) 0.008333333333333333 0.16666666666666666) y)
         y
         1.0)
        y)))))

C

double code(double x, double y) {
	double t_0 = (sin(x) * sinh(y)) / x;
	double tmp;
	if (t_0 <= -4e-149) {
		tmp = fma((x * x), -0.16666666666666666, 1.0) * y;
	} else if (t_0 <= 5e-276) {
		tmp = 0.5 * (1.0 - 1.0);
	} else {
		tmp = fma((fma((y * y), 0.008333333333333333, 0.16666666666666666) * y), y, 1.0) * y;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(sin(x) * sinh(y)) / x)
	tmp = 0.0
	if (t_0 <= -4e-149)
		tmp = Float64(fma(Float64(x * x), -0.16666666666666666, 1.0) * y);
	elseif (t_0 <= 5e-276)
		tmp = Float64(0.5 * Float64(1.0 - 1.0));
	else
		tmp = Float64(fma(Float64(fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666) * y), y, 1.0) * y);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sin[x], $MachinePrecision] * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[t$95$0, -4e-149], N[(N[(N[(x * x), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t$95$0, 5e-276], N[(0.5 * N[(1.0 - 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision] * y), $MachinePrecision] * y + 1.0), $MachinePrecision] * y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -3.99999999999999992e-149

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sin.f64 24.7

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

   5. Applied rewrites 24.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 30.4%

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

   if -3.99999999999999992e-149 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 4.99999999999999967e-276

   1. Initial program 73.3%

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

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. rec-exp N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-neg.f64 49.3

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

   5. Applied rewrites 49.3%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 49.3%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 49.4%

       \[\leadsto \left(1 - 1\right) \cdot 0.5 \]

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

   1. Initial program 99.4%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 84.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 49.4%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot y \]
   9. Step-by-step derivation

   10. Applied rewrites 49.4%

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

4. Final simplification 43.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin x \cdot \sinh y}{x} \leq -4 \cdot 10^{-149}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right) \cdot y\\ \mathbf{elif}\;\frac{\sin x \cdot \sinh y}{x} \leq 5 \cdot 10^{-276}:\\ \;\;\;\;0.5 \cdot \left(1 - 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right) \cdot y, y, 1\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 46.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin x \cdot \sinh y}{x}\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{-149}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right) \cdot y\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-276}:\\ \;\;\;\;0.5 \cdot \left(1 - 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(y \cdot y\right) \cdot 0.008333333333333333, y \cdot y, 1\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sin x) (sinh y)) x)))
   (if (<= t_0 -4e-149)
     (* (fma (* x x) -0.16666666666666666 1.0) y)
     (if (<= t_0 5e-276)
       (* 0.5 (- 1.0 1.0))
       (* (fma (* (* y y) 0.008333333333333333) (* y y) 1.0) y)))))

C

double code(double x, double y) {
	double t_0 = (sin(x) * sinh(y)) / x;
	double tmp;
	if (t_0 <= -4e-149) {
		tmp = fma((x * x), -0.16666666666666666, 1.0) * y;
	} else if (t_0 <= 5e-276) {
		tmp = 0.5 * (1.0 - 1.0);
	} else {
		tmp = fma(((y * y) * 0.008333333333333333), (y * y), 1.0) * y;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(sin(x) * sinh(y)) / x)
	tmp = 0.0
	if (t_0 <= -4e-149)
		tmp = Float64(fma(Float64(x * x), -0.16666666666666666, 1.0) * y);
	elseif (t_0 <= 5e-276)
		tmp = Float64(0.5 * Float64(1.0 - 1.0));
	else
		tmp = Float64(fma(Float64(Float64(y * y) * 0.008333333333333333), Float64(y * y), 1.0) * y);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sin[x], $MachinePrecision] * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[t$95$0, -4e-149], N[(N[(N[(x * x), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t$95$0, 5e-276], N[(0.5 * N[(1.0 - 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(y * y), $MachinePrecision] * 0.008333333333333333), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -3.99999999999999992e-149

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sin.f64 24.7

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

   5. Applied rewrites 24.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 30.4%

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

   if -3.99999999999999992e-149 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 4.99999999999999967e-276

   1. Initial program 73.3%

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

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. rec-exp N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-neg.f64 49.3

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

   5. Applied rewrites 49.3%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 49.3%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 49.4%

       \[\leadsto \left(1 - 1\right) \cdot 0.5 \]

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

   1. Initial program 99.4%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 84.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 49.4%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 49.4%

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

4. Final simplification 43.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin x \cdot \sinh y}{x} \leq -4 \cdot 10^{-149}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right) \cdot y\\ \mathbf{elif}\;\frac{\sin x \cdot \sinh y}{x} \leq 5 \cdot 10^{-276}:\\ \;\;\;\;0.5 \cdot \left(1 - 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(y \cdot y\right) \cdot 0.008333333333333333, y \cdot y, 1\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 43.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin x \cdot \sinh y}{x}\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{-149}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right) \cdot y\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-276}:\\ \;\;\;\;0.5 \cdot \left(1 - 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sin x) (sinh y)) x)))
   (if (<= t_0 -4e-149)
     (* (fma (* x x) -0.16666666666666666 1.0) y)
     (if (<= t_0 5e-276)
       (* 0.5 (- 1.0 1.0))
       (* (fma 0.16666666666666666 (* y y) 1.0) y)))))

C

double code(double x, double y) {
	double t_0 = (sin(x) * sinh(y)) / x;
	double tmp;
	if (t_0 <= -4e-149) {
		tmp = fma((x * x), -0.16666666666666666, 1.0) * y;
	} else if (t_0 <= 5e-276) {
		tmp = 0.5 * (1.0 - 1.0);
	} else {
		tmp = fma(0.16666666666666666, (y * y), 1.0) * y;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(sin(x) * sinh(y)) / x)
	tmp = 0.0
	if (t_0 <= -4e-149)
		tmp = Float64(fma(Float64(x * x), -0.16666666666666666, 1.0) * y);
	elseif (t_0 <= 5e-276)
		tmp = Float64(0.5 * Float64(1.0 - 1.0));
	else
		tmp = Float64(fma(0.16666666666666666, Float64(y * y), 1.0) * y);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sin[x], $MachinePrecision] * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[t$95$0, -4e-149], N[(N[(N[(x * x), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t$95$0, 5e-276], N[(0.5 * N[(1.0 - 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -3.99999999999999992e-149

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sin.f64 24.7

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

   5. Applied rewrites 24.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 30.4%

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

   if -3.99999999999999992e-149 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 4.99999999999999967e-276

   1. Initial program 73.3%

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

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. rec-exp N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-neg.f64 49.3

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

   5. Applied rewrites 49.3%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 49.3%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 49.4%

       \[\leadsto \left(1 - 1\right) \cdot 0.5 \]

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

   1. Initial program 99.4%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 84.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 49.4%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 41.5%

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

4. Final simplification 40.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin x \cdot \sinh y}{x} \leq -4 \cdot 10^{-149}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right) \cdot y\\ \mathbf{elif}\;\frac{\sin x \cdot \sinh y}{x} \leq 5 \cdot 10^{-276}:\\ \;\;\;\;0.5 \cdot \left(1 - 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 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 91.9%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 99.9

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

4. Applied rewrites 99.9%

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

Alternative 14: 53.1% accurate, 9.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{+100}:\\ \;\;\;\;\mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(1 - 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1.35e+100)
   (* (fma 0.16666666666666666 (* y y) 1.0) y)
   (* 0.5 (- 1.0 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.35e+100) {
		tmp = fma(0.16666666666666666, (y * y), 1.0) * y;
	} else {
		tmp = 0.5 * (1.0 - 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.35e+100)
		tmp = Float64(fma(0.16666666666666666, Float64(y * y), 1.0) * y);
	else
		tmp = Float64(0.5 * Float64(1.0 - 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.35e+100], N[(N[(0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision], N[(0.5 * N[(1.0 - 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.34999999999999999e100

   1. Initial program 90.2%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 88.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 56.6%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 49.7%

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

   if 1.34999999999999999e100 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. rec-exp N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-neg.f64 51.1

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

   5. Applied rewrites 51.1%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 46.9%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 36.7%

       \[\leadsto \left(1 - 1\right) \cdot 0.5 \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{+100}:\\ \;\;\;\;\mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(1 - 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 32.8% accurate, 14.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.8 \cdot 10^{+38}:\\ \;\;\;\;1 \cdot y\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(1 - 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 2.8e+38) (* 1.0 y) (* 0.5 (- 1.0 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 2.8e+38) {
		tmp = 1.0 * y;
	} else {
		tmp = 0.5 * (1.0 - 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 (x <= 2.8d+38) then
        tmp = 1.0d0 * y
    else
        tmp = 0.5d0 * (1.0d0 - 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 2.8e+38) {
		tmp = 1.0 * y;
	} else {
		tmp = 0.5 * (1.0 - 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 2.8e+38:
		tmp = 1.0 * y
	else:
		tmp = 0.5 * (1.0 - 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 2.8e+38)
		tmp = Float64(1.0 * y);
	else
		tmp = Float64(0.5 * Float64(1.0 - 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 2.8e+38)
		tmp = 1.0 * y;
	else
		tmp = 0.5 * (1.0 - 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 2.8e+38], N[(1.0 * y), $MachinePrecision], N[(0.5 * N[(1.0 - 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.8e38

   1. Initial program 89.5%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sin.f64 49.5

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

   5. Applied rewrites 49.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 29.9%

      \[\leadsto 1 \cdot y \]

   if 2.8e38 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. rec-exp N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-neg.f64 47.1

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

   5. Applied rewrites 47.1%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 42.2%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 32.6%

       \[\leadsto \left(1 - 1\right) \cdot 0.5 \]
3. Recombined 2 regimes into one program.

4. Final simplification 30.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.8 \cdot 10^{+38}:\\ \;\;\;\;1 \cdot y\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(1 - 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 27.2% accurate, 36.2× speedup

Math

\[\begin{array}{l} \\ 1 \cdot y \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* 1.0 y))

C

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

Fortran

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

Java

public static double code(double x, double y) {
	return 1.0 * y;
}

Python

def code(x, y):
	return 1.0 * y

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.9%

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

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

   2. associate-*l/ N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-sin.f64 50.9

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

5. Applied rewrites 50.9%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 23.9%

   \[\leadsto 1 \cdot y \]
9. Add Preprocessing

Developer Target 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024249 
(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))