Linear.Quaternion:$csinh from linear-1.19.1.3

Percentage Accurate: 99.9% → 99.9%

Time: 13.0s

Alternatives: 16

Speedup: 1.0×

• 49.3% 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

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

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 2: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ t_1 := \cosh x \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\cosh x \cdot \left(-0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ \mathbf{elif}\;t\_1 \leq 0.9999999994777551:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(0.5, x \cdot x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\cosh x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)) (t_1 (* (cosh x) t_0)))
   (if (<= t_1 (- INFINITY))
     (* (cosh x) (* -0.16666666666666666 (* y y)))
     (if (<= t_1 0.9999999994777551) (* 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) * (-0.16666666666666666 * (y * y));
	} else if (t_1 <= 0.9999999994777551) {
		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(-0.16666666666666666 * Float64(y * y)));
	elseif (t_1 <= 0.9999999994777551)
		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[(-0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.9999999994777551], 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. *-lowering-*.f64 N/A

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 100.0

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

   8. Simplified 100.0%

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

   if -inf.0 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < 0.99999999947775509

   1. Initial program 99.7%

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

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

   5. Simplified 99.7%

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

   if 0.99999999947775509 < (*.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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -\infty:\\ \;\;\;\;\cosh x \cdot \left(-0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ \mathbf{elif}\;\cosh x \cdot \frac{\sin y}{y} \leq 0.9999999994777551:\\ \;\;\;\;\frac{\sin y}{y} \cdot \mathsf{fma}\left(0.5, x \cdot x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\cosh x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ t_1 := \cosh x \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\cosh x \cdot \left(-0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ \mathbf{elif}\;t\_1 \leq 0.9999999994777551:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\cosh x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)) (t_1 (* (cosh x) t_0)))
   (if (<= t_1 (- INFINITY))
     (* (cosh x) (* -0.16666666666666666 (* y y)))
     (if (<= t_1 0.9999999994777551) 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) * (-0.16666666666666666 * (y * y));
	} else if (t_1 <= 0.9999999994777551) {
		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) * (-0.16666666666666666 * (y * y));
	} else if (t_1 <= 0.9999999994777551) {
		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) * (-0.16666666666666666 * (y * y))
	elif t_1 <= 0.9999999994777551:
		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(-0.16666666666666666 * Float64(y * y)));
	elseif (t_1 <= 0.9999999994777551)
		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) * (-0.16666666666666666 * (y * y));
	elseif (t_1 <= 0.9999999994777551)
		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[(-0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.9999999994777551], 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. *-lowering-*.f64 N/A

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 100.0

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

   8. Simplified 100.0%

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

   if -inf.0 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < 0.99999999947775509

   1. Initial program 99.7%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   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 0.99999999947775509 < (*.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: 98.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ t_1 := \cosh x \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(0.5, x \cdot x, 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)\\ \mathbf{elif}\;t\_1 \leq 0.9999999994777551:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\cosh x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)) (t_1 (* (cosh x) t_0)))
   (if (<= t_1 (- INFINITY))
     (*
      (fma 0.5 (* x x) 1.0)
      (fma
       (* y y)
       (fma
        y
        (* y (fma (* y y) -0.0001984126984126984 0.008333333333333333))
        -0.16666666666666666)
       1.0))
     (if (<= t_1 0.9999999994777551) 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(0.5, (x * x), 1.0) * fma((y * y), fma(y, (y * fma((y * y), -0.0001984126984126984, 0.008333333333333333)), -0.16666666666666666), 1.0);
	} else if (t_1 <= 0.9999999994777551) {
		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(0.5, Float64(x * x), 1.0) * fma(Float64(y * y), fma(y, Float64(y * fma(Float64(y * y), -0.0001984126984126984, 0.008333333333333333)), -0.16666666666666666), 1.0));
	elseif (t_1 <= 0.9999999994777551)
		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[(0.5 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.9999999994777551], 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 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 53.2

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

   5. Simplified 53.2%

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

   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. metadata-eval N/A

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

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

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

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

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

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

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 93.7

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

   8. Simplified 93.7%

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

   if -inf.0 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < 0.99999999947775509

   1. Initial program 99.7%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   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 0.99999999947775509 < (*.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: 74.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq 0.9999999994777551:\\ \;\;\;\;\mathsf{fma}\left(0.5, x \cdot x, 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\cosh x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cosh x) (/ (sin y) y)) 0.9999999994777551)
   (*
    (fma 0.5 (* x x) 1.0)
    (fma
     (* y y)
     (fma
      y
      (* y (fma (* y y) -0.0001984126984126984 0.008333333333333333))
      -0.16666666666666666)
     1.0))
   (cosh x)))

C

double code(double x, double y) {
	double tmp;
	if ((cosh(x) * (sin(y) / y)) <= 0.9999999994777551) {
		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 = cosh(x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(cosh(x) * Float64(sin(y) / y)) <= 0.9999999994777551)
		tmp = Float64(fma(0.5, Float64(x * x), 1.0) * fma(Float64(y * y), fma(y, Float64(y * fma(Float64(y * y), -0.0001984126984126984, 0.008333333333333333)), -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], 0.9999999994777551], N[(N[(0.5 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + -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)) < 0.99999999947775509

   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 86.2

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

   5. Simplified 86.2%

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

   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. metadata-eval N/A

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

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

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

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

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

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

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 33.4

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

   8. Simplified 33.4%

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

   if 0.99999999947775509 < (*.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 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 69.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq 0.9999999994777551:\\ \;\;\;\;\mathsf{fma}\left(0.5, x \cdot x, 1\right) \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cosh x) (/ (sin y) y)) 0.9999999994777551)
   (*
    (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)) <= 0.9999999994777551) {
		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)) <= 0.9999999994777551)
		tmp = Float64(fma(0.5, Float64(x * x), 1.0) * fma(Float64(y * y), fma(y, Float64(y * fma(Float64(y * y), -0.0001984126984126984, 0.008333333333333333)), -0.16666666666666666), 1.0));
	else
		tmp = fma(x, Float64(x * fma(x, Float64(x * fma(x, Float64(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], 0.9999999994777551], N[(N[(0.5 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(x * N[(x * N[(x * N[(x * N[(x * N[(x * 0.001388888888888889), $MachinePrecision] + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < 0.99999999947775509

   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 86.2

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

   5. Simplified 86.2%

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

   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. metadata-eval N/A

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

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

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

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

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

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

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 33.4

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

   8. Simplified 33.4%

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

   if 0.99999999947775509 < (*.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%

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

   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. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

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

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

   10. Simplified 91.4%

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

Alternative 7: 69.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)\\ \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -1 \cdot 10^{-147}:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (fma
          x
          (*
           x
           (fma
            x
            (* x (fma x (* x 0.001388888888888889) 0.041666666666666664))
            0.5))
          1.0)))
   (if (<= (* (cosh x) (/ (sin y) y)) -1e-147)
     (* t_0 (fma -0.16666666666666666 (* y y) 1.0))
     t_0)))

C

double code(double x, double y) {
	double t_0 = fma(x, (x * fma(x, (x * fma(x, (x * 0.001388888888888889), 0.041666666666666664)), 0.5)), 1.0);
	double tmp;
	if ((cosh(x) * (sin(y) / y)) <= -1e-147) {
		tmp = t_0 * fma(-0.16666666666666666, (y * y), 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(x, Float64(x * fma(x, Float64(x * fma(x, Float64(x * 0.001388888888888889), 0.041666666666666664)), 0.5)), 1.0)
	tmp = 0.0
	if (Float64(cosh(x) * Float64(sin(y) / y)) <= -1e-147)
		tmp = Float64(t_0 * fma(-0.16666666666666666, Float64(y * y), 1.0));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x * N[(x * N[(x * N[(x * N[(x * N[(x * 0.001388888888888889), $MachinePrecision] + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[N[(N[Cosh[x], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], -1e-147], N[(t$95$0 * N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < -9.9999999999999997e-148

   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 62.1

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

   5. Simplified 62.1%

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

   6. Taylor expanded in x around 0

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

   8. Simplified 58.1%

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

   if -9.9999999999999997e-148 < (*.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 77.4%

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

      1. *-rgt-identity N/A

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

      2. cosh-lowering-cosh.f64 77.4

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

   7. Applied egg-rr 77.4%

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

   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. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

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

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

   10. Simplified 70.9%

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

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -1 \cdot 10^{-147}:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right) \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 69.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -1 \cdot 10^{-147}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cosh x) (/ (sin y) y)) -1e-147)
   (*
    (fma -0.16666666666666666 (* y y) 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-147) {
		tmp = fma(-0.16666666666666666, (y * y), 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-147)
		tmp = Float64(fma(-0.16666666666666666, Float64(y * y), 1.0) * fma(Float64(x * x), fma(Float64(x * x), 0.041666666666666664, 0.5), 1.0));
	else
		tmp = fma(x, Float64(x * fma(x, Float64(x * fma(x, Float64(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-147], N[(N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(x * N[(x * N[(x * N[(x * N[(x * N[(x * 0.001388888888888889), $MachinePrecision] + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < -9.9999999999999997e-148

   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 62.1

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

   5. Simplified 62.1%

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

   6. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

      2. *-rgt-identity N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. +-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. distribute-lft-out N/A

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

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

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

   8. Simplified 56.2%

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

   if -9.9999999999999997e-148 < (*.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 77.4%

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

      1. *-rgt-identity N/A

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

      2. cosh-lowering-cosh.f64 77.4

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

   7. Applied egg-rr 77.4%

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

   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. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

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

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

   10. Simplified 70.9%

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

Alternative 9: 68.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -1 \cdot 10^{-147}:\\ \;\;\;\;\mathsf{fma}\left(0.5, x \cdot x, 1\right) \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cosh x) (/ (sin y) y)) -1e-147)
   (* (fma 0.5 (* x x) 1.0) (fma -0.16666666666666666 (* y y) 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-147) {
		tmp = fma(0.5, (x * x), 1.0) * fma(-0.16666666666666666, (y * y), 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-147)
		tmp = Float64(fma(0.5, Float64(x * x), 1.0) * fma(-0.16666666666666666, Float64(y * y), 1.0));
	else
		tmp = fma(x, Float64(x * fma(x, Float64(x * fma(x, Float64(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-147], N[(N[(0.5 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(x * N[(x * N[(x * N[(x * N[(x * N[(x * 0.001388888888888889), $MachinePrecision] + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < -9.9999999999999997e-148

   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 62.1

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

   5. Simplified 62.1%

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

   6. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

      2. *-lft-identity N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-in N/A

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

      5. lft-mult-inverse N/A

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

      6. distribute-rgt-in N/A

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

      7. +-commutative N/A

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

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

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

      9. +-commutative N/A

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

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

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

      11. unpow2 N/A

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

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

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

      13. distribute-rgt-in N/A

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

      14. lft-mult-inverse N/A

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

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

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 54.2

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

   8. Simplified 54.2%

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

   if -9.9999999999999997e-148 < (*.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 77.4%

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

      1. *-rgt-identity N/A

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

      2. cosh-lowering-cosh.f64 77.4

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

   7. Applied egg-rr 77.4%

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

   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. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

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

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

   10. Simplified 70.9%

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

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -1 \cdot 10^{-147}:\\ \;\;\;\;\mathsf{fma}\left(0.5, x \cdot x, 1\right) \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 68.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq 2:\\ \;\;\;\;\mathsf{fma}\left(0.5, x \cdot x, 1\right) \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, x \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot 0.001388888888888889, 0.041666666666666664\right)\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cosh x) (/ (sin y) y)) 2.0)
   (* (fma 0.5 (* x x) 1.0) (fma -0.16666666666666666 (* y y) 1.0))
   (fma
    (* x x)
    (* x (* x (fma x (* x 0.001388888888888889) 0.041666666666666664)))
    1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((cosh(x) * (sin(y) / y)) <= 2.0) {
		tmp = fma(0.5, (x * x), 1.0) * fma(-0.16666666666666666, (y * y), 1.0);
	} else {
		tmp = fma((x * x), (x * (x * fma(x, (x * 0.001388888888888889), 0.041666666666666664))), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(cosh(x) * Float64(sin(y) / y)) <= 2.0)
		tmp = Float64(fma(0.5, Float64(x * x), 1.0) * fma(-0.16666666666666666, Float64(y * y), 1.0));
	else
		tmp = fma(Float64(x * x), Float64(x * Float64(x * fma(x, Float64(x * 0.001388888888888889), 0.041666666666666664))), 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], 2.0], N[(N[(0.5 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(x * N[(x * 0.001388888888888889), $MachinePrecision] + 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

      \[\leadsto \cosh x \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {y}^{2}\right)} \]
   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 62.0

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

   5. Simplified 62.0%

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

   6. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

      2. *-lft-identity N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-in N/A

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

      5. lft-mult-inverse N/A

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

      6. distribute-rgt-in N/A

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

      7. +-commutative N/A

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

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

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

      9. +-commutative N/A

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

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

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

      11. unpow2 N/A

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

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

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

      13. distribute-rgt-in N/A

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

      14. lft-mult-inverse N/A

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

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

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 59.5

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

   8. Simplified 59.5%

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

   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

      \[\leadsto \color{blue}{1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)} \]
   7. 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. accelerator-lowering-fma.f64 N/A

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

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

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 84.2

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

   8. Simplified 84.2%

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

   9. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{4} \cdot \left(\frac{1}{720} + \frac{1}{24} \cdot \frac{1}{{x}^{2}}\right)}, 1\right) \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, {x}^{4} \cdot \color{blue}{\left(\frac{1}{24} \cdot \frac{1}{{x}^{2}} + \frac{1}{720}\right)}, 1\right) \]

       2. distribute-rgt-in N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{\left(\frac{1}{24} \cdot \frac{1}{{x}^{2}}\right) \cdot {x}^{4} + \frac{1}{720} \cdot {x}^{4}}, 1\right) \]

       3. associate-*r/ N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{\frac{1}{24} \cdot 1}{{x}^{2}}} \cdot {x}^{4} + \frac{1}{720} \cdot {x}^{4}, 1\right) \]

       4. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{\color{blue}{\frac{1}{24}}}{{x}^{2}} \cdot {x}^{4} + \frac{1}{720} \cdot {x}^{4}, 1\right) \]

       5. associate-*l/ N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{\frac{1}{24} \cdot {x}^{4}}{{x}^{2}}} + \frac{1}{720} \cdot {x}^{4}, 1\right) \]

       6. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{24} \cdot \frac{{x}^{4}}{{x}^{2}}} + \frac{1}{720} \cdot {x}^{4}, 1\right) \]

       7. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{1}{24} \cdot \frac{{x}^{\color{blue}{\left(2 \cdot 2\right)}}}{{x}^{2}} + \frac{1}{720} \cdot {x}^{4}, 1\right) \]

       8. pow-sqr N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{1}{24} \cdot \frac{\color{blue}{{x}^{2} \cdot {x}^{2}}}{{x}^{2}} + \frac{1}{720} \cdot {x}^{4}, 1\right) \]

       9. *-rgt-identity N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{1}{24} \cdot \frac{{x}^{2} \cdot {x}^{2}}{\color{blue}{{x}^{2} \cdot 1}} + \frac{1}{720} \cdot {x}^{4}, 1\right) \]

       10. times-frac N/A

           \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{1}{24} \cdot \color{blue}{\left(\frac{{x}^{2}}{{x}^{2}} \cdot \frac{{x}^{2}}{1}\right)} + \frac{1}{720} \cdot {x}^{4}, 1\right) \]

       11. *-rgt-identity N/A

           \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{1}{24} \cdot \left(\frac{\color{blue}{{x}^{2} \cdot 1}}{{x}^{2}} \cdot \frac{{x}^{2}}{1}\right) + \frac{1}{720} \cdot {x}^{4}, 1\right) \]

       12. associate-*r/ N/A

           \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{1}{24} \cdot \left(\color{blue}{\left({x}^{2} \cdot \frac{1}{{x}^{2}}\right)} \cdot \frac{{x}^{2}}{1}\right) + \frac{1}{720} \cdot {x}^{4}, 1\right) \]

       13. rgt-mult-inverse N/A

           \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{1}{24} \cdot \left(\color{blue}{1} \cdot \frac{{x}^{2}}{1}\right) + \frac{1}{720} \cdot {x}^{4}, 1\right) \]

       14. /-rgt-identity N/A

           \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{1}{24} \cdot \left(1 \cdot \color{blue}{{x}^{2}}\right) + \frac{1}{720} \cdot {x}^{4}, 1\right) \]

       15. *-lft-identity N/A

           \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{1}{24} \cdot \color{blue}{{x}^{2}} + \frac{1}{720} \cdot {x}^{4}, 1\right) \]

       16. metadata-eval N/A

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

       17. pow-sqr N/A

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

       18. associate-*l* N/A

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

       19. distribute-rgt-in N/A

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

   11. Simplified 84.2%

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

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq 2:\\ \;\;\;\;\mathsf{fma}\left(0.5, x \cdot x, 1\right) \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, x \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot 0.001388888888888889, 0.041666666666666664\right)\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 68.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq 2:\\ \;\;\;\;\mathsf{fma}\left(0.5, x \cdot x, 1\right) \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, x \cdot \left(0.001388888888888889 \cdot \left(x \cdot \left(x \cdot x\right)\right)\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cosh x) (/ (sin y) y)) 2.0)
   (* (fma 0.5 (* x x) 1.0) (fma -0.16666666666666666 (* y y) 1.0))
   (fma (* x x) (* x (* 0.001388888888888889 (* x (* x x)))) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((cosh(x) * (sin(y) / y)) <= 2.0) {
		tmp = fma(0.5, (x * x), 1.0) * fma(-0.16666666666666666, (y * y), 1.0);
	} else {
		tmp = fma((x * x), (x * (0.001388888888888889 * (x * (x * x)))), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(cosh(x) * Float64(sin(y) / y)) <= 2.0)
		tmp = Float64(fma(0.5, Float64(x * x), 1.0) * fma(-0.16666666666666666, Float64(y * y), 1.0));
	else
		tmp = fma(Float64(x * x), Float64(x * Float64(0.001388888888888889 * Float64(x * 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], 2.0], N[(N[(0.5 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] * N[(x * N[(0.001388888888888889 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

      \[\leadsto \cosh x \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {y}^{2}\right)} \]
   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 62.0

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

   5. Simplified 62.0%

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

   6. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

      2. *-lft-identity N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-in N/A

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

      5. lft-mult-inverse N/A

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

      6. distribute-rgt-in N/A

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

      7. +-commutative N/A

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

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

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

      9. +-commutative N/A

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

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

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

      11. unpow2 N/A

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

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

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

      13. distribute-rgt-in N/A

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

      14. lft-mult-inverse N/A

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

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

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 59.5

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

   8. Simplified 59.5%

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

   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

      \[\leadsto \color{blue}{1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)} \]
   7. 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. accelerator-lowering-fma.f64 N/A

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

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

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 84.2

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

   8. Simplified 84.2%

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

   9. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{720} \cdot {x}^{4}}, 1\right) \]
   10. Step-by-step derivation

       1. metadata-eval N/A

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

       2. pow-sqr N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

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

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

       6. unpow2 N/A

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

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

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

       8. *-commutative N/A

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

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

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

       10. unpow2 N/A

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

       11. *-lowering-*.f64 84.2

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

   11. Simplified 84.2%

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

   12. Taylor expanded in x around 0

       \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{720} \cdot {x}^{4}}, 1\right) \]
   13. Step-by-step derivation

       1. metadata-eval N/A

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

       2. pow-sqr N/A

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

       3. unpow2 N/A

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

       4. associate-*r* N/A

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

       5. unpow2 N/A

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

       6. unpow3 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{1}{720} \cdot \left(\color{blue}{{x}^{3}} \cdot x\right), 1\right) \]

       7. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{\left(\frac{1}{720} \cdot {x}^{3}\right) \cdot x}, 1\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(\frac{1}{720} \cdot {x}^{3}\right)}, 1\right) \]

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

          \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(\frac{1}{720} \cdot {x}^{3}\right)}, 1\right) \]

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

           \[\leadsto \mathsf{fma}\left(x \cdot x, x \cdot \color{blue}{\left(\frac{1}{720} \cdot {x}^{3}\right)}, 1\right) \]

       11. cube-mult N/A

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

       12. unpow2 N/A

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

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

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

       14. unpow2 N/A

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

       15. *-lowering-*.f64 84.2

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

   14. Simplified 84.2%

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

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq 2:\\ \;\;\;\;\mathsf{fma}\left(0.5, x \cdot x, 1\right) \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, x \cdot \left(0.001388888888888889 \cdot \left(x \cdot \left(x \cdot x\right)\right)\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 65.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -1 \cdot 10^{-147}:\\ \;\;\;\;\mathsf{fma}\left(0.5, x \cdot x, 1\right) \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cosh x) (/ (sin y) y)) -1e-147)
   (* (fma 0.5 (* x x) 1.0) (fma -0.16666666666666666 (* y y) 1.0))
   (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-147) {
		tmp = fma(0.5, (x * x), 1.0) * fma(-0.16666666666666666, (y * y), 1.0);
	} 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-147)
		tmp = Float64(fma(0.5, Float64(x * x), 1.0) * fma(-0.16666666666666666, Float64(y * y), 1.0));
	else
		tmp = fma(x, Float64(x * fma(x, Float64(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-147], N[(N[(0.5 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(x * N[(x * N[(x * N[(x * 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < -9.9999999999999997e-148

   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 62.1

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

   5. Simplified 62.1%

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

   6. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

      2. *-lft-identity N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-in N/A

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

      5. lft-mult-inverse N/A

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

      6. distribute-rgt-in N/A

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

      7. +-commutative N/A

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

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

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

      9. +-commutative N/A

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

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

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

      11. unpow2 N/A

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

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

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

      13. distribute-rgt-in N/A

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

      14. lft-mult-inverse N/A

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

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

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 54.2

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

   8. Simplified 54.2%

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

   if -9.9999999999999997e-148 < (*.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 77.4%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)} \]
   7. 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. accelerator-lowering-fma.f64 N/A

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

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

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 70.9

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

   8. Simplified 70.9%

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

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)} \]
   10. 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. unpow2 N/A

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

       3. associate-*l* N/A

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

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

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

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

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

       6. +-commutative N/A

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

       7. *-commutative N/A

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

       8. unpow2 N/A

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

       9. associate-*l* N/A

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

       10. *-commutative N/A

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

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

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

       12. *-commutative N/A

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

       13. *-lowering-*.f64 69.0

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

   11. Simplified 69.0%

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

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -1 \cdot 10^{-147}:\\ \;\;\;\;\mathsf{fma}\left(0.5, x \cdot x, 1\right) \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 61.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -1 \cdot 10^{-147}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cosh x) (/ (sin y) y)) -1e-147)
   (* -0.16666666666666666 (* y y))
   (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-147) {
		tmp = -0.16666666666666666 * (y * y);
	} 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-147)
		tmp = Float64(-0.16666666666666666 * Float64(y * y));
	else
		tmp = fma(x, Float64(x * fma(x, Float64(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-147], N[(-0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision], N[(x * N[(x * N[(x * N[(x * 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < -9.9999999999999997e-148

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

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

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

      2. sin-lowering-sin.f64 40.8

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

   5. Simplified 40.8%

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

   6. Taylor expanded in y 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. unpow2 N/A

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

      4. associate-*l* N/A

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

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

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

      6. *-lowering-*.f64 36.4

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

   8. Simplified 36.4%

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

   9. Taylor expanded in y around inf

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

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

          \[\leadsto \color{blue}{\frac{-1}{6} \cdot {y}^{2}} \]

       2. unpow2 N/A

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

       3. *-lowering-*.f64 36.4

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

   11. Simplified 36.4%

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

   if -9.9999999999999997e-148 < (*.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 77.4%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)} \]
   7. 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. accelerator-lowering-fma.f64 N/A

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

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

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 70.9

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

   8. Simplified 70.9%

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

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)} \]
   10. 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. unpow2 N/A

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

       3. associate-*l* N/A

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

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

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

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

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

       6. +-commutative N/A

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

       7. *-commutative N/A

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

       8. unpow2 N/A

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

       9. associate-*l* N/A

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

       10. *-commutative N/A

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

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

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

       12. *-commutative N/A

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

       13. *-lowering-*.f64 69.0

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

   11. Simplified 69.0%

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

Alternative 14: 52.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -1 \cdot 10^{-147}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, x \cdot x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cosh x) (/ (sin y) y)) -1e-147)
   (* -0.16666666666666666 (* y y))
   (fma 0.5 (* x x) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((cosh(x) * (sin(y) / y)) <= -1e-147) {
		tmp = -0.16666666666666666 * (y * y);
	} 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-147)
		tmp = Float64(-0.16666666666666666 * Float64(y * y));
	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-147], N[(-0.16666666666666666 * N[(y * y), $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)) < -9.9999999999999997e-148

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

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

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

      2. sin-lowering-sin.f64 40.8

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

   5. Simplified 40.8%

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

   6. Taylor expanded in y 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. unpow2 N/A

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

      4. associate-*l* N/A

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

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

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

      6. *-lowering-*.f64 36.4

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

   8. Simplified 36.4%

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

   9. Taylor expanded in y around inf

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

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

          \[\leadsto \color{blue}{\frac{-1}{6} \cdot {y}^{2}} \]

       2. unpow2 N/A

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

       3. *-lowering-*.f64 36.4

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

   11. Simplified 36.4%

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

   if -9.9999999999999997e-148 < (*.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 77.4%

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

   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 59.0

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

   8. Simplified 59.0%

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

Alternative 15: 33.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -1 \cdot 10^{-147}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cosh x) (/ (sin y) y)) -1e-147)
   (* -0.16666666666666666 (* y y))
   1.0))

C

double code(double x, double y) {
	double tmp;
	if ((cosh(x) * (sin(y) / y)) <= -1e-147) {
		tmp = -0.16666666666666666 * (y * y);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((cosh(x) * (sin(y) / y)) <= (-1d-147)) then
        tmp = (-0.16666666666666666d0) * (y * y)
    else
        tmp = 1.0d0
    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)) <= -1e-147) {
		tmp = -0.16666666666666666 * (y * y);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (math.cosh(x) * (math.sin(y) / y)) <= -1e-147:
		tmp = -0.16666666666666666 * (y * y)
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(cosh(x) * Float64(sin(y) / y)) <= -1e-147)
		tmp = Float64(-0.16666666666666666 * Float64(y * y));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((cosh(x) * (sin(y) / y)) <= -1e-147)
		tmp = -0.16666666666666666 * (y * y);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Cosh[x], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], -1e-147], N[(-0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < -9.9999999999999997e-148

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

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

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

      2. sin-lowering-sin.f64 40.8

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

   5. Simplified 40.8%

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

   6. Taylor expanded in y 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. unpow2 N/A

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

      4. associate-*l* N/A

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

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

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

      6. *-lowering-*.f64 36.4

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

   8. Simplified 36.4%

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

   9. Taylor expanded in y around inf

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

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

          \[\leadsto \color{blue}{\frac{-1}{6} \cdot {y}^{2}} \]

       2. unpow2 N/A

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

       3. *-lowering-*.f64 36.4

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

   11. Simplified 36.4%

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

   if -9.9999999999999997e-148 < (*.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 77.4%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \]
   7. Step-by-step derivation

   8. Simplified 37.7%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 26.6% 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 63.0%

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

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1} \]
7. Step-by-step derivation

8. Simplified 30.8%

   \[\leadsto \color{blue}{1} \]
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 2024198 
(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)))