Linear.Quaternion:$csinh from linear-1.19.1.3

Percentage Accurate: 99.9% → 99.9%

Time: 12.6s

Alternatives: 21

Speedup: 1.0×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{0.5 \cdot \sin y}{y} \cdot \left(\cosh x \cdot 2\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* (/ (* 0.5 (sin y)) y) (* (cosh x) 2.0)))

C

double code(double x, double y) {
	return ((0.5 * sin(y)) / y) * (cosh(x) * 2.0);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((0.5d0 * sin(y)) / y) * (cosh(x) * 2.0d0)
end function

Java

public static double code(double x, double y) {
	return ((0.5 * Math.sin(y)) / y) * (Math.cosh(x) * 2.0);
}

Python

def code(x, y):
	return ((0.5 * math.sin(y)) / y) * (math.cosh(x) * 2.0)

Julia

function code(x, y)
	return Float64(Float64(Float64(0.5 * sin(y)) / y) * Float64(cosh(x) * 2.0))
end

MATLAB

function tmp = code(x, y)
	tmp = ((0.5 * sin(y)) / y) * (cosh(x) * 2.0);
end

Wolfram

code[x_, y_] := N[(N[(N[(0.5 * N[Sin[y], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] * N[(N[Cosh[x], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

   1. *-commutative N/A

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

   2. cosh-def N/A

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

   3. clear-num N/A

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

   4. associate-/r/ N/A

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

   5. associate-*r* N/A

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

   6. div-inv N/A

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

   7. associate-/l/ N/A

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

   8. *-lowering-*.f64 N/A

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

   9. associate-/l/ N/A

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

   10. clear-num N/A

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

   11. associate-/r/ N/A

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

   12. associate-*r/ N/A

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

   13. /-lowering-/.f64 N/A

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

   14. *-lowering-*.f64 N/A

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

   15. metadata-eval N/A

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

   16. sin-lowering-sin.f64 N/A

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

   17. cosh-undef N/A

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

   18. *-commutative N/A

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

   19. *-lowering-*.f64 N/A

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

   20. cosh-lowering-cosh.f64 99.9

       \[\leadsto \frac{0.5 \cdot \sin y}{y} \cdot \left(\color{blue}{\cosh x} \cdot 2\right) \]

4. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{\frac{0.5 \cdot \sin y}{y} \cdot \left(\cosh x \cdot 2\right)} \]
5. Add Preprocessing

Alternative 2: 99.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ t_1 := \cosh x \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\cosh x \cdot \left(\left(y \cdot y\right) \cdot -0.16666666666666666\right)\\ \mathbf{elif}\;t\_1 \leq 0.2:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(0.5, x \cdot x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\cosh x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)) (t_1 (* (cosh x) t_0)))
   (if (<= t_1 (- INFINITY))
     (* (cosh x) (* (* y y) -0.16666666666666666))
     (if (<= t_1 0.2) (* t_0 (fma 0.5 (* x x) 1.0)) (cosh x)))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double t_1 = cosh(x) * t_0;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = cosh(x) * ((y * y) * -0.16666666666666666);
	} else if (t_1 <= 0.2) {
		tmp = t_0 * fma(0.5, (x * x), 1.0);
	} else {
		tmp = cosh(x);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(y) / y)
	t_1 = Float64(cosh(x) * t_0)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(cosh(x) * Float64(Float64(y * y) * -0.16666666666666666));
	elseif (t_1 <= 0.2)
		tmp = Float64(t_0 * fma(0.5, Float64(x * x), 1.0));
	else
		tmp = cosh(x);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cosh[x], $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[Cosh[x], $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.2], N[(t$95$0 * N[(0.5 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[Cosh[x], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. *-lowering-*.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 100.0

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

   8. Simplified 100.0%

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

   if -inf.0 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < 0.20000000000000001

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{{x}^{2} \cdot \sin y}{y} \cdot \frac{1}{2}} + \frac{\sin y}{y} \]

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. distribute-lft1-in N/A

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

      8. +-commutative N/A

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

      9. *-lowering-*.f64 N/A

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

      10. +-commutative N/A

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

      11. accelerator-lowering-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 N/A

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

      14. /-lowering-/.f64 N/A

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

      15. sin-lowering-sin.f64 99.5

          \[\leadsto \mathsf{fma}\left(0.5, x \cdot x, 1\right) \cdot \frac{\color{blue}{\sin y}}{y} \]

   5. Simplified 99.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, x \cdot x, 1\right) \cdot \frac{\sin y}{y}} \]

   if 0.20000000000000001 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \cosh x \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 100.0%

      \[\leadsto \cosh x \cdot \color{blue}{1} \]
   6. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto \color{blue}{\cosh x} \]

      2. cosh-lowering-cosh.f64 100.0

         \[\leadsto \color{blue}{\cosh x} \]

   7. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\cosh x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -\infty:\\ \;\;\;\;\cosh x \cdot \left(\left(y \cdot y\right) \cdot -0.16666666666666666\right)\\ \mathbf{elif}\;\cosh x \cdot \frac{\sin y}{y} \leq 0.2:\\ \;\;\;\;\frac{\sin y}{y} \cdot \mathsf{fma}\left(0.5, x \cdot x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\cosh x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cosh x \cdot \frac{\sin y}{y}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\cosh x \cdot \left(\left(y \cdot y\right) \cdot -0.16666666666666666\right)\\ \mathbf{elif}\;t\_0 \leq 0.2:\\ \;\;\;\;\frac{0.5 \cdot \sin y}{y} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\cosh x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (cosh x) (/ (sin y) y))))
   (if (<= t_0 (- INFINITY))
     (* (cosh x) (* (* y y) -0.16666666666666666))
     (if (<= t_0 0.2) (* (/ (* 0.5 (sin y)) y) 2.0) (cosh x)))))

C

double code(double x, double y) {
	double t_0 = cosh(x) * (sin(y) / y);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = cosh(x) * ((y * y) * -0.16666666666666666);
	} else if (t_0 <= 0.2) {
		tmp = ((0.5 * sin(y)) / y) * 2.0;
	} else {
		tmp = cosh(x);
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double t_0 = Math.cosh(x) * (Math.sin(y) / y);
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = Math.cosh(x) * ((y * y) * -0.16666666666666666);
	} else if (t_0 <= 0.2) {
		tmp = ((0.5 * Math.sin(y)) / y) * 2.0;
	} else {
		tmp = Math.cosh(x);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.cosh(x) * (math.sin(y) / y)
	tmp = 0
	if t_0 <= -math.inf:
		tmp = math.cosh(x) * ((y * y) * -0.16666666666666666)
	elif t_0 <= 0.2:
		tmp = ((0.5 * math.sin(y)) / y) * 2.0
	else:
		tmp = math.cosh(x)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(cosh(x) * Float64(sin(y) / y))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(cosh(x) * Float64(Float64(y * y) * -0.16666666666666666));
	elseif (t_0 <= 0.2)
		tmp = Float64(Float64(Float64(0.5 * sin(y)) / y) * 2.0);
	else
		tmp = cosh(x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = cosh(x) * (sin(y) / y);
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = cosh(x) * ((y * y) * -0.16666666666666666);
	elseif (t_0 <= 0.2)
		tmp = ((0.5 * sin(y)) / y) * 2.0;
	else
		tmp = cosh(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cosh[x], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[Cosh[x], $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.2], N[(N[(N[(0.5 * N[Sin[y], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] * 2.0), $MachinePrecision], N[Cosh[x], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. *-lowering-*.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 100.0

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

   8. Simplified 100.0%

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

   if -inf.0 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < 0.20000000000000001

   1. Initial program 99.5%

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

      1. *-commutative N/A

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

      2. cosh-def N/A

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

      3. clear-num N/A

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

      4. associate-/r/ N/A

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

      5. associate-*r* N/A

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

      6. div-inv N/A

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

      7. associate-/l/ N/A

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

      8. *-lowering-*.f64 N/A

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

      9. associate-/l/ N/A

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

      10. clear-num N/A

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

      11. associate-/r/ N/A

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

      12. associate-*r/ N/A

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

      13. /-lowering-/.f64 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. sin-lowering-sin.f64 N/A

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

      17. cosh-undef N/A

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

      18. *-commutative N/A

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

      19. *-lowering-*.f64 N/A

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

      20. cosh-lowering-cosh.f64 99.6

          \[\leadsto \frac{0.5 \cdot \sin y}{y} \cdot \left(\color{blue}{\cosh x} \cdot 2\right) \]

   4. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \sin y}{y} \cdot \left(\cosh x \cdot 2\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\frac{1}{2} \cdot \sin y}{y} \cdot \color{blue}{2} \]
   6. Step-by-step derivation

   7. Simplified 99.5%

      \[\leadsto \frac{0.5 \cdot \sin y}{y} \cdot \color{blue}{2} \]

   if 0.20000000000000001 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \cosh x \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 100.0%

      \[\leadsto \cosh x \cdot \color{blue}{1} \]
   6. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto \color{blue}{\cosh x} \]

      2. cosh-lowering-cosh.f64 100.0

         \[\leadsto \color{blue}{\cosh x} \]

   7. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\cosh x} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 99.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ t_1 := \cosh x \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\cosh x \cdot \left(\left(y \cdot y\right) \cdot -0.16666666666666666\right)\\ \mathbf{elif}\;t\_1 \leq 0.2:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\cosh x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)) (t_1 (* (cosh x) t_0)))
   (if (<= t_1 (- INFINITY))
     (* (cosh x) (* (* y y) -0.16666666666666666))
     (if (<= t_1 0.2) t_0 (cosh x)))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double t_1 = cosh(x) * t_0;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = cosh(x) * ((y * y) * -0.16666666666666666);
	} else if (t_1 <= 0.2) {
		tmp = t_0;
	} else {
		tmp = cosh(x);
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double t_0 = Math.sin(y) / y;
	double t_1 = Math.cosh(x) * t_0;
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = Math.cosh(x) * ((y * y) * -0.16666666666666666);
	} else if (t_1 <= 0.2) {
		tmp = t_0;
	} else {
		tmp = Math.cosh(x);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sin(y) / y
	t_1 = math.cosh(x) * t_0
	tmp = 0
	if t_1 <= -math.inf:
		tmp = math.cosh(x) * ((y * y) * -0.16666666666666666)
	elif t_1 <= 0.2:
		tmp = t_0
	else:
		tmp = math.cosh(x)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sin(y) / y)
	t_1 = Float64(cosh(x) * t_0)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(cosh(x) * Float64(Float64(y * y) * -0.16666666666666666));
	elseif (t_1 <= 0.2)
		tmp = t_0;
	else
		tmp = cosh(x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sin(y) / y;
	t_1 = cosh(x) * t_0;
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = cosh(x) * ((y * y) * -0.16666666666666666);
	elseif (t_1 <= 0.2)
		tmp = t_0;
	else
		tmp = cosh(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cosh[x], $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[Cosh[x], $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.2], t$95$0, N[Cosh[x], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. *-lowering-*.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 100.0

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

   8. Simplified 100.0%

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

   if -inf.0 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < 0.20000000000000001

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 N/A

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

      2. sin-lowering-sin.f64 99.5

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

   5. Simplified 99.5%

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

   if 0.20000000000000001 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \cosh x \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 100.0%

      \[\leadsto \cosh x \cdot \color{blue}{1} \]
   6. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto \color{blue}{\cosh x} \]

      2. cosh-lowering-cosh.f64 100.0

         \[\leadsto \color{blue}{\cosh x} \]

   7. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\cosh x} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 98.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ t_1 := \cosh x \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot -0.16666666666666666, 1\right)\\ \mathbf{elif}\;t\_1 \leq 0.2:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\cosh x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)) (t_1 (* (cosh x) t_0)))
   (if (<= t_1 (- INFINITY))
     (*
      (fma
       (* x x)
       (fma
        (* x x)
        (fma (* x x) 0.001388888888888889 0.041666666666666664)
        0.5)
       1.0)
      (fma y (* y -0.16666666666666666) 1.0))
     (if (<= t_1 0.2) t_0 (cosh x)))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double t_1 = cosh(x) * t_0;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma((x * x), fma((x * x), fma((x * x), 0.001388888888888889, 0.041666666666666664), 0.5), 1.0) * fma(y, (y * -0.16666666666666666), 1.0);
	} else if (t_1 <= 0.2) {
		tmp = t_0;
	} else {
		tmp = cosh(x);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(y) / y)
	t_1 = Float64(cosh(x) * t_0)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(fma(Float64(x * x), fma(Float64(x * x), fma(Float64(x * x), 0.001388888888888889, 0.041666666666666664), 0.5), 1.0) * fma(y, Float64(y * -0.16666666666666666), 1.0));
	elseif (t_1 <= 0.2)
		tmp = t_0;
	else
		tmp = cosh(x);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cosh[x], $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] * N[(y * N[(y * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.2], t$95$0, N[Cosh[x], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. *-lowering-*.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)} \cdot \mathsf{fma}\left(y, y \cdot \frac{-1}{6}, 1\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) + 1\right)} \cdot \mathsf{fma}\left(y, y \cdot \frac{-1}{6}, 1\right) \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right), 1\right)} \cdot \mathsf{fma}\left(y, y \cdot \frac{-1}{6}, 1\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \frac{-1}{6}, 1\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \frac{-1}{6}, 1\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right) + \frac{1}{2}}, 1\right) \cdot \mathsf{fma}\left(y, y \cdot \frac{-1}{6}, 1\right) \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{24} + \frac{1}{720} \cdot {x}^{2}, \frac{1}{2}\right)}, 1\right) \cdot \mathsf{fma}\left(y, y \cdot \frac{-1}{6}, 1\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24} + \frac{1}{720} \cdot {x}^{2}, \frac{1}{2}\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \frac{-1}{6}, 1\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24} + \frac{1}{720} \cdot {x}^{2}, \frac{1}{2}\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \frac{-1}{6}, 1\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{720} \cdot {x}^{2} + \frac{1}{24}}, \frac{1}{2}\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \frac{-1}{6}, 1\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{1}{720}} + \frac{1}{24}, \frac{1}{2}\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \frac{-1}{6}, 1\right) \]

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{720}, \frac{1}{24}\right)}, \frac{1}{2}\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \frac{-1}{6}, 1\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{720}, \frac{1}{24}\right), \frac{1}{2}\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \frac{-1}{6}, 1\right) \]

      13. *-lowering-*.f64 94.8

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot -0.16666666666666666, 1\right) \]

   8. Simplified 94.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)} \cdot \mathsf{fma}\left(y, y \cdot -0.16666666666666666, 1\right) \]

   if -inf.0 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < 0.20000000000000001

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 N/A

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

      2. sin-lowering-sin.f64 99.5

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

   5. Simplified 99.5%

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

   if 0.20000000000000001 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \cosh x \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 100.0%

      \[\leadsto \cosh x \cdot \color{blue}{1} \]
   6. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto \color{blue}{\cosh x} \]

      2. cosh-lowering-cosh.f64 100.0

         \[\leadsto \color{blue}{\cosh x} \]

   7. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\cosh x} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 75.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -1 \cdot 10^{-137}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot -0.16666666666666666, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\cosh x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cosh x) (/ (sin y) y)) -1e-137)
   (*
    (fma
     (* x x)
     (fma (* x x) (fma (* x x) 0.001388888888888889 0.041666666666666664) 0.5)
     1.0)
    (fma y (* y -0.16666666666666666) 1.0))
   (cosh x)))

C

double code(double x, double y) {
	double tmp;
	if ((cosh(x) * (sin(y) / y)) <= -1e-137) {
		tmp = fma((x * x), fma((x * x), fma((x * x), 0.001388888888888889, 0.041666666666666664), 0.5), 1.0) * fma(y, (y * -0.16666666666666666), 1.0);
	} else {
		tmp = cosh(x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(cosh(x) * Float64(sin(y) / y)) <= -1e-137)
		tmp = Float64(fma(Float64(x * x), fma(Float64(x * x), fma(Float64(x * x), 0.001388888888888889, 0.041666666666666664), 0.5), 1.0) * fma(y, Float64(y * -0.16666666666666666), 1.0));
	else
		tmp = cosh(x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Cosh[x], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], -1e-137], N[(N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] * N[(y * N[(y * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[Cosh[x], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < -9.99999999999999978e-138

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. *-lowering-*.f64 70.0

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

   5. Simplified 70.0%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)} \cdot \mathsf{fma}\left(y, y \cdot \frac{-1}{6}, 1\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) + 1\right)} \cdot \mathsf{fma}\left(y, y \cdot \frac{-1}{6}, 1\right) \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right), 1\right)} \cdot \mathsf{fma}\left(y, y \cdot \frac{-1}{6}, 1\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \frac{-1}{6}, 1\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \frac{-1}{6}, 1\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right) + \frac{1}{2}}, 1\right) \cdot \mathsf{fma}\left(y, y \cdot \frac{-1}{6}, 1\right) \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{24} + \frac{1}{720} \cdot {x}^{2}, \frac{1}{2}\right)}, 1\right) \cdot \mathsf{fma}\left(y, y \cdot \frac{-1}{6}, 1\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24} + \frac{1}{720} \cdot {x}^{2}, \frac{1}{2}\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \frac{-1}{6}, 1\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24} + \frac{1}{720} \cdot {x}^{2}, \frac{1}{2}\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \frac{-1}{6}, 1\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{720} \cdot {x}^{2} + \frac{1}{24}}, \frac{1}{2}\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \frac{-1}{6}, 1\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{1}{720}} + \frac{1}{24}, \frac{1}{2}\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \frac{-1}{6}, 1\right) \]

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{720}, \frac{1}{24}\right)}, \frac{1}{2}\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \frac{-1}{6}, 1\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{720}, \frac{1}{24}\right), \frac{1}{2}\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \frac{-1}{6}, 1\right) \]

      13. *-lowering-*.f64 66.5

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot -0.16666666666666666, 1\right) \]

   8. Simplified 66.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)} \cdot \mathsf{fma}\left(y, y \cdot -0.16666666666666666, 1\right) \]

   if -9.99999999999999978e-138 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y))

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

      \[\leadsto \cosh x \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 74.4%

      \[\leadsto \cosh x \cdot \color{blue}{1} \]
   6. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto \color{blue}{\cosh x} \]

      2. cosh-lowering-cosh.f64 74.4

         \[\leadsto \color{blue}{\cosh x} \]

   7. Applied egg-rr 74.4%

      \[\leadsto \color{blue}{\cosh x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 70.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right)\\ \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -1 \cdot 10^{-137}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, t\_0, 0.5\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot -0.16666666666666666, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot t\_0, 0.5\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma (* x x) 0.001388888888888889 0.041666666666666664)))
   (if (<= (* (cosh x) (/ (sin y) y)) -1e-137)
     (*
      (fma (* x x) (fma (* x x) t_0 0.5) 1.0)
      (fma y (* y -0.16666666666666666) 1.0))
     (fma (* x x) (fma x (* x t_0) 0.5) 1.0))))

C

double code(double x, double y) {
	double t_0 = fma((x * x), 0.001388888888888889, 0.041666666666666664);
	double tmp;
	if ((cosh(x) * (sin(y) / y)) <= -1e-137) {
		tmp = fma((x * x), fma((x * x), t_0, 0.5), 1.0) * fma(y, (y * -0.16666666666666666), 1.0);
	} else {
		tmp = fma((x * x), fma(x, (x * t_0), 0.5), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(Float64(x * x), 0.001388888888888889, 0.041666666666666664)
	tmp = 0.0
	if (Float64(cosh(x) * Float64(sin(y) / y)) <= -1e-137)
		tmp = Float64(fma(Float64(x * x), fma(Float64(x * x), t_0, 0.5), 1.0) * fma(y, Float64(y * -0.16666666666666666), 1.0));
	else
		tmp = fma(Float64(x * x), fma(x, Float64(x * t_0), 0.5), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision]}, If[LessEqual[N[(N[Cosh[x], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], -1e-137], N[(N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * t$95$0 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] * N[(y * N[(y * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * t$95$0), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < -9.99999999999999978e-138

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. *-lowering-*.f64 70.0

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

   5. Simplified 70.0%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)} \cdot \mathsf{fma}\left(y, y \cdot \frac{-1}{6}, 1\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) + 1\right)} \cdot \mathsf{fma}\left(y, y \cdot \frac{-1}{6}, 1\right) \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right), 1\right)} \cdot \mathsf{fma}\left(y, y \cdot \frac{-1}{6}, 1\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \frac{-1}{6}, 1\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \frac{-1}{6}, 1\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right) + \frac{1}{2}}, 1\right) \cdot \mathsf{fma}\left(y, y \cdot \frac{-1}{6}, 1\right) \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{24} + \frac{1}{720} \cdot {x}^{2}, \frac{1}{2}\right)}, 1\right) \cdot \mathsf{fma}\left(y, y \cdot \frac{-1}{6}, 1\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24} + \frac{1}{720} \cdot {x}^{2}, \frac{1}{2}\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \frac{-1}{6}, 1\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24} + \frac{1}{720} \cdot {x}^{2}, \frac{1}{2}\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \frac{-1}{6}, 1\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{720} \cdot {x}^{2} + \frac{1}{24}}, \frac{1}{2}\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \frac{-1}{6}, 1\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{1}{720}} + \frac{1}{24}, \frac{1}{2}\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \frac{-1}{6}, 1\right) \]

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{720}, \frac{1}{24}\right)}, \frac{1}{2}\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \frac{-1}{6}, 1\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{720}, \frac{1}{24}\right), \frac{1}{2}\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \frac{-1}{6}, 1\right) \]

      13. *-lowering-*.f64 66.5

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot -0.16666666666666666, 1\right) \]

   8. Simplified 66.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)} \cdot \mathsf{fma}\left(y, y \cdot -0.16666666666666666, 1\right) \]

   if -9.99999999999999978e-138 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y))

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

      \[\leadsto \cosh x \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 74.4%

      \[\leadsto \cosh x \cdot \color{blue}{1} \]
   6. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto \color{blue}{\cosh x} \]

      2. cosh-lowering-cosh.f64 74.4

         \[\leadsto \color{blue}{\cosh x} \]

   7. Applied egg-rr 74.4%

      \[\leadsto \color{blue}{\cosh x} \]

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) + 1} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right), 1\right)} \]

      3. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right), 1\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right), 1\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right) + \frac{1}{2}}, 1\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right) + \frac{1}{2}, 1\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(x \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)} + \frac{1}{2}, 1\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, x \cdot \color{blue}{\left(\left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right) \cdot x\right)} + \frac{1}{2}, 1\right) \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right) \cdot x, \frac{1}{2}\right)}, 1\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)}, \frac{1}{2}\right), 1\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)}, \frac{1}{2}\right), 1\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\left(\frac{1}{720} \cdot {x}^{2} + \frac{1}{24}\right)}, \frac{1}{2}\right), 1\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{1}{720}} + \frac{1}{24}\right), \frac{1}{2}\right), 1\right) \]

      14. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{720}, \frac{1}{24}\right)}, \frac{1}{2}\right), 1\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{720}, \frac{1}{24}\right), \frac{1}{2}\right), 1\right) \]

      16. *-lowering-*.f64 68.9

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right) \]

   10. Simplified 68.9%

       \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 69.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -1 \cdot 10^{-137}:\\ \;\;\;\;\mathsf{fma}\left(0.5, x \cdot x, 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cosh x) (/ (sin y) y)) -1e-137)
   (*
    (fma 0.5 (* x x) 1.0)
    (fma
     (* y y)
     (fma
      y
      (* y (fma y (* y -0.0001984126984126984) 0.008333333333333333))
      -0.16666666666666666)
     1.0))
   (fma
    (* x x)
    (fma x (* x (fma (* x x) 0.001388888888888889 0.041666666666666664)) 0.5)
    1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((cosh(x) * (sin(y) / y)) <= -1e-137) {
		tmp = fma(0.5, (x * x), 1.0) * fma((y * y), fma(y, (y * fma(y, (y * -0.0001984126984126984), 0.008333333333333333)), -0.16666666666666666), 1.0);
	} else {
		tmp = fma((x * x), fma(x, (x * fma((x * x), 0.001388888888888889, 0.041666666666666664)), 0.5), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(cosh(x) * Float64(sin(y) / y)) <= -1e-137)
		tmp = Float64(fma(0.5, Float64(x * x), 1.0) * fma(Float64(y * y), fma(y, Float64(y * fma(y, Float64(y * -0.0001984126984126984), 0.008333333333333333)), -0.16666666666666666), 1.0));
	else
		tmp = fma(Float64(x * x), fma(x, Float64(x * fma(Float64(x * x), 0.001388888888888889, 0.041666666666666664)), 0.5), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Cosh[x], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], -1e-137], N[(N[(0.5 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * N[(y * N[(y * -0.0001984126984126984), $MachinePrecision] + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < -9.99999999999999978e-138

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{{x}^{2} \cdot \sin y}{y} \cdot \frac{1}{2}} + \frac{\sin y}{y} \]

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. distribute-lft1-in N/A

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

      8. +-commutative N/A

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

      9. *-lowering-*.f64 N/A

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

      10. +-commutative N/A

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

      11. accelerator-lowering-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 N/A

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

      14. /-lowering-/.f64 N/A

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

      15. sin-lowering-sin.f64 63.8

          \[\leadsto \mathsf{fma}\left(0.5, x \cdot x, 1\right) \cdot \frac{\color{blue}{\sin y}}{y} \]

   5. Simplified 63.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, x \cdot x, 1\right) \cdot \frac{\sin y}{y}} \]

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. sub-neg N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. *-lowering-*.f64 N/A

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

      13. +-commutative N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

      17. accelerator-lowering-fma.f64 N/A

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

      18. *-commutative N/A

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

      19. *-lowering-*.f64 66.0

          \[\leadsto \mathsf{fma}\left(0.5, x \cdot x, 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, \color{blue}{y \cdot -0.0001984126984126984}, 0.008333333333333333\right), -0.16666666666666666\right), 1\right) \]

   8. Simplified 66.0%

      \[\leadsto \mathsf{fma}\left(0.5, x \cdot x, 1\right) \cdot \color{blue}{\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)} \]

   if -9.99999999999999978e-138 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y))

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

      \[\leadsto \cosh x \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 74.4%

      \[\leadsto \cosh x \cdot \color{blue}{1} \]
   6. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto \color{blue}{\cosh x} \]

      2. cosh-lowering-cosh.f64 74.4

         \[\leadsto \color{blue}{\cosh x} \]

   7. Applied egg-rr 74.4%

      \[\leadsto \color{blue}{\cosh x} \]

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) + 1} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right), 1\right)} \]

      3. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right), 1\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right), 1\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right) + \frac{1}{2}}, 1\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right) + \frac{1}{2}, 1\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(x \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)} + \frac{1}{2}, 1\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, x \cdot \color{blue}{\left(\left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right) \cdot x\right)} + \frac{1}{2}, 1\right) \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right) \cdot x, \frac{1}{2}\right)}, 1\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)}, \frac{1}{2}\right), 1\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)}, \frac{1}{2}\right), 1\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\left(\frac{1}{720} \cdot {x}^{2} + \frac{1}{24}\right)}, \frac{1}{2}\right), 1\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{1}{720}} + \frac{1}{24}\right), \frac{1}{2}\right), 1\right) \]

      14. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{720}, \frac{1}{24}\right)}, \frac{1}{2}\right), 1\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{720}, \frac{1}{24}\right), \frac{1}{2}\right), 1\right) \]

      16. *-lowering-*.f64 68.9

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right) \]

   10. Simplified 68.9%

       \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 69.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -1 \cdot 10^{-41}:\\ \;\;\;\;\left(0.5 \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cosh x) (/ (sin y) y)) -1e-41)
   (*
    (* 0.5 (* x x))
    (fma
     (* y y)
     (fma
      (* y y)
      (fma y (* y -0.0001984126984126984) 0.008333333333333333)
      -0.16666666666666666)
     1.0))
   (fma
    (* x x)
    (fma x (* x (fma (* x x) 0.001388888888888889 0.041666666666666664)) 0.5)
    1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((cosh(x) * (sin(y) / y)) <= -1e-41) {
		tmp = (0.5 * (x * x)) * fma((y * y), fma((y * y), fma(y, (y * -0.0001984126984126984), 0.008333333333333333), -0.16666666666666666), 1.0);
	} else {
		tmp = fma((x * x), fma(x, (x * fma((x * x), 0.001388888888888889, 0.041666666666666664)), 0.5), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(cosh(x) * Float64(sin(y) / y)) <= -1e-41)
		tmp = Float64(Float64(0.5 * Float64(x * x)) * fma(Float64(y * y), fma(Float64(y * y), fma(y, Float64(y * -0.0001984126984126984), 0.008333333333333333), -0.16666666666666666), 1.0));
	else
		tmp = fma(Float64(x * x), fma(x, Float64(x * fma(Float64(x * x), 0.001388888888888889, 0.041666666666666664)), 0.5), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Cosh[x], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], -1e-41], N[(N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * -0.0001984126984126984), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < -1.00000000000000001e-41

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{{x}^{2} \cdot \sin y}{y} \cdot \frac{1}{2}} + \frac{\sin y}{y} \]

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. distribute-lft1-in N/A

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

      8. +-commutative N/A

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

      9. *-lowering-*.f64 N/A

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

      10. +-commutative N/A

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

      11. accelerator-lowering-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 N/A

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

      14. /-lowering-/.f64 N/A

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

      15. sin-lowering-sin.f64 51.7

          \[\leadsto \mathsf{fma}\left(0.5, x \cdot x, 1\right) \cdot \frac{\color{blue}{\sin y}}{y} \]

   5. Simplified 51.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, x \cdot x, 1\right) \cdot \frac{\sin y}{y}} \]

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. sub-neg N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. *-lowering-*.f64 N/A

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

      13. +-commutative N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

      17. accelerator-lowering-fma.f64 N/A

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

      18. *-commutative N/A

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

      19. *-lowering-*.f64 87.7

          \[\leadsto \mathsf{fma}\left(0.5, x \cdot x, 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, \color{blue}{y \cdot -0.0001984126984126984}, 0.008333333333333333\right), -0.16666666666666666\right), 1\right) \]

   8. Simplified 87.7%

      \[\leadsto \mathsf{fma}\left(0.5, x \cdot x, 1\right) \cdot \color{blue}{\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)} \]

   9. Taylor expanded in x around inf

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

       1. associate-*r* N/A

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

       2. *-lowering-*.f64 N/A

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

       3. *-lowering-*.f64 N/A

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

       4. unpow2 N/A

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

       5. *-lowering-*.f64 N/A

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

       6. +-commutative N/A

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

       7. accelerator-lowering-fma.f64 N/A

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

       8. unpow2 N/A

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

       9. *-lowering-*.f64 N/A

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

       10. sub-neg N/A

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

       11. metadata-eval N/A

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

       12. accelerator-lowering-fma.f64 N/A

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

       13. unpow2 N/A

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

       14. *-lowering-*.f64 N/A

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

       15. +-commutative N/A

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

       16. unpow2 N/A

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

       17. associate-*r* N/A

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

       18. *-commutative N/A

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

       19. accelerator-lowering-fma.f64 N/A

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

       20. *-commutative N/A

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

       21. *-lowering-*.f64 87.7

           \[\leadsto \left(0.5 \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, \color{blue}{y \cdot -0.0001984126984126984}, 0.008333333333333333\right), -0.16666666666666666\right), 1\right) \]

   11. Simplified 87.7%

       \[\leadsto \color{blue}{\left(0.5 \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)} \]

   if -1.00000000000000001e-41 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y))

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

      \[\leadsto \cosh x \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 70.1%

      \[\leadsto \cosh x \cdot \color{blue}{1} \]
   6. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto \color{blue}{\cosh x} \]

      2. cosh-lowering-cosh.f64 70.1

         \[\leadsto \color{blue}{\cosh x} \]

   7. Applied egg-rr 70.1%

      \[\leadsto \color{blue}{\cosh x} \]

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) + 1} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right), 1\right)} \]

      3. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right), 1\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right), 1\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right) + \frac{1}{2}}, 1\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right) + \frac{1}{2}, 1\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(x \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)} + \frac{1}{2}, 1\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, x \cdot \color{blue}{\left(\left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right) \cdot x\right)} + \frac{1}{2}, 1\right) \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right) \cdot x, \frac{1}{2}\right)}, 1\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)}, \frac{1}{2}\right), 1\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)}, \frac{1}{2}\right), 1\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\left(\frac{1}{720} \cdot {x}^{2} + \frac{1}{24}\right)}, \frac{1}{2}\right), 1\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{1}{720}} + \frac{1}{24}\right), \frac{1}{2}\right), 1\right) \]

      14. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{720}, \frac{1}{24}\right)}, \frac{1}{2}\right), 1\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{720}, \frac{1}{24}\right), \frac{1}{2}\right), 1\right) \]

      16. *-lowering-*.f64 64.9

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right) \]

   10. Simplified 64.9%

       \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 69.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -1 \cdot 10^{-137}:\\ \;\;\;\;\mathsf{fma}\left(y, y \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cosh x) (/ (sin y) y)) -1e-137)
   (*
    (fma y (* y -0.16666666666666666) 1.0)
    (fma (* x x) (fma (* x x) 0.041666666666666664 0.5) 1.0))
   (fma
    (* x x)
    (fma x (* x (fma (* x x) 0.001388888888888889 0.041666666666666664)) 0.5)
    1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((cosh(x) * (sin(y) / y)) <= -1e-137) {
		tmp = fma(y, (y * -0.16666666666666666), 1.0) * fma((x * x), fma((x * x), 0.041666666666666664, 0.5), 1.0);
	} else {
		tmp = fma((x * x), fma(x, (x * fma((x * x), 0.001388888888888889, 0.041666666666666664)), 0.5), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(cosh(x) * Float64(sin(y) / y)) <= -1e-137)
		tmp = Float64(fma(y, Float64(y * -0.16666666666666666), 1.0) * fma(Float64(x * x), fma(Float64(x * x), 0.041666666666666664, 0.5), 1.0));
	else
		tmp = fma(Float64(x * x), fma(x, Float64(x * fma(Float64(x * x), 0.001388888888888889, 0.041666666666666664)), 0.5), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Cosh[x], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], -1e-137], N[(N[(y * N[(y * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < -9.99999999999999978e-138

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. *-lowering-*.f64 70.0

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

   5. Simplified 70.0%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)} \cdot \mathsf{fma}\left(y, y \cdot \frac{-1}{6}, 1\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) + 1\right)} \cdot \mathsf{fma}\left(y, y \cdot \frac{-1}{6}, 1\right) \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right)} \cdot \mathsf{fma}\left(y, y \cdot \frac{-1}{6}, 1\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right) \cdot \mathsf{fma}\left(y, y \cdot \frac{-1}{6}, 1\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right) \cdot \mathsf{fma}\left(y, y \cdot \frac{-1}{6}, 1\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{24} \cdot {x}^{2} + \frac{1}{2}}, 1\right) \cdot \mathsf{fma}\left(y, y \cdot \frac{-1}{6}, 1\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{1}{24}} + \frac{1}{2}, 1\right) \cdot \mathsf{fma}\left(y, y \cdot \frac{-1}{6}, 1\right) \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{24}, \frac{1}{2}\right)}, 1\right) \cdot \mathsf{fma}\left(y, y \cdot \frac{-1}{6}, 1\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24}, \frac{1}{2}\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot \frac{-1}{6}, 1\right) \]

      9. *-lowering-*.f64 62.7

         \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.041666666666666664, 0.5\right), 1\right) \cdot \mathsf{fma}\left(y, y \cdot -0.16666666666666666, 1\right) \]

   8. Simplified 62.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right), 1\right)} \cdot \mathsf{fma}\left(y, y \cdot -0.16666666666666666, 1\right) \]

   if -9.99999999999999978e-138 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y))

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

      \[\leadsto \cosh x \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 74.4%

      \[\leadsto \cosh x \cdot \color{blue}{1} \]
   6. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto \color{blue}{\cosh x} \]

      2. cosh-lowering-cosh.f64 74.4

         \[\leadsto \color{blue}{\cosh x} \]

   7. Applied egg-rr 74.4%

      \[\leadsto \color{blue}{\cosh x} \]

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) + 1} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right), 1\right)} \]

      3. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right), 1\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right), 1\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right) + \frac{1}{2}}, 1\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right) + \frac{1}{2}, 1\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(x \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)} + \frac{1}{2}, 1\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, x \cdot \color{blue}{\left(\left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right) \cdot x\right)} + \frac{1}{2}, 1\right) \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right) \cdot x, \frac{1}{2}\right)}, 1\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)}, \frac{1}{2}\right), 1\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)}, \frac{1}{2}\right), 1\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\left(\frac{1}{720} \cdot {x}^{2} + \frac{1}{24}\right)}, \frac{1}{2}\right), 1\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{1}{720}} + \frac{1}{24}\right), \frac{1}{2}\right), 1\right) \]

      14. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{720}, \frac{1}{24}\right)}, \frac{1}{2}\right), 1\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{720}, \frac{1}{24}\right), \frac{1}{2}\right), 1\right) \]

      16. *-lowering-*.f64 68.9

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right) \]

   10. Simplified 68.9%

       \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -1 \cdot 10^{-137}:\\ \;\;\;\;\mathsf{fma}\left(y, y \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 60.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-307}:\\ \;\;\;\;y \cdot \left(y \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, -0.16666666666666666\right)\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-89}:\\ \;\;\;\;\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.008333333333333333, -0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, x \cdot x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)))
   (if (<= t_0 -1e-307)
     (* y (* y (fma (* x x) -0.08333333333333333 -0.16666666666666666)))
     (if (<= t_0 5e-89)
       (fma
        y
        (* y (fma y (* y 0.008333333333333333) -0.16666666666666666))
        1.0)
       (fma 0.5 (* x x) 1.0)))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double tmp;
	if (t_0 <= -1e-307) {
		tmp = y * (y * fma((x * x), -0.08333333333333333, -0.16666666666666666));
	} else if (t_0 <= 5e-89) {
		tmp = fma(y, (y * fma(y, (y * 0.008333333333333333), -0.16666666666666666)), 1.0);
	} else {
		tmp = fma(0.5, (x * x), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(y) / y)
	tmp = 0.0
	if (t_0 <= -1e-307)
		tmp = Float64(y * Float64(y * fma(Float64(x * x), -0.08333333333333333, -0.16666666666666666)));
	elseif (t_0 <= 5e-89)
		tmp = fma(y, Float64(y * fma(y, Float64(y * 0.008333333333333333), -0.16666666666666666)), 1.0);
	else
		tmp = fma(0.5, Float64(x * x), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-307], N[(y * N[(y * N[(N[(x * x), $MachinePrecision] * -0.08333333333333333 + -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e-89], N[(y * N[(y * N[(y * N[(y * 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], N[(0.5 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 y) y) < -9.99999999999999909e-308

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. *-lowering-*.f64 52.2

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

   5. Simplified 52.2%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 42.7

         \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{x \cdot x}, 1\right) \cdot \mathsf{fma}\left(y, y \cdot -0.16666666666666666, 1\right) \]

   8. Simplified 42.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, x \cdot x, 1\right)} \cdot \mathsf{fma}\left(y, y \cdot -0.16666666666666666, 1\right) \]

   9. Taylor expanded in y around inf

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. unpow2 N/A

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

       4. associate-*r* N/A

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

       5. *-commutative N/A

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

       6. *-lowering-*.f64 N/A

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

       7. *-commutative N/A

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

       8. *-lowering-*.f64 N/A

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

       9. +-commutative N/A

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

       10. distribute-rgt-in N/A

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

       11. *-commutative N/A

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

       12. associate-*l* N/A

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

       13. metadata-eval N/A

           \[\leadsto y \cdot \left(y \cdot \left({x}^{2} \cdot \color{blue}{\frac{-1}{12}} + 1 \cdot \frac{-1}{6}\right)\right) \]

       14. metadata-eval N/A

           \[\leadsto y \cdot \left(y \cdot \left({x}^{2} \cdot \frac{-1}{12} + \color{blue}{\frac{-1}{6}}\right)\right) \]

       15. accelerator-lowering-fma.f64 N/A

           \[\leadsto y \cdot \left(y \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{-1}{12}, \frac{-1}{6}\right)}\right) \]

       16. unpow2 N/A

           \[\leadsto y \cdot \left(y \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{12}, \frac{-1}{6}\right)\right) \]

       17. *-lowering-*.f64 42.7

           \[\leadsto y \cdot \left(y \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, -0.08333333333333333, -0.16666666666666666\right)\right) \]

   11. Simplified 42.7%

       \[\leadsto \color{blue}{y \cdot \left(y \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, -0.16666666666666666\right)\right)} \]

   if -9.99999999999999909e-308 < (/.f64 (sin.f64 y) y) < 4.99999999999999967e-89

   1. Initial program 99.7%

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

      1. *-commutative N/A

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

      2. div-inv N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. *-commutative N/A

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

      7. div-inv N/A

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

      8. /-lowering-/.f64 N/A

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

      9. cosh-lowering-cosh.f64 N/A

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

      10. sin-lowering-sin.f64 99.6

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in x around 0

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

   7. Simplified 54.2%

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

   8. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 N/A

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

      8. sub-neg N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. metadata-eval N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. *-lowering-*.f64 47.4

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

   10. Simplified 47.4%

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

   if 4.99999999999999967e-89 < (/.f64 (sin.f64 y) y)

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \cosh x \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 94.6%

      \[\leadsto \cosh x \cdot \color{blue}{1} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + \frac{1}{2} \cdot {x}^{2}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot {x}^{2} + 1} \]

      2. accelerator-lowering-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 73.4

         \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{x \cdot x}, 1\right) \]

   8. Simplified 73.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, x \cdot x, 1\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 68.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -1 \cdot 10^{-137}:\\ \;\;\;\;x \cdot \left(\mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right) \cdot \left(0.5 \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cosh x) (/ (sin y) y)) -1e-137)
   (* x (* (fma -0.16666666666666666 (* y y) 1.0) (* 0.5 x)))
   (fma
    (* x x)
    (fma x (* x (fma (* x x) 0.001388888888888889 0.041666666666666664)) 0.5)
    1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((cosh(x) * (sin(y) / y)) <= -1e-137) {
		tmp = x * (fma(-0.16666666666666666, (y * y), 1.0) * (0.5 * x));
	} else {
		tmp = fma((x * x), fma(x, (x * fma((x * x), 0.001388888888888889, 0.041666666666666664)), 0.5), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(cosh(x) * Float64(sin(y) / y)) <= -1e-137)
		tmp = Float64(x * Float64(fma(-0.16666666666666666, Float64(y * y), 1.0) * Float64(0.5 * x)));
	else
		tmp = fma(Float64(x * x), fma(x, Float64(x * fma(Float64(x * x), 0.001388888888888889, 0.041666666666666664)), 0.5), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Cosh[x], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], -1e-137], N[(x * N[(N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * N[(0.5 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < -9.99999999999999978e-138

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. *-lowering-*.f64 70.0

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

   5. Simplified 70.0%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 57.1

         \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{x \cdot x}, 1\right) \cdot \mathsf{fma}\left(y, y \cdot -0.16666666666666666, 1\right) \]

   8. Simplified 57.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, x \cdot x, 1\right)} \cdot \mathsf{fma}\left(y, y \cdot -0.16666666666666666, 1\right) \]

   9. Taylor expanded in x around inf

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

       1. *-lowering-*.f64 N/A

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

       2. unpow2 N/A

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

       3. *-lowering-*.f64 57.1

          \[\leadsto \left(0.5 \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \mathsf{fma}\left(y, y \cdot -0.16666666666666666, 1\right) \]

   11. Simplified 57.1%

       \[\leadsto \color{blue}{\left(0.5 \cdot \left(x \cdot x\right)\right)} \cdot \mathsf{fma}\left(y, y \cdot -0.16666666666666666, 1\right) \]
   12. Step-by-step derivation

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. associate-*r* N/A

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

       4. *-lowering-*.f64 N/A

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

       5. *-lowering-*.f64 N/A

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

       6. associate-*r* N/A

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

       7. *-commutative N/A

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

       8. accelerator-lowering-fma.f64 N/A

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

       9. *-lowering-*.f64 N/A

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

       10. *-commutative N/A

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

       11. *-lowering-*.f64 57.1

           \[\leadsto \left(\mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right) \cdot \color{blue}{\left(x \cdot 0.5\right)}\right) \cdot x \]

   13. Applied egg-rr 57.1%

       \[\leadsto \color{blue}{\left(\mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right) \cdot \left(x \cdot 0.5\right)\right) \cdot x} \]

   if -9.99999999999999978e-138 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y))

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

      \[\leadsto \cosh x \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 74.4%

      \[\leadsto \cosh x \cdot \color{blue}{1} \]
   6. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto \color{blue}{\cosh x} \]

      2. cosh-lowering-cosh.f64 74.4

         \[\leadsto \color{blue}{\cosh x} \]

   7. Applied egg-rr 74.4%

      \[\leadsto \color{blue}{\cosh x} \]

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) + 1} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right), 1\right)} \]

      3. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right), 1\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right), 1\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right) + \frac{1}{2}}, 1\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right) + \frac{1}{2}, 1\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(x \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)} + \frac{1}{2}, 1\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, x \cdot \color{blue}{\left(\left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right) \cdot x\right)} + \frac{1}{2}, 1\right) \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right) \cdot x, \frac{1}{2}\right)}, 1\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)}, \frac{1}{2}\right), 1\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)}, \frac{1}{2}\right), 1\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\left(\frac{1}{720} \cdot {x}^{2} + \frac{1}{24}\right)}, \frac{1}{2}\right), 1\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{1}{720}} + \frac{1}{24}\right), \frac{1}{2}\right), 1\right) \]

      14. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{720}, \frac{1}{24}\right)}, \frac{1}{2}\right), 1\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{720}, \frac{1}{24}\right), \frac{1}{2}\right), 1\right) \]

      16. *-lowering-*.f64 68.9

          \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right) \]

   10. Simplified 68.9%

       \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -1 \cdot 10^{-137}:\\ \;\;\;\;x \cdot \left(\mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right) \cdot \left(0.5 \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 66.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -1 \cdot 10^{-137}:\\ \;\;\;\;x \cdot \left(\mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right) \cdot \left(0.5 \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cosh x) (/ (sin y) y)) -1e-137)
   (* x (* (fma -0.16666666666666666 (* y y) 1.0) (* 0.5 x)))
   (fma (* x x) (fma (* x x) 0.041666666666666664 0.5) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((cosh(x) * (sin(y) / y)) <= -1e-137) {
		tmp = x * (fma(-0.16666666666666666, (y * y), 1.0) * (0.5 * x));
	} else {
		tmp = fma((x * x), fma((x * x), 0.041666666666666664, 0.5), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(cosh(x) * Float64(sin(y) / y)) <= -1e-137)
		tmp = Float64(x * Float64(fma(-0.16666666666666666, Float64(y * y), 1.0) * Float64(0.5 * x)));
	else
		tmp = fma(Float64(x * x), fma(Float64(x * x), 0.041666666666666664, 0.5), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Cosh[x], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], -1e-137], N[(x * N[(N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * N[(0.5 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < -9.99999999999999978e-138

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. *-lowering-*.f64 70.0

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

   5. Simplified 70.0%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 57.1

         \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{x \cdot x}, 1\right) \cdot \mathsf{fma}\left(y, y \cdot -0.16666666666666666, 1\right) \]

   8. Simplified 57.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, x \cdot x, 1\right)} \cdot \mathsf{fma}\left(y, y \cdot -0.16666666666666666, 1\right) \]

   9. Taylor expanded in x around inf

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

       1. *-lowering-*.f64 N/A

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

       2. unpow2 N/A

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

       3. *-lowering-*.f64 57.1

          \[\leadsto \left(0.5 \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \mathsf{fma}\left(y, y \cdot -0.16666666666666666, 1\right) \]

   11. Simplified 57.1%

       \[\leadsto \color{blue}{\left(0.5 \cdot \left(x \cdot x\right)\right)} \cdot \mathsf{fma}\left(y, y \cdot -0.16666666666666666, 1\right) \]
   12. Step-by-step derivation

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. associate-*r* N/A

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

       4. *-lowering-*.f64 N/A

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

       5. *-lowering-*.f64 N/A

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

       6. associate-*r* N/A

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

       7. *-commutative N/A

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

       8. accelerator-lowering-fma.f64 N/A

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

       9. *-lowering-*.f64 N/A

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

       10. *-commutative N/A

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

       11. *-lowering-*.f64 57.1

           \[\leadsto \left(\mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right) \cdot \color{blue}{\left(x \cdot 0.5\right)}\right) \cdot x \]

   13. Applied egg-rr 57.1%

       \[\leadsto \color{blue}{\left(\mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right) \cdot \left(x \cdot 0.5\right)\right) \cdot x} \]

   if -9.99999999999999978e-138 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y))

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

      \[\leadsto \cosh x \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 74.4%

      \[\leadsto \cosh x \cdot \color{blue}{1} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) + 1} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right)} \]

      3. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{24} \cdot {x}^{2} + \frac{1}{2}}, 1\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{1}{24}} + \frac{1}{2}, 1\right) \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{24}, \frac{1}{2}\right)}, 1\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

      9. *-lowering-*.f64 67.0

         \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.041666666666666664, 0.5\right), 1\right) \]

   8. Simplified 67.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right), 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -1 \cdot 10^{-137}:\\ \;\;\;\;x \cdot \left(\mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right) \cdot \left(0.5 \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 66.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -1 \cdot 10^{-137}:\\ \;\;\;\;y \cdot \left(y \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cosh x) (/ (sin y) y)) -1e-137)
   (* y (* y (fma (* x x) -0.08333333333333333 -0.16666666666666666)))
   (fma (* x x) (fma (* x x) 0.041666666666666664 0.5) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((cosh(x) * (sin(y) / y)) <= -1e-137) {
		tmp = y * (y * fma((x * x), -0.08333333333333333, -0.16666666666666666));
	} else {
		tmp = fma((x * x), fma((x * x), 0.041666666666666664, 0.5), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(cosh(x) * Float64(sin(y) / y)) <= -1e-137)
		tmp = Float64(y * Float64(y * fma(Float64(x * x), -0.08333333333333333, -0.16666666666666666)));
	else
		tmp = fma(Float64(x * x), fma(Float64(x * x), 0.041666666666666664, 0.5), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Cosh[x], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], -1e-137], N[(y * N[(y * N[(N[(x * x), $MachinePrecision] * -0.08333333333333333 + -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < -9.99999999999999978e-138

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. *-lowering-*.f64 70.0

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

   5. Simplified 70.0%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 57.1

         \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{x \cdot x}, 1\right) \cdot \mathsf{fma}\left(y, y \cdot -0.16666666666666666, 1\right) \]

   8. Simplified 57.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, x \cdot x, 1\right)} \cdot \mathsf{fma}\left(y, y \cdot -0.16666666666666666, 1\right) \]

   9. Taylor expanded in y around inf

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. unpow2 N/A

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

       4. associate-*r* N/A

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

       5. *-commutative N/A

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

       6. *-lowering-*.f64 N/A

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

       7. *-commutative N/A

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

       8. *-lowering-*.f64 N/A

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

       9. +-commutative N/A

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

       10. distribute-rgt-in N/A

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

       11. *-commutative N/A

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

       12. associate-*l* N/A

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

       13. metadata-eval N/A

           \[\leadsto y \cdot \left(y \cdot \left({x}^{2} \cdot \color{blue}{\frac{-1}{12}} + 1 \cdot \frac{-1}{6}\right)\right) \]

       14. metadata-eval N/A

           \[\leadsto y \cdot \left(y \cdot \left({x}^{2} \cdot \frac{-1}{12} + \color{blue}{\frac{-1}{6}}\right)\right) \]

       15. accelerator-lowering-fma.f64 N/A

           \[\leadsto y \cdot \left(y \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{-1}{12}, \frac{-1}{6}\right)}\right) \]

       16. unpow2 N/A

           \[\leadsto y \cdot \left(y \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{12}, \frac{-1}{6}\right)\right) \]

       17. *-lowering-*.f64 57.1

           \[\leadsto y \cdot \left(y \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, -0.08333333333333333, -0.16666666666666666\right)\right) \]

   11. Simplified 57.1%

       \[\leadsto \color{blue}{y \cdot \left(y \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, -0.16666666666666666\right)\right)} \]

   if -9.99999999999999978e-138 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y))

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

      \[\leadsto \cosh x \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 74.4%

      \[\leadsto \cosh x \cdot \color{blue}{1} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) + 1} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right)} \]

      3. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{24} \cdot {x}^{2} + \frac{1}{2}}, 1\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{1}{24}} + \frac{1}{2}, 1\right) \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{24}, \frac{1}{2}\right)}, 1\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

      9. *-lowering-*.f64 67.0

         \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.041666666666666664, 0.5\right), 1\right) \]

   8. Simplified 67.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right), 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 56.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -1 \cdot 10^{-137}:\\ \;\;\;\;y \cdot \left(y \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, x \cdot x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cosh x) (/ (sin y) y)) -1e-137)
   (* y (* y (fma (* x x) -0.08333333333333333 -0.16666666666666666)))
   (fma 0.5 (* x x) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((cosh(x) * (sin(y) / y)) <= -1e-137) {
		tmp = y * (y * fma((x * x), -0.08333333333333333, -0.16666666666666666));
	} else {
		tmp = fma(0.5, (x * x), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(cosh(x) * Float64(sin(y) / y)) <= -1e-137)
		tmp = Float64(y * Float64(y * fma(Float64(x * x), -0.08333333333333333, -0.16666666666666666)));
	else
		tmp = fma(0.5, Float64(x * x), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Cosh[x], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], -1e-137], N[(y * N[(y * N[(N[(x * x), $MachinePrecision] * -0.08333333333333333 + -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < -9.99999999999999978e-138

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. *-lowering-*.f64 70.0

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

   5. Simplified 70.0%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 57.1

         \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{x \cdot x}, 1\right) \cdot \mathsf{fma}\left(y, y \cdot -0.16666666666666666, 1\right) \]

   8. Simplified 57.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, x \cdot x, 1\right)} \cdot \mathsf{fma}\left(y, y \cdot -0.16666666666666666, 1\right) \]

   9. Taylor expanded in y around inf

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. unpow2 N/A

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

       4. associate-*r* N/A

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

       5. *-commutative N/A

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

       6. *-lowering-*.f64 N/A

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

       7. *-commutative N/A

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

       8. *-lowering-*.f64 N/A

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

       9. +-commutative N/A

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

       10. distribute-rgt-in N/A

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

       11. *-commutative N/A

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

       12. associate-*l* N/A

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

       13. metadata-eval N/A

           \[\leadsto y \cdot \left(y \cdot \left({x}^{2} \cdot \color{blue}{\frac{-1}{12}} + 1 \cdot \frac{-1}{6}\right)\right) \]

       14. metadata-eval N/A

           \[\leadsto y \cdot \left(y \cdot \left({x}^{2} \cdot \frac{-1}{12} + \color{blue}{\frac{-1}{6}}\right)\right) \]

       15. accelerator-lowering-fma.f64 N/A

           \[\leadsto y \cdot \left(y \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{-1}{12}, \frac{-1}{6}\right)}\right) \]

       16. unpow2 N/A

           \[\leadsto y \cdot \left(y \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{12}, \frac{-1}{6}\right)\right) \]

       17. *-lowering-*.f64 57.1

           \[\leadsto y \cdot \left(y \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, -0.08333333333333333, -0.16666666666666666\right)\right) \]

   11. Simplified 57.1%

       \[\leadsto \color{blue}{y \cdot \left(y \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, -0.16666666666666666\right)\right)} \]

   if -9.99999999999999978e-138 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y))

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

      \[\leadsto \cosh x \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 74.4%

      \[\leadsto \cosh x \cdot \color{blue}{1} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + \frac{1}{2} \cdot {x}^{2}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot {x}^{2} + 1} \]

      2. accelerator-lowering-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 56.0

         \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{x \cdot x}, 1\right) \]

   8. Simplified 56.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, x \cdot x, 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 56.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -1 \cdot 10^{-137}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(\left(y \cdot y\right) \cdot -0.08333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, x \cdot x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cosh x) (/ (sin y) y)) -1e-137)
   (* (* x x) (* (* y y) -0.08333333333333333))
   (fma 0.5 (* x x) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((cosh(x) * (sin(y) / y)) <= -1e-137) {
		tmp = (x * x) * ((y * y) * -0.08333333333333333);
	} else {
		tmp = fma(0.5, (x * x), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(cosh(x) * Float64(sin(y) / y)) <= -1e-137)
		tmp = Float64(Float64(x * x) * Float64(Float64(y * y) * -0.08333333333333333));
	else
		tmp = fma(0.5, Float64(x * x), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Cosh[x], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], -1e-137], N[(N[(x * x), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * -0.08333333333333333), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < -9.99999999999999978e-138

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. *-lowering-*.f64 70.0

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

   5. Simplified 70.0%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 57.1

         \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{x \cdot x}, 1\right) \cdot \mathsf{fma}\left(y, y \cdot -0.16666666666666666, 1\right) \]

   8. Simplified 57.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, x \cdot x, 1\right)} \cdot \mathsf{fma}\left(y, y \cdot -0.16666666666666666, 1\right) \]

   9. Taylor expanded in x around inf

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

       1. *-lowering-*.f64 N/A

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

       2. unpow2 N/A

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

       3. *-lowering-*.f64 57.1

          \[\leadsto \left(0.5 \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \mathsf{fma}\left(y, y \cdot -0.16666666666666666, 1\right) \]

   11. Simplified 57.1%

       \[\leadsto \color{blue}{\left(0.5 \cdot \left(x \cdot x\right)\right)} \cdot \mathsf{fma}\left(y, y \cdot -0.16666666666666666, 1\right) \]

   12. Taylor expanded in y around inf

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

       1. *-commutative N/A

          \[\leadsto \color{blue}{\left({x}^{2} \cdot {y}^{2}\right) \cdot \frac{-1}{12}} \]

       2. associate-*l* N/A

          \[\leadsto \color{blue}{{x}^{2} \cdot \left({y}^{2} \cdot \frac{-1}{12}\right)} \]

       3. *-commutative N/A

          \[\leadsto {x}^{2} \cdot \color{blue}{\left(\frac{-1}{12} \cdot {y}^{2}\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{-1}{12} \cdot {y}^{2}\right)} \]

       5. unpow2 N/A

          \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{-1}{12} \cdot {y}^{2}\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{-1}{12} \cdot {y}^{2}\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \left(x \cdot x\right) \cdot \color{blue}{\left(\frac{-1}{12} \cdot {y}^{2}\right)} \]

       8. unpow2 N/A

          \[\leadsto \left(x \cdot x\right) \cdot \left(\frac{-1}{12} \cdot \color{blue}{\left(y \cdot y\right)}\right) \]

       9. *-lowering-*.f64 57.1

          \[\leadsto \left(x \cdot x\right) \cdot \left(-0.08333333333333333 \cdot \color{blue}{\left(y \cdot y\right)}\right) \]

   14. Simplified 57.1%

       \[\leadsto \color{blue}{\left(x \cdot x\right) \cdot \left(-0.08333333333333333 \cdot \left(y \cdot y\right)\right)} \]

   if -9.99999999999999978e-138 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y))

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

      \[\leadsto \cosh x \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 74.4%

      \[\leadsto \cosh x \cdot \color{blue}{1} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + \frac{1}{2} \cdot {x}^{2}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot {x}^{2} + 1} \]

      2. accelerator-lowering-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 56.0

         \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{x \cdot x}, 1\right) \]

   8. Simplified 56.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, x \cdot x, 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -1 \cdot 10^{-137}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(\left(y \cdot y\right) \cdot -0.08333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, x \cdot x, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 52.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -1 \cdot 10^{-137}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, x \cdot x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cosh x) (/ (sin y) y)) -1e-137)
   (fma (* y y) -0.16666666666666666 1.0)
   (fma 0.5 (* x x) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((cosh(x) * (sin(y) / y)) <= -1e-137) {
		tmp = fma((y * y), -0.16666666666666666, 1.0);
	} else {
		tmp = fma(0.5, (x * x), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(cosh(x) * Float64(sin(y) / y)) <= -1e-137)
		tmp = fma(Float64(y * y), -0.16666666666666666, 1.0);
	else
		tmp = fma(0.5, Float64(x * x), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Cosh[x], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], -1e-137], N[(N[(y * y), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision], N[(0.5 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < -9.99999999999999978e-138

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. *-lowering-*.f64 70.0

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

   5. Simplified 70.0%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{-1}{6} \cdot {y}^{2} + 1} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{{y}^{2} \cdot \frac{-1}{6}} + 1 \]

      3. accelerator-lowering-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 33.2

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

   8. Simplified 33.2%

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

   if -9.99999999999999978e-138 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y))

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

      \[\leadsto \cosh x \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 74.4%

      \[\leadsto \cosh x \cdot \color{blue}{1} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + \frac{1}{2} \cdot {x}^{2}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot {x}^{2} + 1} \]

      2. accelerator-lowering-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 56.0

         \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{x \cdot x}, 1\right) \]

   8. Simplified 56.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, x \cdot x, 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 45.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cosh x) (/ (sin y) y)) 2.0) 1.0 (* 0.5 (* x x))))

C

double code(double x, double y) {
	double tmp;
	if ((cosh(x) * (sin(y) / y)) <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = 0.5 * (x * x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((cosh(x) * (sin(y) / y)) <= 2.0d0) then
        tmp = 1.0d0
    else
        tmp = 0.5d0 * (x * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((Math.cosh(x) * (Math.sin(y) / y)) <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = 0.5 * (x * x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (math.cosh(x) * (math.sin(y) / y)) <= 2.0:
		tmp = 1.0
	else:
		tmp = 0.5 * (x * x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(cosh(x) * Float64(sin(y) / y)) <= 2.0)
		tmp = 1.0;
	else
		tmp = Float64(0.5 * Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((cosh(x) * (sin(y) / y)) <= 2.0)
		tmp = 1.0;
	else
		tmp = 0.5 * (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Cosh[x], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], 2.0], 1.0, N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

      \[\leadsto \cosh x \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 36.3%

      \[\leadsto \cosh x \cdot \color{blue}{1} \]

   6. Taylor expanded in x around 0

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

   8. Simplified 36.4%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \cosh x \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 100.0%

      \[\leadsto \cosh x \cdot \color{blue}{1} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + \frac{1}{2} \cdot {x}^{2}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot {x}^{2} + 1} \]

      2. accelerator-lowering-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 60.0

         \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{x \cdot x}, 1\right) \]

   8. Simplified 60.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, x \cdot x, 1\right)} \]

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot {x}^{2}} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{2} \cdot {x}^{2}} \]

       2. unpow2 N/A

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

       3. *-lowering-*.f64 60.0

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

   11. Simplified 60.0%

       \[\leadsto \color{blue}{0.5 \cdot \left(x \cdot x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

Alternative 20: 45.7% accurate, 18.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(0.5, x \cdot x, 1\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (fma 0.5 (* x x) 1.0))

C

double code(double x, double y) {
	return fma(0.5, (x * x), 1.0);
}

Julia

function code(x, y)
	return fma(0.5, Float64(x * x), 1.0)
end

Wolfram

code[x_, y_] := N[(0.5 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in y around 0

   \[\leadsto \cosh x \cdot \color{blue}{1} \]
4. Step-by-step derivation

5. Simplified 59.7%

   \[\leadsto \cosh x \cdot \color{blue}{1} \]

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1 + \frac{1}{2} \cdot {x}^{2}} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{\frac{1}{2} \cdot {x}^{2} + 1} \]

   2. accelerator-lowering-fma.f64 N/A

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

   3. unpow2 N/A

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

   4. *-lowering-*.f64 45.0

      \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{x \cdot x}, 1\right) \]

8. Simplified 45.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, x \cdot x, 1\right)} \]
9. Add Preprocessing

Alternative 21: 27.6% accurate, 217.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

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

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in y around 0

   \[\leadsto \cosh x \cdot \color{blue}{1} \]
4. Step-by-step derivation

5. Simplified 59.7%

   \[\leadsto \cosh x \cdot \color{blue}{1} \]

6. Taylor expanded in x around 0

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

8. Simplified 24.2%

   \[\leadsto \color{blue}{1} \]
9. Add Preprocessing

Developer Target 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024195 
(FPCore (x y)
  :name "Linear.Quaternion:$csinh from linear-1.19.1.3"
  :precision binary64

  :alt
  (! :herbie-platform default (/ (* (cosh x) (sin y)) y))

  (* (cosh x) (/ (sin y) y)))