Linear.Quaternion:$csinh from linear-1.19.1.3

Percentage Accurate: 99.9% → 99.9%

Time: 8.3s

Alternatives: 16

Speedup: 1.0×

• 48.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Final simplification 99.9%

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

Alternative 2: 99.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ t_1 := t\_0 \cdot \cosh x\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\left(\left(y \cdot y\right) \cdot -0.16666666666666666\right) \cdot \cosh x\\ \mathbf{elif}\;t\_1 \leq 0.9546743324871534:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, 0.5, 1\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \cosh x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)) (t_1 (* t_0 (cosh x))))
   (if (<= t_1 (- INFINITY))
     (* (* (* y y) -0.16666666666666666) (cosh x))
     (if (<= t_1 0.9546743324871534)
       (* (fma (* x x) 0.5 1.0) t_0)
       (* 1.0 (cosh x))))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double t_1 = t_0 * cosh(x);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = ((y * y) * -0.16666666666666666) * cosh(x);
	} else if (t_1 <= 0.9546743324871534) {
		tmp = fma((x * x), 0.5, 1.0) * t_0;
	} else {
		tmp = 1.0 * cosh(x);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(y) / y)
	t_1 = Float64(t_0 * cosh(x))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(y * y) * -0.16666666666666666) * cosh(x));
	elseif (t_1 <= 0.9546743324871534)
		tmp = Float64(fma(Float64(x * x), 0.5, 1.0) * t_0);
	else
		tmp = Float64(1.0 * 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[(t$95$0 * N[Cosh[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(y * y), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * N[Cosh[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.9546743324871534], N[(N[(N[(x * x), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision], N[(1.0 * N[Cosh[x], $MachinePrecision]), $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. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. neg-sub0 N/A

         \[\leadsto \cosh x \cdot \frac{\color{blue}{0 - \sin y}}{\mathsf{neg}\left(y\right)} \]

      4. div-sub N/A

         \[\leadsto \cosh x \cdot \color{blue}{\left(\frac{0}{\mathsf{neg}\left(y\right)} - \frac{\sin y}{\mathsf{neg}\left(y\right)}\right)} \]

      5. clear-num N/A

         \[\leadsto \cosh x \cdot \left(\frac{0}{\mathsf{neg}\left(y\right)} - \color{blue}{\frac{1}{\frac{\mathsf{neg}\left(y\right)}{\sin y}}}\right) \]

      6. distribute-neg-frac N/A

         \[\leadsto \cosh x \cdot \left(\frac{0}{\mathsf{neg}\left(y\right)} - \frac{1}{\color{blue}{\mathsf{neg}\left(\frac{y}{\sin y}\right)}}\right) \]

      7. frac-sub N/A

         \[\leadsto \cosh x \cdot \color{blue}{\frac{0 \cdot \left(\mathsf{neg}\left(\frac{y}{\sin y}\right)\right) - \left(\mathsf{neg}\left(y\right)\right) \cdot 1}{\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(\frac{y}{\sin y}\right)\right)}} \]

      8. distribute-lft-neg-in N/A

         \[\leadsto \cosh x \cdot \frac{0 \cdot \left(\mathsf{neg}\left(\frac{y}{\sin y}\right)\right) - \color{blue}{\left(\mathsf{neg}\left(y \cdot 1\right)\right)}}{\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(\frac{y}{\sin y}\right)\right)} \]

      9. distribute-rgt-neg-in N/A

         \[\leadsto \cosh x \cdot \frac{0 \cdot \left(\mathsf{neg}\left(\frac{y}{\sin y}\right)\right) - \color{blue}{y \cdot \left(\mathsf{neg}\left(1\right)\right)}}{\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(\frac{y}{\sin y}\right)\right)} \]

      10. metadata-eval N/A

          \[\leadsto \cosh x \cdot \frac{0 \cdot \left(\mathsf{neg}\left(\frac{y}{\sin y}\right)\right) - y \cdot \color{blue}{-1}}{\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(\frac{y}{\sin y}\right)\right)} \]

      11. lower-/.f64 N/A

          \[\leadsto \cosh x \cdot \color{blue}{\frac{0 \cdot \left(\mathsf{neg}\left(\frac{y}{\sin y}\right)\right) - y \cdot -1}{\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(\frac{y}{\sin y}\right)\right)}} \]

   4. Applied rewrites 41.9%

      \[\leadsto \cosh x \cdot \color{blue}{\frac{0 \cdot \frac{-y}{\sin y} - \left(-y\right)}{\left(-y\right) \cdot \frac{-y}{\sin y}}} \]

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 100.0%

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

   if -inf.0 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < 0.95467433248715339

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 99.6

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

   5. Applied rewrites 99.6%

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

   if 0.95467433248715339 < (*.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. Applied rewrites 100.0%

      \[\leadsto \cosh x \cdot \color{blue}{1} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \cdot \cosh x \leq -\infty:\\ \;\;\;\;\left(\left(y \cdot y\right) \cdot -0.16666666666666666\right) \cdot \cosh x\\ \mathbf{elif}\;\frac{\sin y}{y} \cdot \cosh x \leq 0.9546743324871534:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, 0.5, 1\right) \cdot \frac{\sin y}{y}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \cosh x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ t_1 := t\_0 \cdot \cosh x\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\left(\left(y \cdot y\right) \cdot -0.16666666666666666\right) \cdot \cosh x\\ \mathbf{elif}\;t\_1 \leq 0.9546743324871534:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \cosh x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)) (t_1 (* t_0 (cosh x))))
   (if (<= t_1 (- INFINITY))
     (* (* (* y y) -0.16666666666666666) (cosh x))
     (if (<= t_1 0.9546743324871534) t_0 (* 1.0 (cosh x))))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double t_1 = t_0 * cosh(x);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = ((y * y) * -0.16666666666666666) * cosh(x);
	} else if (t_1 <= 0.9546743324871534) {
		tmp = t_0;
	} else {
		tmp = 1.0 * cosh(x);
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double t_0 = Math.sin(y) / y;
	double t_1 = t_0 * Math.cosh(x);
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = ((y * y) * -0.16666666666666666) * Math.cosh(x);
	} else if (t_1 <= 0.9546743324871534) {
		tmp = t_0;
	} else {
		tmp = 1.0 * Math.cosh(x);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sin(y) / y
	t_1 = t_0 * math.cosh(x)
	tmp = 0
	if t_1 <= -math.inf:
		tmp = ((y * y) * -0.16666666666666666) * math.cosh(x)
	elif t_1 <= 0.9546743324871534:
		tmp = t_0
	else:
		tmp = 1.0 * math.cosh(x)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sin(y) / y)
	t_1 = Float64(t_0 * cosh(x))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(y * y) * -0.16666666666666666) * cosh(x));
	elseif (t_1 <= 0.9546743324871534)
		tmp = t_0;
	else
		tmp = Float64(1.0 * cosh(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sin(y) / y;
	t_1 = t_0 * cosh(x);
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = ((y * y) * -0.16666666666666666) * cosh(x);
	elseif (t_1 <= 0.9546743324871534)
		tmp = t_0;
	else
		tmp = 1.0 * 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[(t$95$0 * N[Cosh[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(y * y), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * N[Cosh[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.9546743324871534], t$95$0, N[(1.0 * N[Cosh[x], $MachinePrecision]), $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. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. neg-sub0 N/A

         \[\leadsto \cosh x \cdot \frac{\color{blue}{0 - \sin y}}{\mathsf{neg}\left(y\right)} \]

      4. div-sub N/A

         \[\leadsto \cosh x \cdot \color{blue}{\left(\frac{0}{\mathsf{neg}\left(y\right)} - \frac{\sin y}{\mathsf{neg}\left(y\right)}\right)} \]

      5. clear-num N/A

         \[\leadsto \cosh x \cdot \left(\frac{0}{\mathsf{neg}\left(y\right)} - \color{blue}{\frac{1}{\frac{\mathsf{neg}\left(y\right)}{\sin y}}}\right) \]

      6. distribute-neg-frac N/A

         \[\leadsto \cosh x \cdot \left(\frac{0}{\mathsf{neg}\left(y\right)} - \frac{1}{\color{blue}{\mathsf{neg}\left(\frac{y}{\sin y}\right)}}\right) \]

      7. frac-sub N/A

         \[\leadsto \cosh x \cdot \color{blue}{\frac{0 \cdot \left(\mathsf{neg}\left(\frac{y}{\sin y}\right)\right) - \left(\mathsf{neg}\left(y\right)\right) \cdot 1}{\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(\frac{y}{\sin y}\right)\right)}} \]

      8. distribute-lft-neg-in N/A

         \[\leadsto \cosh x \cdot \frac{0 \cdot \left(\mathsf{neg}\left(\frac{y}{\sin y}\right)\right) - \color{blue}{\left(\mathsf{neg}\left(y \cdot 1\right)\right)}}{\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(\frac{y}{\sin y}\right)\right)} \]

      9. distribute-rgt-neg-in N/A

         \[\leadsto \cosh x \cdot \frac{0 \cdot \left(\mathsf{neg}\left(\frac{y}{\sin y}\right)\right) - \color{blue}{y \cdot \left(\mathsf{neg}\left(1\right)\right)}}{\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(\frac{y}{\sin y}\right)\right)} \]

      10. metadata-eval N/A

          \[\leadsto \cosh x \cdot \frac{0 \cdot \left(\mathsf{neg}\left(\frac{y}{\sin y}\right)\right) - y \cdot \color{blue}{-1}}{\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(\frac{y}{\sin y}\right)\right)} \]

      11. lower-/.f64 N/A

          \[\leadsto \cosh x \cdot \color{blue}{\frac{0 \cdot \left(\mathsf{neg}\left(\frac{y}{\sin y}\right)\right) - y \cdot -1}{\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(\frac{y}{\sin y}\right)\right)}} \]

   4. Applied rewrites 41.9%

      \[\leadsto \cosh x \cdot \color{blue}{\frac{0 \cdot \frac{-y}{\sin y} - \left(-y\right)}{\left(-y\right) \cdot \frac{-y}{\sin y}}} \]

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 100.0%

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

   if -inf.0 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < 0.95467433248715339

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 98.9

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

   5. Applied rewrites 98.9%

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

   if 0.95467433248715339 < (*.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. Applied rewrites 100.0%

      \[\leadsto \cosh x \cdot \color{blue}{1} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \cdot \cosh x \leq -\infty:\\ \;\;\;\;\left(\left(y \cdot y\right) \cdot -0.16666666666666666\right) \cdot \cosh x\\ \mathbf{elif}\;\frac{\sin y}{y} \cdot \cosh x \leq 0.9546743324871534:\\ \;\;\;\;\frac{\sin y}{y}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \cosh x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ t_1 := t\_0 \cdot \cosh x\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right)\\ \mathbf{elif}\;t\_1 \leq 0.9546743324871534:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \cosh x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)) (t_1 (* t_0 (cosh x))))
   (if (<= t_1 (- INFINITY))
     (*
      (fma -0.16666666666666666 (* y y) 1.0)
      (fma (fma 0.041666666666666664 (* x x) 0.5) (* x x) 1.0))
     (if (<= t_1 0.9546743324871534) t_0 (* 1.0 (cosh x))))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double t_1 = t_0 * cosh(x);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma(-0.16666666666666666, (y * y), 1.0) * fma(fma(0.041666666666666664, (x * x), 0.5), (x * x), 1.0);
	} else if (t_1 <= 0.9546743324871534) {
		tmp = t_0;
	} else {
		tmp = 1.0 * cosh(x);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(y) / y)
	t_1 = Float64(t_0 * cosh(x))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(fma(-0.16666666666666666, Float64(y * y), 1.0) * fma(fma(0.041666666666666664, Float64(x * x), 0.5), Float64(x * x), 1.0));
	elseif (t_1 <= 0.9546743324871534)
		tmp = t_0;
	else
		tmp = Float64(1.0 * 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[(t$95$0 * N[Cosh[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(0.041666666666666664 * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.9546743324871534], t$95$0, N[(1.0 * N[Cosh[x], $MachinePrecision]), $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. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. neg-sub0 N/A

         \[\leadsto \cosh x \cdot \frac{\color{blue}{0 - \sin y}}{\mathsf{neg}\left(y\right)} \]

      4. div-sub N/A

         \[\leadsto \cosh x \cdot \color{blue}{\left(\frac{0}{\mathsf{neg}\left(y\right)} - \frac{\sin y}{\mathsf{neg}\left(y\right)}\right)} \]

      5. clear-num N/A

         \[\leadsto \cosh x \cdot \left(\frac{0}{\mathsf{neg}\left(y\right)} - \color{blue}{\frac{1}{\frac{\mathsf{neg}\left(y\right)}{\sin y}}}\right) \]

      6. distribute-neg-frac N/A

         \[\leadsto \cosh x \cdot \left(\frac{0}{\mathsf{neg}\left(y\right)} - \frac{1}{\color{blue}{\mathsf{neg}\left(\frac{y}{\sin y}\right)}}\right) \]

      7. frac-sub N/A

         \[\leadsto \cosh x \cdot \color{blue}{\frac{0 \cdot \left(\mathsf{neg}\left(\frac{y}{\sin y}\right)\right) - \left(\mathsf{neg}\left(y\right)\right) \cdot 1}{\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(\frac{y}{\sin y}\right)\right)}} \]

      8. distribute-lft-neg-in N/A

         \[\leadsto \cosh x \cdot \frac{0 \cdot \left(\mathsf{neg}\left(\frac{y}{\sin y}\right)\right) - \color{blue}{\left(\mathsf{neg}\left(y \cdot 1\right)\right)}}{\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(\frac{y}{\sin y}\right)\right)} \]

      9. distribute-rgt-neg-in N/A

         \[\leadsto \cosh x \cdot \frac{0 \cdot \left(\mathsf{neg}\left(\frac{y}{\sin y}\right)\right) - \color{blue}{y \cdot \left(\mathsf{neg}\left(1\right)\right)}}{\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(\frac{y}{\sin y}\right)\right)} \]

      10. metadata-eval N/A

          \[\leadsto \cosh x \cdot \frac{0 \cdot \left(\mathsf{neg}\left(\frac{y}{\sin y}\right)\right) - y \cdot \color{blue}{-1}}{\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(\frac{y}{\sin y}\right)\right)} \]

      11. lower-/.f64 N/A

          \[\leadsto \cosh x \cdot \color{blue}{\frac{0 \cdot \left(\mathsf{neg}\left(\frac{y}{\sin y}\right)\right) - y \cdot -1}{\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(\frac{y}{\sin y}\right)\right)}} \]

   4. Applied rewrites 41.9%

      \[\leadsto \cosh x \cdot \color{blue}{\frac{0 \cdot \frac{-y}{\sin y} - \left(-y\right)}{\left(-y\right) \cdot \frac{-y}{\sin y}}} \]

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 96.9

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

   10. Applied rewrites 96.9%

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

   if -inf.0 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < 0.95467433248715339

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 98.9

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

   5. Applied rewrites 98.9%

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

   if 0.95467433248715339 < (*.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. Applied rewrites 100.0%

      \[\leadsto \cosh x \cdot \color{blue}{1} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \cdot \cosh x \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right)\\ \mathbf{elif}\;\frac{\sin y}{y} \cdot \cosh x \leq 0.9546743324871534:\\ \;\;\;\;\frac{\sin y}{y}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \cosh x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 52.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y} \cdot \cosh x\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{-153}:\\ \;\;\;\;1 \cdot \left(\left(y \cdot y\right) \cdot -0.16666666666666666\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;1 \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \left(x \cdot x\right)\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (/ (sin y) y) (cosh x))))
   (if (<= t_0 -4e-153)
     (* 1.0 (* (* y y) -0.16666666666666666))
     (if (<= t_0 2.0) (* 1.0 1.0) (* (* 0.5 (* x x)) 1.0)))))

C

double code(double x, double y) {
	double t_0 = (sin(y) / y) * cosh(x);
	double tmp;
	if (t_0 <= -4e-153) {
		tmp = 1.0 * ((y * y) * -0.16666666666666666);
	} else if (t_0 <= 2.0) {
		tmp = 1.0 * 1.0;
	} else {
		tmp = (0.5 * (x * x)) * 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (sin(y) / y) * cosh(x)
    if (t_0 <= (-4d-153)) then
        tmp = 1.0d0 * ((y * y) * (-0.16666666666666666d0))
    else if (t_0 <= 2.0d0) then
        tmp = 1.0d0 * 1.0d0
    else
        tmp = (0.5d0 * (x * x)) * 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (Math.sin(y) / y) * Math.cosh(x);
	double tmp;
	if (t_0 <= -4e-153) {
		tmp = 1.0 * ((y * y) * -0.16666666666666666);
	} else if (t_0 <= 2.0) {
		tmp = 1.0 * 1.0;
	} else {
		tmp = (0.5 * (x * x)) * 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (math.sin(y) / y) * math.cosh(x)
	tmp = 0
	if t_0 <= -4e-153:
		tmp = 1.0 * ((y * y) * -0.16666666666666666)
	elif t_0 <= 2.0:
		tmp = 1.0 * 1.0
	else:
		tmp = (0.5 * (x * x)) * 1.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(sin(y) / y) * cosh(x))
	tmp = 0.0
	if (t_0 <= -4e-153)
		tmp = Float64(1.0 * Float64(Float64(y * y) * -0.16666666666666666));
	elseif (t_0 <= 2.0)
		tmp = Float64(1.0 * 1.0);
	else
		tmp = Float64(Float64(0.5 * Float64(x * x)) * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (sin(y) / y) * cosh(x);
	tmp = 0.0;
	if (t_0 <= -4e-153)
		tmp = 1.0 * ((y * y) * -0.16666666666666666);
	elseif (t_0 <= 2.0)
		tmp = 1.0 * 1.0;
	else
		tmp = (0.5 * (x * x)) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision] * N[Cosh[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -4e-153], N[(1.0 * N[(N[(y * y), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(1.0 * 1.0), $MachinePrecision], N[(N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < -4.00000000000000016e-153

   1. Initial program 99.9%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. neg-sub0 N/A

         \[\leadsto \cosh x \cdot \frac{\color{blue}{0 - \sin y}}{\mathsf{neg}\left(y\right)} \]

      4. div-sub N/A

         \[\leadsto \cosh x \cdot \color{blue}{\left(\frac{0}{\mathsf{neg}\left(y\right)} - \frac{\sin y}{\mathsf{neg}\left(y\right)}\right)} \]

      5. clear-num N/A

         \[\leadsto \cosh x \cdot \left(\frac{0}{\mathsf{neg}\left(y\right)} - \color{blue}{\frac{1}{\frac{\mathsf{neg}\left(y\right)}{\sin y}}}\right) \]

      6. distribute-neg-frac N/A

         \[\leadsto \cosh x \cdot \left(\frac{0}{\mathsf{neg}\left(y\right)} - \frac{1}{\color{blue}{\mathsf{neg}\left(\frac{y}{\sin y}\right)}}\right) \]

      7. frac-sub N/A

         \[\leadsto \cosh x \cdot \color{blue}{\frac{0 \cdot \left(\mathsf{neg}\left(\frac{y}{\sin y}\right)\right) - \left(\mathsf{neg}\left(y\right)\right) \cdot 1}{\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(\frac{y}{\sin y}\right)\right)}} \]

      8. distribute-lft-neg-in N/A

         \[\leadsto \cosh x \cdot \frac{0 \cdot \left(\mathsf{neg}\left(\frac{y}{\sin y}\right)\right) - \color{blue}{\left(\mathsf{neg}\left(y \cdot 1\right)\right)}}{\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(\frac{y}{\sin y}\right)\right)} \]

      9. distribute-rgt-neg-in N/A

         \[\leadsto \cosh x \cdot \frac{0 \cdot \left(\mathsf{neg}\left(\frac{y}{\sin y}\right)\right) - \color{blue}{y \cdot \left(\mathsf{neg}\left(1\right)\right)}}{\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(\frac{y}{\sin y}\right)\right)} \]

      10. metadata-eval N/A

          \[\leadsto \cosh x \cdot \frac{0 \cdot \left(\mathsf{neg}\left(\frac{y}{\sin y}\right)\right) - y \cdot \color{blue}{-1}}{\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(\frac{y}{\sin y}\right)\right)} \]

      11. lower-/.f64 N/A

          \[\leadsto \cosh x \cdot \color{blue}{\frac{0 \cdot \left(\mathsf{neg}\left(\frac{y}{\sin y}\right)\right) - y \cdot -1}{\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(\frac{y}{\sin y}\right)\right)}} \]

   4. Applied rewrites 60.6%

      \[\leadsto \cosh x \cdot \color{blue}{\frac{0 \cdot \frac{-y}{\sin y} - \left(-y\right)}{\left(-y\right) \cdot \frac{-y}{\sin y}}} \]

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 69.1

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

   7. Applied rewrites 69.1%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 42.3%

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

   11. Taylor expanded in y around inf

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

   13. Applied rewrites 42.3%

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

   if -4.00000000000000016e-153 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < 2

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 55.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 54.6%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 70.1

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

   5. Applied rewrites 70.1%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 81.7

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

   8. Applied rewrites 81.7%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 81.7%

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

   12. Taylor expanded in y around 0

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

   14. Applied rewrites 70.3%

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

4. Final simplification 57.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \cdot \cosh x \leq -4 \cdot 10^{-153}:\\ \;\;\;\;1 \cdot \left(\left(y \cdot y\right) \cdot -0.16666666666666666\right)\\ \mathbf{elif}\;\frac{\sin y}{y} \cdot \cosh x \leq 2:\\ \;\;\;\;1 \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \left(x \cdot x\right)\right) \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \cdot \cosh x \leq -4 \cdot 10^{-153}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \cosh x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (/ (sin y) y) (cosh x)) -4e-153)
   (*
    (fma -0.16666666666666666 (* y y) 1.0)
    (fma (fma 0.041666666666666664 (* x x) 0.5) (* x x) 1.0))
   (* 1.0 (cosh x))))

C

double code(double x, double y) {
	double tmp;
	if (((sin(y) / y) * cosh(x)) <= -4e-153) {
		tmp = fma(-0.16666666666666666, (y * y), 1.0) * fma(fma(0.041666666666666664, (x * x), 0.5), (x * x), 1.0);
	} else {
		tmp = 1.0 * cosh(x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(sin(y) / y) * cosh(x)) <= -4e-153)
		tmp = Float64(fma(-0.16666666666666666, Float64(y * y), 1.0) * fma(fma(0.041666666666666664, Float64(x * x), 0.5), Float64(x * x), 1.0));
	else
		tmp = Float64(1.0 * cosh(x));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision] * N[Cosh[x], $MachinePrecision]), $MachinePrecision], -4e-153], N[(N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(0.041666666666666664 * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 * N[Cosh[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < -4.00000000000000016e-153

   1. Initial program 99.9%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. neg-sub0 N/A

         \[\leadsto \cosh x \cdot \frac{\color{blue}{0 - \sin y}}{\mathsf{neg}\left(y\right)} \]

      4. div-sub N/A

         \[\leadsto \cosh x \cdot \color{blue}{\left(\frac{0}{\mathsf{neg}\left(y\right)} - \frac{\sin y}{\mathsf{neg}\left(y\right)}\right)} \]

      5. clear-num N/A

         \[\leadsto \cosh x \cdot \left(\frac{0}{\mathsf{neg}\left(y\right)} - \color{blue}{\frac{1}{\frac{\mathsf{neg}\left(y\right)}{\sin y}}}\right) \]

      6. distribute-neg-frac N/A

         \[\leadsto \cosh x \cdot \left(\frac{0}{\mathsf{neg}\left(y\right)} - \frac{1}{\color{blue}{\mathsf{neg}\left(\frac{y}{\sin y}\right)}}\right) \]

      7. frac-sub N/A

         \[\leadsto \cosh x \cdot \color{blue}{\frac{0 \cdot \left(\mathsf{neg}\left(\frac{y}{\sin y}\right)\right) - \left(\mathsf{neg}\left(y\right)\right) \cdot 1}{\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(\frac{y}{\sin y}\right)\right)}} \]

      8. distribute-lft-neg-in N/A

         \[\leadsto \cosh x \cdot \frac{0 \cdot \left(\mathsf{neg}\left(\frac{y}{\sin y}\right)\right) - \color{blue}{\left(\mathsf{neg}\left(y \cdot 1\right)\right)}}{\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(\frac{y}{\sin y}\right)\right)} \]

      9. distribute-rgt-neg-in N/A

         \[\leadsto \cosh x \cdot \frac{0 \cdot \left(\mathsf{neg}\left(\frac{y}{\sin y}\right)\right) - \color{blue}{y \cdot \left(\mathsf{neg}\left(1\right)\right)}}{\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(\frac{y}{\sin y}\right)\right)} \]

      10. metadata-eval N/A

          \[\leadsto \cosh x \cdot \frac{0 \cdot \left(\mathsf{neg}\left(\frac{y}{\sin y}\right)\right) - y \cdot \color{blue}{-1}}{\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(\frac{y}{\sin y}\right)\right)} \]

      11. lower-/.f64 N/A

          \[\leadsto \cosh x \cdot \color{blue}{\frac{0 \cdot \left(\mathsf{neg}\left(\frac{y}{\sin y}\right)\right) - y \cdot -1}{\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(\frac{y}{\sin y}\right)\right)}} \]

   4. Applied rewrites 60.6%

      \[\leadsto \cosh x \cdot \color{blue}{\frac{0 \cdot \frac{-y}{\sin y} - \left(-y\right)}{\left(-y\right) \cdot \frac{-y}{\sin y}}} \]

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 69.1

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

   7. Applied rewrites 69.1%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 67.0

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

   10. Applied rewrites 67.0%

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

   if -4.00000000000000016e-153 < (*.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. Applied rewrites 72.9%

      \[\leadsto \cosh x \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \cdot \cosh x \leq -4 \cdot 10^{-153}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \cosh x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 61.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ t_1 := \mathsf{fma}\left(x \cdot x, 0.5, 1\right)\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-306}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right) \cdot t\_1\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-78}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), y \cdot y, 1\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;1 \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)) (t_1 (fma (* x x) 0.5 1.0)))
   (if (<= t_0 -1e-306)
     (* (fma -0.16666666666666666 (* y y) 1.0) t_1)
     (if (<= t_0 5e-78)
       (*
        (fma
         (fma 0.008333333333333333 (* y y) -0.16666666666666666)
         (* y y)
         1.0)
        1.0)
       (* 1.0 t_1)))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double t_1 = fma((x * x), 0.5, 1.0);
	double tmp;
	if (t_0 <= -1e-306) {
		tmp = fma(-0.16666666666666666, (y * y), 1.0) * t_1;
	} else if (t_0 <= 5e-78) {
		tmp = fma(fma(0.008333333333333333, (y * y), -0.16666666666666666), (y * y), 1.0) * 1.0;
	} else {
		tmp = 1.0 * t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(y) / y)
	t_1 = fma(Float64(x * x), 0.5, 1.0)
	tmp = 0.0
	if (t_0 <= -1e-306)
		tmp = Float64(fma(-0.16666666666666666, Float64(y * y), 1.0) * t_1);
	elseif (t_0 <= 5e-78)
		tmp = Float64(fma(fma(0.008333333333333333, Float64(y * y), -0.16666666666666666), Float64(y * y), 1.0) * 1.0);
	else
		tmp = Float64(1.0 * t_1);
	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[(x * x), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-306], N[(N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t$95$0, 5e-78], N[(N[(N[(0.008333333333333333 * N[(y * y), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * 1.0), $MachinePrecision], N[(1.0 * t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 y) y) < -1.00000000000000003e-306

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 72.7

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

   5. Applied rewrites 72.7%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 49.4

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

   8. Applied rewrites 49.4%

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

   if -1.00000000000000003e-306 < (/.f64 (sin.f64 y) y) < 4.9999999999999996e-78

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 84.0

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

   5. Applied rewrites 84.0%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 43.1

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

   8. Applied rewrites 43.1%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 41.5%

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

   if 4.9999999999999996e-78 < (/.f64 (sin.f64 y) y)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 87.9

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

   5. Applied rewrites 87.9%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 79.6%

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

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq -1 \cdot 10^{-306}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right) \cdot \mathsf{fma}\left(x \cdot x, 0.5, 1\right)\\ \mathbf{elif}\;\frac{\sin y}{y} \leq 5 \cdot 10^{-78}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), y \cdot y, 1\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(x \cdot x, 0.5, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 61.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-306}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right) \cdot \left(0.5 \cdot \left(x \cdot x\right)\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-78}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), y \cdot y, 1\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(x \cdot x, 0.5, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)))
   (if (<= t_0 -1e-306)
     (* (fma -0.16666666666666666 (* y y) 1.0) (* 0.5 (* x x)))
     (if (<= t_0 5e-78)
       (*
        (fma
         (fma 0.008333333333333333 (* y y) -0.16666666666666666)
         (* y y)
         1.0)
        1.0)
       (* 1.0 (fma (* x x) 0.5 1.0))))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double tmp;
	if (t_0 <= -1e-306) {
		tmp = fma(-0.16666666666666666, (y * y), 1.0) * (0.5 * (x * x));
	} else if (t_0 <= 5e-78) {
		tmp = fma(fma(0.008333333333333333, (y * y), -0.16666666666666666), (y * y), 1.0) * 1.0;
	} else {
		tmp = 1.0 * fma((x * x), 0.5, 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(y) / y)
	tmp = 0.0
	if (t_0 <= -1e-306)
		tmp = Float64(fma(-0.16666666666666666, Float64(y * y), 1.0) * Float64(0.5 * Float64(x * x)));
	elseif (t_0 <= 5e-78)
		tmp = Float64(fma(fma(0.008333333333333333, Float64(y * y), -0.16666666666666666), Float64(y * y), 1.0) * 1.0);
	else
		tmp = Float64(1.0 * fma(Float64(x * x), 0.5, 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-306], N[(N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e-78], N[(N[(N[(0.008333333333333333 * N[(y * y), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * 1.0), $MachinePrecision], N[(1.0 * N[(N[(x * x), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 y) y) < -1.00000000000000003e-306

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 72.7

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

   5. Applied rewrites 72.7%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 0.6

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

   8. Applied rewrites 0.6%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 0.5%

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

   12. Taylor expanded in y around 0

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

   14. Applied rewrites 49.1%

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

   if -1.00000000000000003e-306 < (/.f64 (sin.f64 y) y) < 4.9999999999999996e-78

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 84.0

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

   5. Applied rewrites 84.0%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 43.1

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

   8. Applied rewrites 43.1%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 41.5%

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

   if 4.9999999999999996e-78 < (/.f64 (sin.f64 y) y)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 87.9

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

   5. Applied rewrites 87.9%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 79.6%

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

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq -1 \cdot 10^{-306}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right) \cdot \left(0.5 \cdot \left(x \cdot x\right)\right)\\ \mathbf{elif}\;\frac{\sin y}{y} \leq 5 \cdot 10^{-78}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), y \cdot y, 1\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(x \cdot x, 0.5, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 69.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \cdot \cosh x \leq -4 \cdot 10^{-153}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (/ (sin y) y) (cosh x)) -4e-153)
   (*
    (fma -0.16666666666666666 (* y y) 1.0)
    (fma (fma 0.041666666666666664 (* x x) 0.5) (* x x) 1.0))
   (*
    (fma
     (fma (fma 0.001388888888888889 (* x x) 0.041666666666666664) (* x x) 0.5)
     (* x x)
     1.0)
    1.0)))

C

double code(double x, double y) {
	double tmp;
	if (((sin(y) / y) * cosh(x)) <= -4e-153) {
		tmp = fma(-0.16666666666666666, (y * y), 1.0) * fma(fma(0.041666666666666664, (x * x), 0.5), (x * x), 1.0);
	} else {
		tmp = fma(fma(fma(0.001388888888888889, (x * x), 0.041666666666666664), (x * x), 0.5), (x * x), 1.0) * 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(sin(y) / y) * cosh(x)) <= -4e-153)
		tmp = Float64(fma(-0.16666666666666666, Float64(y * y), 1.0) * fma(fma(0.041666666666666664, Float64(x * x), 0.5), Float64(x * x), 1.0));
	else
		tmp = Float64(fma(fma(fma(0.001388888888888889, Float64(x * x), 0.041666666666666664), Float64(x * x), 0.5), Float64(x * x), 1.0) * 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision] * N[Cosh[x], $MachinePrecision]), $MachinePrecision], -4e-153], N[(N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(0.041666666666666664 * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.001388888888888889 * N[(x * x), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < -4.00000000000000016e-153

   1. Initial program 99.9%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. neg-sub0 N/A

         \[\leadsto \cosh x \cdot \frac{\color{blue}{0 - \sin y}}{\mathsf{neg}\left(y\right)} \]

      4. div-sub N/A

         \[\leadsto \cosh x \cdot \color{blue}{\left(\frac{0}{\mathsf{neg}\left(y\right)} - \frac{\sin y}{\mathsf{neg}\left(y\right)}\right)} \]

      5. clear-num N/A

         \[\leadsto \cosh x \cdot \left(\frac{0}{\mathsf{neg}\left(y\right)} - \color{blue}{\frac{1}{\frac{\mathsf{neg}\left(y\right)}{\sin y}}}\right) \]

      6. distribute-neg-frac N/A

         \[\leadsto \cosh x \cdot \left(\frac{0}{\mathsf{neg}\left(y\right)} - \frac{1}{\color{blue}{\mathsf{neg}\left(\frac{y}{\sin y}\right)}}\right) \]

      7. frac-sub N/A

         \[\leadsto \cosh x \cdot \color{blue}{\frac{0 \cdot \left(\mathsf{neg}\left(\frac{y}{\sin y}\right)\right) - \left(\mathsf{neg}\left(y\right)\right) \cdot 1}{\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(\frac{y}{\sin y}\right)\right)}} \]

      8. distribute-lft-neg-in N/A

         \[\leadsto \cosh x \cdot \frac{0 \cdot \left(\mathsf{neg}\left(\frac{y}{\sin y}\right)\right) - \color{blue}{\left(\mathsf{neg}\left(y \cdot 1\right)\right)}}{\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(\frac{y}{\sin y}\right)\right)} \]

      9. distribute-rgt-neg-in N/A

         \[\leadsto \cosh x \cdot \frac{0 \cdot \left(\mathsf{neg}\left(\frac{y}{\sin y}\right)\right) - \color{blue}{y \cdot \left(\mathsf{neg}\left(1\right)\right)}}{\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(\frac{y}{\sin y}\right)\right)} \]

      10. metadata-eval N/A

          \[\leadsto \cosh x \cdot \frac{0 \cdot \left(\mathsf{neg}\left(\frac{y}{\sin y}\right)\right) - y \cdot \color{blue}{-1}}{\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(\frac{y}{\sin y}\right)\right)} \]

      11. lower-/.f64 N/A

          \[\leadsto \cosh x \cdot \color{blue}{\frac{0 \cdot \left(\mathsf{neg}\left(\frac{y}{\sin y}\right)\right) - y \cdot -1}{\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(\frac{y}{\sin y}\right)\right)}} \]

   4. Applied rewrites 60.6%

      \[\leadsto \cosh x \cdot \color{blue}{\frac{0 \cdot \frac{-y}{\sin y} - \left(-y\right)}{\left(-y\right) \cdot \frac{-y}{\sin y}}} \]

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 69.1

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

   7. Applied rewrites 69.1%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 67.0

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

   10. Applied rewrites 67.0%

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

   if -4.00000000000000016e-153 < (*.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. Applied rewrites 72.9%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 70.4

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

   8. Applied rewrites 70.4%

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

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \cdot \cosh x \leq -4 \cdot 10^{-153}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 66.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right)\\ \mathbf{if}\;\frac{\sin y}{y} \cdot \cosh x \leq -4 \cdot 10^{-153}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma (fma 0.041666666666666664 (* x x) 0.5) (* x x) 1.0)))
   (if (<= (* (/ (sin y) y) (cosh x)) -4e-153)
     (* (fma -0.16666666666666666 (* y y) 1.0) t_0)
     (* t_0 1.0))))

C

double code(double x, double y) {
	double t_0 = fma(fma(0.041666666666666664, (x * x), 0.5), (x * x), 1.0);
	double tmp;
	if (((sin(y) / y) * cosh(x)) <= -4e-153) {
		tmp = fma(-0.16666666666666666, (y * y), 1.0) * t_0;
	} else {
		tmp = t_0 * 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(fma(0.041666666666666664, Float64(x * x), 0.5), Float64(x * x), 1.0)
	tmp = 0.0
	if (Float64(Float64(sin(y) / y) * cosh(x)) <= -4e-153)
		tmp = Float64(fma(-0.16666666666666666, Float64(y * y), 1.0) * t_0);
	else
		tmp = Float64(t_0 * 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(0.041666666666666664 * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision] * N[Cosh[x], $MachinePrecision]), $MachinePrecision], -4e-153], N[(N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision], N[(t$95$0 * 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < -4.00000000000000016e-153

   1. Initial program 99.9%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. neg-sub0 N/A

         \[\leadsto \cosh x \cdot \frac{\color{blue}{0 - \sin y}}{\mathsf{neg}\left(y\right)} \]

      4. div-sub N/A

         \[\leadsto \cosh x \cdot \color{blue}{\left(\frac{0}{\mathsf{neg}\left(y\right)} - \frac{\sin y}{\mathsf{neg}\left(y\right)}\right)} \]

      5. clear-num N/A

         \[\leadsto \cosh x \cdot \left(\frac{0}{\mathsf{neg}\left(y\right)} - \color{blue}{\frac{1}{\frac{\mathsf{neg}\left(y\right)}{\sin y}}}\right) \]

      6. distribute-neg-frac N/A

         \[\leadsto \cosh x \cdot \left(\frac{0}{\mathsf{neg}\left(y\right)} - \frac{1}{\color{blue}{\mathsf{neg}\left(\frac{y}{\sin y}\right)}}\right) \]

      7. frac-sub N/A

         \[\leadsto \cosh x \cdot \color{blue}{\frac{0 \cdot \left(\mathsf{neg}\left(\frac{y}{\sin y}\right)\right) - \left(\mathsf{neg}\left(y\right)\right) \cdot 1}{\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(\frac{y}{\sin y}\right)\right)}} \]

      8. distribute-lft-neg-in N/A

         \[\leadsto \cosh x \cdot \frac{0 \cdot \left(\mathsf{neg}\left(\frac{y}{\sin y}\right)\right) - \color{blue}{\left(\mathsf{neg}\left(y \cdot 1\right)\right)}}{\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(\frac{y}{\sin y}\right)\right)} \]

      9. distribute-rgt-neg-in N/A

         \[\leadsto \cosh x \cdot \frac{0 \cdot \left(\mathsf{neg}\left(\frac{y}{\sin y}\right)\right) - \color{blue}{y \cdot \left(\mathsf{neg}\left(1\right)\right)}}{\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(\frac{y}{\sin y}\right)\right)} \]

      10. metadata-eval N/A

          \[\leadsto \cosh x \cdot \frac{0 \cdot \left(\mathsf{neg}\left(\frac{y}{\sin y}\right)\right) - y \cdot \color{blue}{-1}}{\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(\frac{y}{\sin y}\right)\right)} \]

      11. lower-/.f64 N/A

          \[\leadsto \cosh x \cdot \color{blue}{\frac{0 \cdot \left(\mathsf{neg}\left(\frac{y}{\sin y}\right)\right) - y \cdot -1}{\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(\frac{y}{\sin y}\right)\right)}} \]

   4. Applied rewrites 60.6%

      \[\leadsto \cosh x \cdot \color{blue}{\frac{0 \cdot \frac{-y}{\sin y} - \left(-y\right)}{\left(-y\right) \cdot \frac{-y}{\sin y}}} \]

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 69.1

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

   7. Applied rewrites 69.1%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 67.0

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

   10. Applied rewrites 67.0%

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

   if -4.00000000000000016e-153 < (*.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. Applied rewrites 72.9%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 67.3

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

   8. Applied rewrites 67.3%

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

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \cdot \cosh x \leq -4 \cdot 10^{-153}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 66.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \cdot \cosh x \leq -4 \cdot 10^{-153}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right) \cdot \mathsf{fma}\left(x \cdot x, 0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (/ (sin y) y) (cosh x)) -4e-153)
   (* (fma -0.16666666666666666 (* y y) 1.0) (fma (* x x) 0.5 1.0))
   (* (fma (fma 0.041666666666666664 (* x x) 0.5) (* x x) 1.0) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (((sin(y) / y) * cosh(x)) <= -4e-153) {
		tmp = fma(-0.16666666666666666, (y * y), 1.0) * fma((x * x), 0.5, 1.0);
	} else {
		tmp = fma(fma(0.041666666666666664, (x * x), 0.5), (x * x), 1.0) * 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(sin(y) / y) * cosh(x)) <= -4e-153)
		tmp = Float64(fma(-0.16666666666666666, Float64(y * y), 1.0) * fma(Float64(x * x), 0.5, 1.0));
	else
		tmp = Float64(fma(fma(0.041666666666666664, Float64(x * x), 0.5), Float64(x * x), 1.0) * 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision] * N[Cosh[x], $MachinePrecision]), $MachinePrecision], -4e-153], N[(N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.041666666666666664 * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < -4.00000000000000016e-153

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 64.0

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

   5. Applied rewrites 64.0%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 65.0

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

   8. Applied rewrites 65.0%

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

   if -4.00000000000000016e-153 < (*.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. Applied rewrites 72.9%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 67.3

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

   8. Applied rewrites 67.3%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \cdot \cosh x \leq -4 \cdot 10^{-153}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right) \cdot \mathsf{fma}\left(x \cdot x, 0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right) \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 57.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \cdot \cosh x \leq -4 \cdot 10^{-153}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right) \cdot \left(0.5 \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(x \cdot x, 0.5, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (/ (sin y) y) (cosh x)) -4e-153)
   (* (fma -0.16666666666666666 (* y y) 1.0) (* 0.5 (* x x)))
   (* 1.0 (fma (* x x) 0.5 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (((sin(y) / y) * cosh(x)) <= -4e-153) {
		tmp = fma(-0.16666666666666666, (y * y), 1.0) * (0.5 * (x * x));
	} else {
		tmp = 1.0 * fma((x * x), 0.5, 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(sin(y) / y) * cosh(x)) <= -4e-153)
		tmp = Float64(fma(-0.16666666666666666, Float64(y * y), 1.0) * Float64(0.5 * Float64(x * x)));
	else
		tmp = Float64(1.0 * fma(Float64(x * x), 0.5, 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision] * N[Cosh[x], $MachinePrecision]), $MachinePrecision], -4e-153], N[(N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 * N[(N[(x * x), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < -4.00000000000000016e-153

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 64.0

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

   5. Applied rewrites 64.0%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 0.4

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

   8. Applied rewrites 0.4%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 0.5%

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

   12. Taylor expanded in y around 0

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

   14. Applied rewrites 64.9%

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

   if -4.00000000000000016e-153 < (*.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 x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 87.7

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

   5. Applied rewrites 87.7%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 60.9%

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

4. Final simplification 61.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \cdot \cosh x \leq -4 \cdot 10^{-153}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right) \cdot \left(0.5 \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(x \cdot x, 0.5, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 53.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \cdot \cosh x \leq -4 \cdot 10^{-153}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(x \cdot x, 0.5, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (/ (sin y) y) (cosh x)) -4e-153)
   (* 1.0 (fma -0.16666666666666666 (* y y) 1.0))
   (* 1.0 (fma (* x x) 0.5 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (((sin(y) / y) * cosh(x)) <= -4e-153) {
		tmp = 1.0 * fma(-0.16666666666666666, (y * y), 1.0);
	} else {
		tmp = 1.0 * fma((x * x), 0.5, 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(sin(y) / y) * cosh(x)) <= -4e-153)
		tmp = Float64(1.0 * fma(-0.16666666666666666, Float64(y * y), 1.0));
	else
		tmp = Float64(1.0 * fma(Float64(x * x), 0.5, 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision] * N[Cosh[x], $MachinePrecision]), $MachinePrecision], -4e-153], N[(1.0 * N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 * N[(N[(x * x), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < -4.00000000000000016e-153

   1. Initial program 99.9%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. neg-sub0 N/A

         \[\leadsto \cosh x \cdot \frac{\color{blue}{0 - \sin y}}{\mathsf{neg}\left(y\right)} \]

      4. div-sub N/A

         \[\leadsto \cosh x \cdot \color{blue}{\left(\frac{0}{\mathsf{neg}\left(y\right)} - \frac{\sin y}{\mathsf{neg}\left(y\right)}\right)} \]

      5. clear-num N/A

         \[\leadsto \cosh x \cdot \left(\frac{0}{\mathsf{neg}\left(y\right)} - \color{blue}{\frac{1}{\frac{\mathsf{neg}\left(y\right)}{\sin y}}}\right) \]

      6. distribute-neg-frac N/A

         \[\leadsto \cosh x \cdot \left(\frac{0}{\mathsf{neg}\left(y\right)} - \frac{1}{\color{blue}{\mathsf{neg}\left(\frac{y}{\sin y}\right)}}\right) \]

      7. frac-sub N/A

         \[\leadsto \cosh x \cdot \color{blue}{\frac{0 \cdot \left(\mathsf{neg}\left(\frac{y}{\sin y}\right)\right) - \left(\mathsf{neg}\left(y\right)\right) \cdot 1}{\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(\frac{y}{\sin y}\right)\right)}} \]

      8. distribute-lft-neg-in N/A

         \[\leadsto \cosh x \cdot \frac{0 \cdot \left(\mathsf{neg}\left(\frac{y}{\sin y}\right)\right) - \color{blue}{\left(\mathsf{neg}\left(y \cdot 1\right)\right)}}{\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(\frac{y}{\sin y}\right)\right)} \]

      9. distribute-rgt-neg-in N/A

         \[\leadsto \cosh x \cdot \frac{0 \cdot \left(\mathsf{neg}\left(\frac{y}{\sin y}\right)\right) - \color{blue}{y \cdot \left(\mathsf{neg}\left(1\right)\right)}}{\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(\frac{y}{\sin y}\right)\right)} \]

      10. metadata-eval N/A

          \[\leadsto \cosh x \cdot \frac{0 \cdot \left(\mathsf{neg}\left(\frac{y}{\sin y}\right)\right) - y \cdot \color{blue}{-1}}{\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(\frac{y}{\sin y}\right)\right)} \]

      11. lower-/.f64 N/A

          \[\leadsto \cosh x \cdot \color{blue}{\frac{0 \cdot \left(\mathsf{neg}\left(\frac{y}{\sin y}\right)\right) - y \cdot -1}{\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(\frac{y}{\sin y}\right)\right)}} \]

   4. Applied rewrites 60.6%

      \[\leadsto \cosh x \cdot \color{blue}{\frac{0 \cdot \frac{-y}{\sin y} - \left(-y\right)}{\left(-y\right) \cdot \frac{-y}{\sin y}}} \]

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 69.1

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

   7. Applied rewrites 69.1%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 42.3%

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

   if -4.00000000000000016e-153 < (*.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 x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 87.7

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

   5. Applied rewrites 87.7%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 60.9%

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

4. Final simplification 57.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \cdot \cosh x \leq -4 \cdot 10^{-153}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(x \cdot x, 0.5, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 52.5% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(sin(y) / y) * cosh(x)) <= 2.0)
		tmp = Float64(1.0 * fma(-0.16666666666666666, Float64(y * y), 1.0));
	else
		tmp = Float64(Float64(0.5 * Float64(x * x)) * 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision] * N[Cosh[x], $MachinePrecision]), $MachinePrecision], 2.0], N[(1.0 * N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[(x * x), $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 99.8%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. neg-sub0 N/A

         \[\leadsto \cosh x \cdot \frac{\color{blue}{0 - \sin y}}{\mathsf{neg}\left(y\right)} \]

      4. div-sub N/A

         \[\leadsto \cosh x \cdot \color{blue}{\left(\frac{0}{\mathsf{neg}\left(y\right)} - \frac{\sin y}{\mathsf{neg}\left(y\right)}\right)} \]

      5. clear-num N/A

         \[\leadsto \cosh x \cdot \left(\frac{0}{\mathsf{neg}\left(y\right)} - \color{blue}{\frac{1}{\frac{\mathsf{neg}\left(y\right)}{\sin y}}}\right) \]

      6. distribute-neg-frac N/A

         \[\leadsto \cosh x \cdot \left(\frac{0}{\mathsf{neg}\left(y\right)} - \frac{1}{\color{blue}{\mathsf{neg}\left(\frac{y}{\sin y}\right)}}\right) \]

      7. frac-sub N/A

         \[\leadsto \cosh x \cdot \color{blue}{\frac{0 \cdot \left(\mathsf{neg}\left(\frac{y}{\sin y}\right)\right) - \left(\mathsf{neg}\left(y\right)\right) \cdot 1}{\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(\frac{y}{\sin y}\right)\right)}} \]

      8. distribute-lft-neg-in N/A

         \[\leadsto \cosh x \cdot \frac{0 \cdot \left(\mathsf{neg}\left(\frac{y}{\sin y}\right)\right) - \color{blue}{\left(\mathsf{neg}\left(y \cdot 1\right)\right)}}{\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(\frac{y}{\sin y}\right)\right)} \]

      9. distribute-rgt-neg-in N/A

         \[\leadsto \cosh x \cdot \frac{0 \cdot \left(\mathsf{neg}\left(\frac{y}{\sin y}\right)\right) - \color{blue}{y \cdot \left(\mathsf{neg}\left(1\right)\right)}}{\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(\frac{y}{\sin y}\right)\right)} \]

      10. metadata-eval N/A

          \[\leadsto \cosh x \cdot \frac{0 \cdot \left(\mathsf{neg}\left(\frac{y}{\sin y}\right)\right) - y \cdot \color{blue}{-1}}{\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(\frac{y}{\sin y}\right)\right)} \]

      11. lower-/.f64 N/A

          \[\leadsto \cosh x \cdot \color{blue}{\frac{0 \cdot \left(\mathsf{neg}\left(\frac{y}{\sin y}\right)\right) - y \cdot -1}{\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(\frac{y}{\sin y}\right)\right)}} \]

   4. Applied rewrites 71.7%

      \[\leadsto \cosh x \cdot \color{blue}{\frac{0 \cdot \frac{-y}{\sin y} - \left(-y\right)}{\left(-y\right) \cdot \frac{-y}{\sin y}}} \]

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 57.8

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

   7. Applied rewrites 57.8%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 50.2%

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

   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 x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 70.1

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

   5. Applied rewrites 70.1%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 81.7

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

   8. Applied rewrites 81.7%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 81.7%

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

   12. Taylor expanded in y around 0

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

   14. Applied rewrites 70.3%

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

4. Final simplification 56.7%

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

Alternative 15: 34.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \cdot \cosh x \leq -4 \cdot 10^{-153}:\\ \;\;\;\;1 \cdot \left(\left(y \cdot y\right) \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (/ (sin y) y) (cosh x)) -4e-153)
   (* 1.0 (* (* y y) -0.16666666666666666))
   (* 1.0 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (((sin(y) / y) * cosh(x)) <= -4e-153) {
		tmp = 1.0 * ((y * y) * -0.16666666666666666);
	} else {
		tmp = 1.0 * 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (((sin(y) / y) * cosh(x)) <= (-4d-153)) then
        tmp = 1.0d0 * ((y * y) * (-0.16666666666666666d0))
    else
        tmp = 1.0d0 * 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (((Math.sin(y) / y) * Math.cosh(x)) <= -4e-153) {
		tmp = 1.0 * ((y * y) * -0.16666666666666666);
	} else {
		tmp = 1.0 * 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((math.sin(y) / y) * math.cosh(x)) <= -4e-153:
		tmp = 1.0 * ((y * y) * -0.16666666666666666)
	else:
		tmp = 1.0 * 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(sin(y) / y) * cosh(x)) <= -4e-153)
		tmp = Float64(1.0 * Float64(Float64(y * y) * -0.16666666666666666));
	else
		tmp = Float64(1.0 * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((sin(y) / y) * cosh(x)) <= -4e-153)
		tmp = 1.0 * ((y * y) * -0.16666666666666666);
	else
		tmp = 1.0 * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision] * N[Cosh[x], $MachinePrecision]), $MachinePrecision], -4e-153], N[(1.0 * N[(N[(y * y), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision], N[(1.0 * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < -4.00000000000000016e-153

   1. Initial program 99.9%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. neg-sub0 N/A

         \[\leadsto \cosh x \cdot \frac{\color{blue}{0 - \sin y}}{\mathsf{neg}\left(y\right)} \]

      4. div-sub N/A

         \[\leadsto \cosh x \cdot \color{blue}{\left(\frac{0}{\mathsf{neg}\left(y\right)} - \frac{\sin y}{\mathsf{neg}\left(y\right)}\right)} \]

      5. clear-num N/A

         \[\leadsto \cosh x \cdot \left(\frac{0}{\mathsf{neg}\left(y\right)} - \color{blue}{\frac{1}{\frac{\mathsf{neg}\left(y\right)}{\sin y}}}\right) \]

      6. distribute-neg-frac N/A

         \[\leadsto \cosh x \cdot \left(\frac{0}{\mathsf{neg}\left(y\right)} - \frac{1}{\color{blue}{\mathsf{neg}\left(\frac{y}{\sin y}\right)}}\right) \]

      7. frac-sub N/A

         \[\leadsto \cosh x \cdot \color{blue}{\frac{0 \cdot \left(\mathsf{neg}\left(\frac{y}{\sin y}\right)\right) - \left(\mathsf{neg}\left(y\right)\right) \cdot 1}{\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(\frac{y}{\sin y}\right)\right)}} \]

      8. distribute-lft-neg-in N/A

         \[\leadsto \cosh x \cdot \frac{0 \cdot \left(\mathsf{neg}\left(\frac{y}{\sin y}\right)\right) - \color{blue}{\left(\mathsf{neg}\left(y \cdot 1\right)\right)}}{\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(\frac{y}{\sin y}\right)\right)} \]

      9. distribute-rgt-neg-in N/A

         \[\leadsto \cosh x \cdot \frac{0 \cdot \left(\mathsf{neg}\left(\frac{y}{\sin y}\right)\right) - \color{blue}{y \cdot \left(\mathsf{neg}\left(1\right)\right)}}{\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(\frac{y}{\sin y}\right)\right)} \]

      10. metadata-eval N/A

          \[\leadsto \cosh x \cdot \frac{0 \cdot \left(\mathsf{neg}\left(\frac{y}{\sin y}\right)\right) - y \cdot \color{blue}{-1}}{\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(\frac{y}{\sin y}\right)\right)} \]

      11. lower-/.f64 N/A

          \[\leadsto \cosh x \cdot \color{blue}{\frac{0 \cdot \left(\mathsf{neg}\left(\frac{y}{\sin y}\right)\right) - y \cdot -1}{\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(\frac{y}{\sin y}\right)\right)}} \]

   4. Applied rewrites 60.6%

      \[\leadsto \cosh x \cdot \color{blue}{\frac{0 \cdot \frac{-y}{\sin y} - \left(-y\right)}{\left(-y\right) \cdot \frac{-y}{\sin y}}} \]

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 69.1

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

   7. Applied rewrites 69.1%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 42.3%

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

   11. Taylor expanded in y around inf

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

   13. Applied rewrites 42.3%

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

   if -4.00000000000000016e-153 < (*.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. Applied rewrites 72.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 34.3%

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

4. Final simplification 35.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \cdot \cosh x \leq -4 \cdot 10^{-153}:\\ \;\;\;\;1 \cdot \left(\left(y \cdot y\right) \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 27.6% accurate, 36.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in y around 0

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

5. Applied rewrites 59.9%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 28.3%

   \[\leadsto \color{blue}{1} \cdot 1 \]
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 2024255 
(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)))